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Properties

Label 81.9.d.d
Level $81$
Weight $9$
Character orbit 81.d
Analytic conductor $32.998$
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] = [81,9,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: \(\Q(\sqrt{-3}, \sqrt{-14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 14x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 248 \beta_{2} q^{4} + ( - 10 \beta_{3} + 10 \beta_1) q^{5} + ( - 1750 \beta_{2} + 1750) q^{7} - 8 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 248 \beta_{2} q^{4} + ( - 10 \beta_{3} + 10 \beta_1) q^{5} + ( - 1750 \beta_{2} + 1750) q^{7} - 8 \beta_{3} q^{8} + 5040 q^{10} + 310 \beta_1 q^{11} - 25730 \beta_{2} q^{13} + ( - 1750 \beta_{3} + 1750 \beta_1) q^{14} + ( - 67520 \beta_{2} + 67520) q^{16} - 3336 \beta_{3} q^{17} + 18938 q^{19} + 2480 \beta_1 q^{20} + 156240 \beta_{2} q^{22} + ( - 20956 \beta_{3} + 20956 \beta_1) q^{23} + (340225 \beta_{2} - 340225) q^{25} - 25730 \beta_{3} q^{26} + 434000 q^{28} + 20530 \beta_1 q^{29} + 351478 \beta_{2} q^{31} + ( - 65472 \beta_{3} + 65472 \beta_1) q^{32} + ( - 1681344 \beta_{2} + 1681344) q^{34} - 17500 \beta_{3} q^{35} + 1335170 q^{37} + 18938 \beta_1 q^{38} - 40320 \beta_{2} q^{40} + (83540 \beta_{3} - 83540 \beta_1) q^{41} + ( - 3526150 \beta_{2} + 3526150) q^{43} + 76880 \beta_{3} q^{44} + 10561824 q^{46} - 181784 \beta_1 q^{47} + 2702301 \beta_{2} q^{49} + (340225 \beta_{3} - 340225 \beta_1) q^{50} + ( - 6381040 \beta_{2} + 6381040) q^{52} + 294066 \beta_{3} q^{53} + 1562400 q^{55} - 14000 \beta_1 q^{56} + 10347120 \beta_{2} q^{58} + (610910 \beta_{3} - 610910 \beta_1) q^{59} + (753602 \beta_{2} - 753602) q^{61} + 351478 \beta_{3} q^{62} + 15712768 q^{64} - 257300 \beta_1 q^{65} - 2268890 \beta_{2} q^{67} + ( - 827328 \beta_{3} + 827328 \beta_1) q^{68} + ( - 8820000 \beta_{2} + 8820000) q^{70} - 758220 \beta_{3} q^{71} + 27672770 q^{73} + 1335170 \beta_1 q^{74} + 4696624 \beta_{2} q^{76} + ( - 542500 \beta_{3} + 542500 \beta_1) q^{77} + ( - 22980982 \beta_{2} + 22980982) q^{79} - 675200 \beta_{3} q^{80} - 42104160 q^{82} - 2066606 \beta_1 q^{83} - 16813440 \beta_{2} q^{85} + ( - 3526150 \beta_{3} + 3526150 \beta_1) q^{86} + ( - 1249920 \beta_{2} + 1249920) q^{88} - 3234540 \beta_{3} q^{89} - 45027500 q^{91} + 5197088 \beta_1 q^{92} - 91619136 \beta_{2} q^{94} + ( - 189380 \beta_{3} + 189380 \beta_1) q^{95} + (147271010 \beta_{2} - 147271010) q^{97} + 2702301 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 496 q^{4} + 3500 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 496 q^{4} + 3500 q^{7} + 20160 q^{10} - 51460 q^{13} + 135040 q^{16} + 75752 q^{19} + 312480 q^{22} - 680450 q^{25} + 1736000 q^{28} + 702956 q^{31} + 3362688 q^{34} + 5340680 q^{37} - 80640 q^{40} + 7052300 q^{43} + 42247296 q^{46} + 5404602 q^{49} + 12762080 q^{52} + 6249600 q^{55} + 20694240 q^{58} - 1507204 q^{61} + 62851072 q^{64} - 4537780 q^{67} + 17640000 q^{70} + 110691080 q^{73} + 9393248 q^{76} + 45961964 q^{79} - 168416640 q^{82} - 33626880 q^{85} + 2499840 q^{88} - 180110000 q^{91} - 183238272 q^{94} - 294542020 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 14x^{2} + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 14\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{3} ) / 3 \) 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
−3.24037 − 1.87083i
3.24037 + 1.87083i
−3.24037 + 1.87083i
3.24037 − 1.87083i
−19.4422 − 11.2250i 0 124.000 + 214.774i −194.422 + 112.250i 0 875.000 − 1515.54i 179.600i 0 5040.00
26.2 19.4422 + 11.2250i 0 124.000 + 214.774i 194.422 − 112.250i 0 875.000 − 1515.54i − 179.600i 0 5040.00
53.1 −19.4422 + 11.2250i 0 124.000 − 214.774i −194.422 − 112.250i 0 875.000 + 1515.54i − 179.600i 0 5040.00
53.2 19.4422 − 11.2250i 0 124.000 − 214.774i 194.422 + 112.250i 0 875.000 + 1515.54i 179.600i 0 5040.00
\(n\): 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.d 4
3.b odd 2 1 inner 81.9.d.d 4
9.c even 3 1 3.9.b.a ✓ 2
9.c even 3 1 inner 81.9.d.d 4
9.d odd 6 1 3.9.b.a ✓ 2
9.d odd 6 1 inner 81.9.d.d 4
36.f odd 6 1 48.9.e.b 2
36.h even 6 1 48.9.e.b 2
45.h odd 6 1 75.9.c.c 2
45.j even 6 1 75.9.c.c 2
45.k odd 12 2 75.9.d.b 4
45.l even 12 2 75.9.d.b 4
72.j odd 6 1 192.9.e.e 2
72.l even 6 1 192.9.e.f 2
72.n even 6 1 192.9.e.e 2
72.p odd 6 1 192.9.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.9.b.a ✓ 2 9.c even 3 1
3.9.b.a ✓ 2 9.d odd 6 1
48.9.e.b 2 36.f odd 6 1
48.9.e.b 2 36.h even 6 1
75.9.c.c 2 45.h odd 6 1
75.9.c.c 2 45.j even 6 1
75.9.d.b 4 45.k odd 12 2
75.9.d.b 4 45.l even 12 2
81.9.d.d 4 1.a even 1 1 trivial
81.9.d.d 4 3.b odd 2 1 inner
81.9.d.d 4 9.c even 3 1 inner
81.9.d.d 4 9.d odd 6 1 inner
192.9.e.e 2 72.j odd 6 1
192.9.e.e 2 72.n even 6 1
192.9.e.f 2 72.l even 6 1
192.9.e.f 2 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 504T_{2}^{2} + 254016 \) acting on \(S_{9}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 504 T^{2} + 254016 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 2540160000 \) Copy content Toggle raw display
$7$ \( (T^{2} - 1750 T + 3062500)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} + 25730 T + 662032900)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 5608963584)^{2} \) Copy content Toggle raw display
$19$ \( (T - 18938)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} - 351478 T + 123536784484)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1335170)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots + 12433733822500)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{2} + 43583305427424)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + 753602 T + 567915974404)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 5147861832100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 289748374473600)^{2} \) Copy content Toggle raw display
$73$ \( (T - 27672770)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 528125533684324)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{2} + 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
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