Properties

Label 81.9.d.b
Level $81$
Weight $9$
Character orbit 81.d
Analytic conductor $32.998$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.26");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(32.9976674150\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 238 \beta_{2} q^{4} + (233 \beta_{3} - 233 \beta_1) q^{5} + (1652 \beta_{2} - 1652) q^{7} - 494 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 238 \beta_{2} q^{4} + (233 \beta_{3} - 233 \beta_1) q^{5} + (1652 \beta_{2} - 1652) q^{7} - 494 \beta_{3} q^{8} - 4194 q^{10} - 5036 \beta_1 q^{11} + 26272 \beta_{2} q^{13} + (1652 \beta_{3} - 1652 \beta_1) q^{14} + (52036 \beta_{2} - 52036) q^{16} - 3579 \beta_{3} q^{17} + 46640 q^{19} + 55454 \beta_1 q^{20} - 90648 \beta_{2} q^{22} + ( - 77332 \beta_{3} + 77332 \beta_1) q^{23} + ( - 586577 \beta_{2} + 586577) q^{25} + 26272 \beta_{3} q^{26} + 393176 q^{28} - 144953 \beta_1 q^{29} + 196444 \beta_{2} q^{31} + (178500 \beta_{3} - 178500 \beta_1) q^{32} + ( - 64422 \beta_{2} + 64422) q^{34} - 384916 \beta_{3} q^{35} + 2819414 q^{37} + 46640 \beta_1 q^{38} + 2071836 \beta_{2} q^{40} + ( - 166507 \beta_{3} + 166507 \beta_1) q^{41} + ( - 2213464 \beta_{2} + 2213464) q^{43} + 1198568 \beta_{3} q^{44} + 1391976 q^{46} + 384892 \beta_1 q^{47} + 3035697 \beta_{2} q^{49} + ( - 586577 \beta_{3} + 586577 \beta_1) q^{50} + ( - 6252736 \beta_{2} + 6252736) q^{52} - 1222983 \beta_{3} q^{53} + 21120984 q^{55} + 816088 \beta_1 q^{56} - 2609154 \beta_{2} q^{58} + (2768264 \beta_{3} - 2768264 \beta_1) q^{59} + ( - 17405302 \beta_{2} + 17405302) q^{61} + 196444 \beta_{3} q^{62} + 10108216 q^{64} - 6121376 \beta_1 q^{65} + 14322664 \beta_{2} q^{67} + (851802 \beta_{3} - 851802 \beta_1) q^{68} + ( - 6928488 \beta_{2} + 6928488) q^{70} + 3687708 \beta_{3} q^{71} - 8906992 q^{73} + 2819414 \beta_1 q^{74} - 11100320 \beta_{2} q^{76} + ( - 8319472 \beta_{3} + 8319472 \beta_1) q^{77} + (32758844 \beta_{2} - 32758844) q^{79} - 12124388 \beta_{3} q^{80} + 2997126 q^{82} + 20055628 \beta_1 q^{83} + 15010326 \beta_{2} q^{85} + ( - 2213464 \beta_{3} + 2213464 \beta_1) q^{86} + (44780112 \beta_{2} - 44780112) q^{88} + 13891371 \beta_{3} q^{89} - 43401344 q^{91} - 18405016 \beta_1 q^{92} + 6928056 \beta_{2} q^{94} + (10867120 \beta_{3} - 10867120 \beta_1) q^{95} + ( - 24451744 \beta_{2} + 24451744) q^{97} + 3035697 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 476 q^{4} - 3304 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 476 q^{4} - 3304 q^{7} - 16776 q^{10} + 52544 q^{13} - 104072 q^{16} + 186560 q^{19} - 181296 q^{22} + 1173154 q^{25} + 1572704 q^{28} + 392888 q^{31} + 128844 q^{34} + 11277656 q^{37} + 4143672 q^{40} + 4426928 q^{43} + 5567904 q^{46} + 6071394 q^{49} + 12505472 q^{52} + 84483936 q^{55} - 5218308 q^{58} + 34810604 q^{61} + 40432864 q^{64} + 28645328 q^{67} + 13856976 q^{70} - 35627968 q^{73} - 22200640 q^{76} - 65517688 q^{79} + 11988504 q^{82} + 30020652 q^{85} - 89560224 q^{88} - 173605376 q^{91} + 13856112 q^{94} + 48903488 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) 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

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 −119.000 206.114i 856.097 494.268i 0 −826.000 + 1430.67i 2095.86i 0 −4194.00
26.2 3.67423 + 2.12132i 0 −119.000 206.114i −856.097 + 494.268i 0 −826.000 + 1430.67i 2095.86i 0 −4194.00
53.1 −3.67423 + 2.12132i 0 −119.000 + 206.114i 856.097 + 494.268i 0 −826.000 1430.67i 2095.86i 0 −4194.00
53.2 3.67423 2.12132i 0 −119.000 + 206.114i −856.097 494.268i 0 −826.000 1430.67i 2095.86i 0 −4194.00
\(n\): 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 81.9.d.b 4
3.b odd 2 1 inner 81.9.d.b 4
9.c even 3 1 9.9.b.a 2
9.c even 3 1 inner 81.9.d.b 4
9.d odd 6 1 9.9.b.a 2
9.d odd 6 1 inner 81.9.d.b 4
36.f odd 6 1 144.9.e.a 2
36.h even 6 1 144.9.e.a 2
45.h odd 6 1 225.9.c.a 2
45.j even 6 1 225.9.c.a 2
45.k odd 12 2 225.9.d.a 4
45.l even 12 2 225.9.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.9.b.a 2 9.c even 3 1
9.9.b.a 2 9.d odd 6 1
81.9.d.b 4 1.a even 1 1 trivial
81.9.d.b 4 3.b odd 2 1 inner
81.9.d.b 4 9.c even 3 1 inner
81.9.d.b 4 9.d odd 6 1 inner
144.9.e.a 2 36.f odd 6 1
144.9.e.a 2 36.h even 6 1
225.9.c.a 2 45.h odd 6 1
225.9.c.a 2 45.j even 6 1
225.9.d.a 4 45.k odd 12 2
225.9.d.a 4 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 18T_{2}^{2} + 324 \) acting on \(S_{9}^{\mathrm{new}}(81, [\chi])\). 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
$5$ \( T^{4} + \cdots + 954923748804 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1652 T + 2729104)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{2} - 26272 T + 690217984)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 230566338)^{2} \) Copy content Toggle raw display
$19$ \( (T - 46640)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{2} - 196444 T + 38590245136)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2819414)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots + 4899422879296)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 71\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{2} + 26922373529202)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 302944537711204)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 205138704056896)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 244785425278752)^{2} \) Copy content Toggle raw display
$73$ \( (T + 8906992)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 52\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{2} + 34\!\cdots\!38)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 597887784641536)^{2} \) Copy content Toggle raw display
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