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Properties

Label 81.11.d.d
Level $81$
Weight $11$
Character orbit 81.d
Analytic conductor $51.464$
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] = [81,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.26");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 11 \)
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: \(51.4639374666\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 3)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 304 \beta_{2} q^{4} + ( - 106 \beta_{3} + 106 \beta_1) q^{5} + (17234 \beta_{2} - 17234) q^{7} - 1328 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 304 \beta_{2} q^{4} + ( - 106 \beta_{3} + 106 \beta_1) q^{5} + (17234 \beta_{2} - 17234) q^{7} - 1328 \beta_{3} q^{8} + 76320 q^{10} - 6962 \beta_1 q^{11} + 169654 \beta_{2} q^{13} + (17234 \beta_{3} - 17234 \beta_1) q^{14} + ( - 644864 \beta_{2} + 644864) q^{16} + 12792 \beta_{3} q^{17} - 949462 q^{19} - 32224 \beta_1 q^{20} - 5012640 \beta_{2} q^{22} + (99044 \beta_{3} - 99044 \beta_1) q^{23} + (1675705 \beta_{2} - 1675705) q^{25} + 169654 \beta_{3} q^{26} + 5239136 q^{28} + 118594 \beta_1 q^{29} + 29793118 \beta_{2} q^{31} + (715008 \beta_{3} - 715008 \beta_1) q^{32} + (9210240 \beta_{2} - 9210240) q^{34} + 1826804 \beta_{3} q^{35} - 60811846 q^{37} - 949462 \beta_1 q^{38} - 101352960 \beta_{2} q^{40} + (6770372 \beta_{3} - 6770372 \beta_1) q^{41} + (107419706 \beta_{2} - 107419706) q^{43} + 2116448 \beta_{3} q^{44} - 71311680 q^{46} - 9987608 \beta_1 q^{47} - 14535507 \beta_{2} q^{49} + (1675705 \beta_{3} - 1675705 \beta_1) q^{50} + ( - 51574816 \beta_{2} + 51574816) q^{52} - 7158798 \beta_{3} q^{53} - 531339840 q^{55} + 22886752 \beta_1 q^{56} + 85387680 \beta_{2} q^{58} + ( - 24192682 \beta_{3} + 24192682 \beta_1) q^{59} + (1030793642 \beta_{2} - 1030793642) q^{61} + 29793118 \beta_{3} q^{62} - 1175146496 q^{64} + 17983324 \beta_1 q^{65} - 1876742474 \beta_{2} q^{67} + ( - 3888768 \beta_{3} + 3888768 \beta_1) q^{68} + (1315298880 \beta_{2} - 1315298880) q^{70} - 100003596 \beta_{3} q^{71} - 2846528494 q^{73} - 60811846 \beta_1 q^{74} + 288636448 \beta_{2} q^{76} + ( - 119983108 \beta_{3} + 119983108 \beta_1) q^{77} + (1488647618 \beta_{2} - 1488647618) q^{79} - 68355584 \beta_{3} q^{80} - 4874667840 q^{82} + 47175562 \beta_1 q^{83} + 976285440 \beta_{2} q^{85} + (107419706 \beta_{3} - 107419706 \beta_1) q^{86} + (6656785920 \beta_{2} - 6656785920) q^{88} + 224371428 \beta_{3} q^{89} - 2923817036 q^{91} + 30109376 \beta_1 q^{92} - 7191077760 \beta_{2} q^{94} + (100642972 \beta_{3} - 100642972 \beta_1) q^{95} + ( - 1592948926 \beta_{2} + 1592948926) q^{97} - 14535507 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 608 q^{4} - 34468 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 608 q^{4} - 34468 q^{7} + 305280 q^{10} + 339308 q^{13} + 1289728 q^{16} - 3797848 q^{19} - 10025280 q^{22} - 3351410 q^{25} + 20956544 q^{28} + 59586236 q^{31} - 18420480 q^{34} - 243247384 q^{37} - 202705920 q^{40} - 214839412 q^{43} - 285246720 q^{46} - 29071014 q^{49} + 103149632 q^{52} - 2125359360 q^{55} + 170775360 q^{58} - 2061587284 q^{61} - 4700585984 q^{64} - 3753484948 q^{67} - 2630597760 q^{70} - 11386113976 q^{73} + 577272896 q^{76} - 2977295236 q^{79} - 19498671360 q^{82} + 1952570880 q^{85} - 13313571840 q^{88} - 11695268144 q^{91} - 14382155520 q^{94} + 3185897852 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 12\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\nu^{3} ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{3} ) / 12 \) 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
−1.93649 − 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 − 1.11803i
−23.2379 − 13.4164i 0 −152.000 − 263.272i −2463.22 + 1422.14i 0 −8617.00 + 14925.1i 35634.0i 0 76320.0
26.2 23.2379 + 13.4164i 0 −152.000 − 263.272i 2463.22 − 1422.14i 0 −8617.00 + 14925.1i − 35634.0i 0 76320.0
53.1 −23.2379 + 13.4164i 0 −152.000 + 263.272i −2463.22 − 1422.14i 0 −8617.00 − 14925.1i − 35634.0i 0 76320.0
53.2 23.2379 − 13.4164i 0 −152.000 + 263.272i 2463.22 + 1422.14i 0 −8617.00 − 14925.1i 35634.0i 0 76320.0
\(n\): 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 81.11.d.d 4
3.b odd 2 1 inner 81.11.d.d 4
9.c even 3 1 3.11.b.a ✓ 2
9.c even 3 1 inner 81.11.d.d 4
9.d odd 6 1 3.11.b.a ✓ 2
9.d odd 6 1 inner 81.11.d.d 4
36.f odd 6 1 48.11.e.c 2
36.h even 6 1 48.11.e.c 2
45.h odd 6 1 75.11.c.d 2
45.j even 6 1 75.11.c.d 2
45.k odd 12 2 75.11.d.b 4
45.l even 12 2 75.11.d.b 4
72.j odd 6 1 192.11.e.e 2
72.l even 6 1 192.11.e.d 2
72.n even 6 1 192.11.e.e 2
72.p odd 6 1 192.11.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.11.b.a ✓ 2 9.c even 3 1
3.11.b.a ✓ 2 9.d odd 6 1
48.11.e.c 2 36.f odd 6 1
48.11.e.c 2 36.h even 6 1
75.11.c.d 2 45.h odd 6 1
75.11.c.d 2 45.j even 6 1
75.11.d.b 4 45.k odd 12 2
75.11.d.b 4 45.l even 12 2
81.11.d.d 4 1.a even 1 1 trivial
81.11.d.d 4 3.b odd 2 1 inner
81.11.d.d 4 9.c even 3 1 inner
81.11.d.d 4 9.d odd 6 1 inner
192.11.e.d 2 72.l even 6 1
192.11.e.d 2 72.p odd 6 1
192.11.e.e 2 72.j odd 6 1
192.11.e.e 2 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 720T_{2}^{2} + 518400 \) acting on \(S_{11}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 720 T^{2} + 518400 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 65446805606400 \) Copy content Toggle raw display
$7$ \( (T^{2} + 17234 T + 297010756)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} - 169654 T + 28782479716)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 117817390080)^{2} \) Copy content Toggle raw display
$19$ \( (T + 949462)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots + 887629880161924)^{2} \) Copy content Toggle raw display
$37$ \( (T + 60811846)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots + 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{2} + 36\!\cdots\!80)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 35\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 72\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2846528494)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 22\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 25\!\cdots\!76)^{2} \) Copy content Toggle raw display
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