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Properties

Label 81.7.b.a
Level $81$
Weight $7$
Character orbit 81.b
Analytic conductor $18.634$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,7,Mod(80,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.80");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 81.b (of order \(2\), degree \(1\), 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: \(18.6343807732\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 75 x^{8} - 2 x^{7} + 4610 x^{6} - 2412 x^{5} + 66932 x^{4} - 174032 x^{3} + \cdots + 1982464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{30} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} - 25) q^{4} + \beta_{6} q^{5} + ( - \beta_{3} + 24) q^{7} + (\beta_{6} + \beta_{5} + 23 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} - 25) q^{4} + \beta_{6} q^{5} + ( - \beta_{3} + 24) q^{7} + (\beta_{6} + \beta_{5} + 23 \beta_1) q^{8} + (\beta_{4} - \beta_{3} + 2 \beta_{2} - 14) q^{10} + (\beta_{9} + 2 \beta_{6} - 7 \beta_1) q^{11} + ( - \beta_{7} + 6 \beta_{2} + 166) q^{13} + ( - \beta_{9} + \beta_{8} + \cdots - 44 \beta_1) q^{14}+ \cdots + ( - 62 \beta_{9} + 334 \beta_{8} + \cdots + 63325 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 254 q^{4} + 242 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 254 q^{4} + 242 q^{7} - 132 q^{10} + 1682 q^{13} + 3970 q^{16} + 6176 q^{19} - 6942 q^{22} - 16904 q^{25} - 22660 q^{28} + 49202 q^{31} - 14850 q^{34} + 12764 q^{37} - 108300 q^{40} + 186542 q^{43} + 112512 q^{46} - 172476 q^{49} - 373840 q^{52} - 419982 q^{55} + 761316 q^{58} + 118610 q^{61} + 403066 q^{64} - 317026 q^{67} + 609576 q^{70} + 933896 q^{73} - 1333282 q^{76} - 937414 q^{79} - 2038470 q^{82} + 2379888 q^{85} + 699546 q^{88} - 211778 q^{91} - 1618248 q^{94} - 672058 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 75 x^{8} - 2 x^{7} + 4610 x^{6} - 2412 x^{5} + 66932 x^{4} - 174032 x^{3} + \cdots + 1982464 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5928919 \nu^{9} - 4480309 \nu^{8} + 11157967 \nu^{7} - 1123354358 \nu^{6} + \cdots - 38990423194624 ) / 25215567395904 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2602307 \nu^{9} - 113956723 \nu^{8} + 283803049 \nu^{7} - 6917962454 \nu^{6} + \cdots - 189953925305392 ) / 2101297282992 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14324248607 \nu^{9} - 273474941299 \nu^{8} + 681074535337 \nu^{7} - 18991543430810 \nu^{6} + \cdots - 11\!\cdots\!92 ) / 33620756527872 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9107923153 \nu^{9} - 188239357757 \nu^{8} + 468799929191 \nu^{7} - 13270108377094 \nu^{6} + \cdots - 75\!\cdots\!44 ) / 11206918842624 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 296706639850 \nu^{9} + 106099180831 \nu^{8} + 20820677882093 \nu^{7} + \cdots - 51\!\cdots\!84 ) / 277371241354944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 316256271215 \nu^{9} - 75259061246 \nu^{8} - 22664179335040 \nu^{7} + \cdots + 11\!\cdots\!76 ) / 277371241354944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 51176926039 \nu^{9} + 771417044363 \nu^{8} - 1921172384369 \nu^{7} + \cdots + 28\!\cdots\!88 ) / 33620756527872 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 689649836999 \nu^{9} + 188911531879 \nu^{8} - 50118641275309 \nu^{7} + \cdots + 30\!\cdots\!40 ) / 201724539167232 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 34197121964459 \nu^{9} + 14539475193317 \nu^{8} + \cdots - 12\!\cdots\!20 ) / 22\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{4} - 7\beta_{3} + 5\beta_{2} + 729\beta _1 + 143 ) / 1458 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{9} - 25\beta_{8} + 230\beta_{6} + 36\beta_{5} - 243\beta_{2} + 575\beta _1 - 21627 ) / 1458 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 31\beta_{7} - 125\beta_{4} + 352\beta_{3} - 965\beta_{2} - 15467 ) / 729 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 666 \beta_{9} + 1162 \beta_{8} + 81 \beta_{7} - 14591 \beta_{6} - 2259 \beta_{5} - 81 \beta_{4} + \cdots - 1084023 ) / 1458 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 324 \beta_{9} + 1620 \beta_{8} - 1471 \beta_{7} - 26649 \beta_{6} - 18873 \beta_{5} + \cdots + 2303915 ) / 1458 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -95\beta_{7} + 143\beta_{4} - 650\beta_{3} + 13009\beta_{2} + 772647 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 52812 \beta_{9} - 175932 \beta_{8} - 85447 \beta_{7} + 2410965 \beta_{6} + 1286037 \beta_{5} + \cdots + 208349891 ) / 1458 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2507364 \beta_{9} - 3874516 \beta_{8} + 569403 \beta_{7} + 54353129 \beta_{6} + 9544185 \beta_{5} + \cdots - 3793867983 ) / 1458 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5384335\beta_{7} - 27462671\beta_{4} + 74443258\beta_{3} - 381795797\beta_{2} - 16277117675 ) / 729 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
4.07727 + 7.06203i
−3.54866 + 6.14646i
−2.32209 + 4.02197i
1.22025 + 2.11353i
1.07323 + 1.85889i
1.07323 − 1.85889i
1.22025 − 2.11353i
−2.32209 − 4.02197i
−3.54866 − 6.14646i
4.07727 − 7.06203i
− 14.1241i 0 −135.489 119.902i 0 257.781 1009.72i 0 1693.51
80.2 − 12.2929i 0 −87.1157 − 181.936i 0 −166.708 284.159i 0 −2236.53
80.3 − 8.04394i 0 −0.705045 92.7497i 0 −120.015 − 509.141i 0 746.074
80.4 − 4.22706i 0 46.1320 75.0401i 0 −362.133 − 465.535i 0 317.199
80.5 − 3.71778i 0 50.1781 − 157.690i 0 512.075 − 424.489i 0 −586.255
80.6 3.71778i 0 50.1781 157.690i 0 512.075 424.489i 0 −586.255
80.7 4.22706i 0 46.1320 − 75.0401i 0 −362.133 465.535i 0 317.199
80.8 8.04394i 0 −0.705045 − 92.7497i 0 −120.015 509.141i 0 746.074
80.9 12.2929i 0 −87.1157 181.936i 0 −166.708 − 284.159i 0 −2236.53
80.10 14.1241i 0 −135.489 − 119.902i 0 257.781 − 1009.72i 0 1693.51
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.7.b.a 10
3.b odd 2 1 inner 81.7.b.a 10
9.c even 3 1 9.7.d.a ✓ 10
9.c even 3 1 27.7.d.a 10
9.d odd 6 1 9.7.d.a ✓ 10
9.d odd 6 1 27.7.d.a 10
36.f odd 6 1 144.7.q.a 10
36.f odd 6 1 432.7.q.a 10
36.h even 6 1 144.7.q.a 10
36.h even 6 1 432.7.q.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.7.d.a ✓ 10 9.c even 3 1
9.7.d.a ✓ 10 9.d odd 6 1
27.7.d.a 10 9.c even 3 1
27.7.d.a 10 9.d odd 6 1
81.7.b.a 10 1.a even 1 1 trivial
81.7.b.a 10 3.b odd 2 1 inner
144.7.q.a 10 36.f odd 6 1
144.7.q.a 10 36.h even 6 1
432.7.q.a 10 36.f odd 6 1
432.7.q.a 10 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 447T_{2}^{8} + 66240T_{2}^{6} + 3727404T_{2}^{4} + 74862144T_{2}^{2} + 481738752 \) acting on \(S_{7}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 481738752 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{5} - 121 T^{4} + \cdots + 956410103836)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 96\!\cdots\!27 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots - 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 51\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 62\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 44\!\cdots\!63 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 21\!\cdots\!51)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 10\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 73\!\cdots\!43 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 22\!\cdots\!88)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 85\!\cdots\!21)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 36\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 59\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 38\!\cdots\!35)^{2} \) Copy content Toggle raw display
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