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Properties

Label 81.13.d.c
Level $81$
Weight $13$
Character orbit 81.d
Analytic conductor $74.033$
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] = [81,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.26");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 13 \)
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: \(74.0334991994\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-26})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 26x^{2} + 676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 4328 \beta_{2} q^{4} + (230 \beta_{3} - 230 \beta_1) q^{5} + (40250 \beta_{2} - 40250) q^{7} + 232 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 4328 \beta_{2} q^{4} + (230 \beta_{3} - 230 \beta_1) q^{5} + (40250 \beta_{2} - 40250) q^{7} + 232 \beta_{3} q^{8} - 1937520 q^{10} - 12650 \beta_1 q^{11} - 1284050 \beta_{2} q^{13} + (40250 \beta_{3} - 40250 \beta_1) q^{14} + ( - 15773120 \beta_{2} + 15773120) q^{16} - 161736 \beta_{3} q^{17} + 53343578 q^{19} - 995440 \beta_1 q^{20} - 106563600 \beta_{2} q^{22} + (1170884 \beta_{3} - 1170884 \beta_1) q^{23} + ( - 201488975 \beta_{2} + 201488975) q^{25} - 1284050 \beta_{3} q^{26} - 174202000 q^{28} + 1310050 \beta_1 q^{29} - 66526202 \beta_{2} q^{31} + ( - 16723392 \beta_{3} + 16723392 \beta_1) q^{32} + ( - 1362464064 \beta_{2} + 1362464064) q^{34} - 9257500 \beta_{3} q^{35} + 2228726450 q^{37} + 53343578 \beta_1 q^{38} - 449504640 \beta_{2} q^{40} + ( - 89469100 \beta_{3} + 89469100 \beta_1) q^{41} + (8977216250 \beta_{2} - 8977216250) q^{43} - 54749200 \beta_{3} q^{44} - 9863526816 q^{46} - 11733464 \beta_1 q^{47} + 12221224701 \beta_{2} q^{49} + ( - 201488975 \beta_{3} + 201488975 \beta_1) q^{50} + ( - 5557368400 \beta_{2} + 5557368400) q^{52} - 448279614 \beta_{3} q^{53} + 24509628000 q^{55} - 9338000 \beta_1 q^{56} + 11035861200 \beta_{2} q^{58} + ( - 502355650 \beta_{3} + 502355650 \beta_1) q^{59} + ( - 40679935918 \beta_{2} + 40679935918) q^{61} - 66526202 \beta_{3} q^{62} + 76271154688 q^{64} + 295331500 \beta_1 q^{65} - 121176846650 \beta_{2} q^{67} + ( - 699993408 \beta_{3} + 699993408 \beta_1) q^{68} + ( - 77985180000 \beta_{2} + 77985180000) q^{70} - 488726700 \beta_{3} q^{71} - 60956187550 q^{73} + 2228726450 \beta_1 q^{74} + 230871005584 \beta_{2} q^{76} + ( - 509162500 \beta_{3} + 509162500 \beta_1) q^{77} + ( - 252324997702 \beta_{2} + 252324997702) q^{79} + 3627817600 \beta_{3} q^{80} + 753687698400 q^{82} - 4475910446 \beta_1 q^{83} + 313366734720 \beta_{2} q^{85} + (8977216250 \beta_{3} - 8977216250 \beta_1) q^{86} + ( - 24722755200 \beta_{2} + 24722755200) q^{88} - 1225929900 \beta_{3} q^{89} + 51683012500 q^{91} - 5067585952 \beta_1 q^{92} - 98842700736 \beta_{2} q^{94} + (12269022940 \beta_{3} - 12269022940 \beta_1) q^{95} + (653817778850 \beta_{2} - 653817778850) q^{97} + 12221224701 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8656 q^{4} - 80500 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8656 q^{4} - 80500 q^{7} - 7750080 q^{10} - 2568100 q^{13} + 31546240 q^{16} + 213374312 q^{19} - 213127200 q^{22} + 402977950 q^{25} - 696808000 q^{28} - 133052404 q^{31} + 2724928128 q^{34} + 8914905800 q^{37} - 899009280 q^{40} - 17954432500 q^{43} - 39454107264 q^{46} + 24442449402 q^{49} + 11114736800 q^{52} + 98038512000 q^{55} + 22071722400 q^{58} + 81359871836 q^{61} + 305084618752 q^{64} - 242353693300 q^{67} + 155970360000 q^{70} - 243824750200 q^{73} + 461742011168 q^{76} + 504649995404 q^{79} + 3014750793600 q^{82} + 626733469440 q^{85} + 49445510400 q^{88} + 206732050000 q^{91} - 197685401472 q^{94} - 1307635557700 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 26x^{2} + 676 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 18\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{3} ) / 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 26\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13\beta_{3} ) / 9 \) 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
−4.41588 − 2.54951i
4.41588 + 2.54951i
−4.41588 + 2.54951i
4.41588 − 2.54951i
−79.4858 − 45.8912i 0 2164.00 + 3748.16i 18281.7 − 10555.0i 0 −20125.0 + 34857.5i − 21293.5i 0 −1.93752e6
26.2 79.4858 + 45.8912i 0 2164.00 + 3748.16i −18281.7 + 10555.0i 0 −20125.0 + 34857.5i 21293.5i 0 −1.93752e6
53.1 −79.4858 + 45.8912i 0 2164.00 − 3748.16i 18281.7 + 10555.0i 0 −20125.0 − 34857.5i 21293.5i 0 −1.93752e6
53.2 79.4858 − 45.8912i 0 2164.00 − 3748.16i −18281.7 − 10555.0i 0 −20125.0 − 34857.5i − 21293.5i 0 −1.93752e6
\(n\): 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 81.13.d.c 4
3.b odd 2 1 inner 81.13.d.c 4
9.c even 3 1 3.13.b.b ✓ 2
9.c even 3 1 inner 81.13.d.c 4
9.d odd 6 1 3.13.b.b ✓ 2
9.d odd 6 1 inner 81.13.d.c 4
36.f odd 6 1 48.13.e.b 2
36.h even 6 1 48.13.e.b 2
45.h odd 6 1 75.13.c.c 2
45.j even 6 1 75.13.c.c 2
45.k odd 12 2 75.13.d.b 4
45.l even 12 2 75.13.d.b 4
72.j odd 6 1 192.13.e.d 2
72.l even 6 1 192.13.e.c 2
72.n even 6 1 192.13.e.d 2
72.p odd 6 1 192.13.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.13.b.b ✓ 2 9.c even 3 1
3.13.b.b ✓ 2 9.d odd 6 1
48.13.e.b 2 36.f odd 6 1
48.13.e.b 2 36.h even 6 1
75.13.c.c 2 45.h odd 6 1
75.13.c.c 2 45.j even 6 1
75.13.d.b 4 45.k odd 12 2
75.13.d.b 4 45.l even 12 2
81.13.d.c 4 1.a even 1 1 trivial
81.13.d.c 4 3.b odd 2 1 inner
81.13.d.c 4 9.c even 3 1 inner
81.13.d.c 4 9.d odd 6 1 inner
192.13.e.c 2 72.l even 6 1
192.13.e.c 2 72.p odd 6 1
192.13.e.d 2 72.j odd 6 1
192.13.e.d 2 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 8424T_{2}^{2} + 70963776 \) acting on \(S_{13}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 8424 T^{2} + 70963776 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} + 40250 T + 1620062500)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots + 1648784402500)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 220359487855104)^{2} \) Copy content Toggle raw display
$19$ \( (T - 53343578)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots + 44\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2228726450)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{2} + 16\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 16\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T + 60956187550)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 63\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
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