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Properties

Label 810.2.k.d
Level $810$
Weight $2$
Character orbit 810.k
Analytic conductor $6.468$
Analytic rank $0$
Dimension $18$
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] = [810,2,Mod(91,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.k (of order \(9\), degree \(6\), 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: \(6.46788256372\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} + 24 x^{16} - 66 x^{15} + 153 x^{14} - 315 x^{13} + 651 x^{12} - 1350 x^{11} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{2} q^{4} + (\beta_{4} + \beta_{2}) q^{5} + ( - \beta_{11} + \beta_{6} + \cdots + \beta_1) q^{7}+ \cdots + (\beta_{13} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{2} q^{4} + (\beta_{4} + \beta_{2}) q^{5} + ( - \beta_{11} + \beta_{6} + \cdots + \beta_1) q^{7}+ \cdots + ( - \beta_{16} + \beta_{15} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 3 q^{7} + 9 q^{8} + 9 q^{10} - 9 q^{11} - 15 q^{13} + 3 q^{14} + 3 q^{17} + 9 q^{19} + 9 q^{22} + 24 q^{23} - 12 q^{26} + 6 q^{28} + 9 q^{29} - 3 q^{31} + 24 q^{34} - 3 q^{35} - 12 q^{37} + 18 q^{38} - 6 q^{41} - 33 q^{43} + 9 q^{44} - 15 q^{46} + 33 q^{47} + 51 q^{49} - 15 q^{52} - 24 q^{53} - 18 q^{55} + 3 q^{56} + 9 q^{58} + 12 q^{59} + 21 q^{61} - 6 q^{62} - 9 q^{64} + 12 q^{65} + 27 q^{67} + 21 q^{68} - 6 q^{70} + 36 q^{71} - 12 q^{73} + 12 q^{74} - 18 q^{76} - 72 q^{77} - 75 q^{79} + 18 q^{80} + 42 q^{82} - 33 q^{83} - 24 q^{85} + 33 q^{86} - 9 q^{88} + 33 q^{89} + 27 q^{91} - 12 q^{92} + 12 q^{94} + 18 q^{95} - 18 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} + 24 x^{16} - 66 x^{15} + 153 x^{14} - 315 x^{13} + 651 x^{12} - 1350 x^{11} + \cdots + 19683 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 256 \nu^{17} - 252 \nu^{16} + 819 \nu^{15} - 2946 \nu^{14} + 8433 \nu^{13} - 18684 \nu^{12} + \cdots - 3726648 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 463 \nu^{17} + 1365 \nu^{16} - 5913 \nu^{15} + 26250 \nu^{14} - 52794 \nu^{13} + 112482 \nu^{12} + \cdots + 17445699 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1259 \nu^{17} - 3681 \nu^{16} + 13977 \nu^{15} - 36627 \nu^{14} + 86877 \nu^{13} - 179712 \nu^{12} + \cdots - 16015401 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1330 \nu^{17} - 5085 \nu^{16} + 18540 \nu^{15} - 38388 \nu^{14} + 80928 \nu^{13} - 154008 \nu^{12} + \cdots + 4494285 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 155 \nu^{17} - 404 \nu^{16} + 1671 \nu^{15} - 7425 \nu^{14} + 14568 \nu^{13} - 30951 \nu^{12} + \cdots - 4166235 ) / 130491 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1495 \nu^{17} - 1329 \nu^{16} + 7407 \nu^{15} + 3399 \nu^{14} - 3645 \nu^{13} + 36558 \nu^{12} + \cdots + 26204634 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1537 \nu^{17} - 10986 \nu^{16} + 39195 \nu^{15} - 107463 \nu^{14} + 225909 \nu^{13} + \cdots - 36577575 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2093 \nu^{17} + 3042 \nu^{16} - 4248 \nu^{15} - 21513 \nu^{14} + 46989 \nu^{13} + \cdots - 38565558 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4 \nu^{17} - 25 \nu^{16} + 84 \nu^{15} - 225 \nu^{14} + 471 \nu^{13} - 981 \nu^{12} + 2001 \nu^{11} + \cdots - 69984 ) / 2187 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2600 \nu^{17} - 14172 \nu^{16} + 50355 \nu^{15} - 126681 \nu^{14} + 272574 \nu^{13} + \cdots - 32857488 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2 \nu^{17} + 12 \nu^{16} - 42 \nu^{15} + 111 \nu^{14} - 237 \nu^{13} + 486 \nu^{12} - 996 \nu^{11} + \cdots + 33534 ) / 729 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3458 \nu^{17} + 25572 \nu^{16} - 91701 \nu^{15} + 251565 \nu^{14} - 529713 \nu^{13} + \cdots + 78745122 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1334 \nu^{17} - 5391 \nu^{16} + 17655 \nu^{15} - 38283 \nu^{14} + 79776 \nu^{13} - 161046 \nu^{12} + \cdots - 2659392 ) / 391473 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 4597 \nu^{17} - 31350 \nu^{16} + 111285 \nu^{15} - 296175 \nu^{14} + 622152 \nu^{13} + \cdots - 78627024 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2059 \nu^{17} - 8138 \nu^{16} + 26538 \nu^{15} - 55365 \nu^{14} + 114774 \nu^{13} - 227628 \nu^{12} + \cdots - 258066 ) / 391473 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2492 \nu^{17} + 10743 \nu^{16} - 37734 \nu^{15} + 90852 \nu^{14} - 197352 \nu^{13} + \cdots + 20177262 ) / 391473 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 1499 \nu^{17} + 7184 \nu^{16} - 25138 \nu^{15} + 60936 \nu^{14} - 129453 \nu^{13} + \cdots + 11744919 ) / 130491 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{17} + \beta_{15} + \beta_{13} - \beta_{12} + 2 \beta_{10} + \beta_{9} + \beta_{6} + \cdots + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{16} + \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \cdots - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} - 2\beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} - \beta_{15} + \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{16} - 3 \beta_{15} - 3 \beta_{12} + \beta_{11} - 6 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{17} - 3 \beta_{16} + 4 \beta_{15} - 6 \beta_{14} - 5 \beta_{13} - 5 \beta_{12} - \beta_{11} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{17} - 3 \beta_{16} + 11 \beta_{15} + 9 \beta_{14} + 2 \beta_{13} + 19 \beta_{12} - 15 \beta_{11} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 33 \beta_{17} - 26 \beta_{16} + 21 \beta_{15} + 41 \beta_{14} + 13 \beta_{13} + 52 \beta_{12} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18 \beta_{17} - 3 \beta_{16} + 9 \beta_{15} + 30 \beta_{14} + 21 \beta_{13} + 9 \beta_{12} + 18 \beta_{11} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 78 \beta_{17} + 87 \beta_{16} - 27 \beta_{15} - 21 \beta_{14} - 84 \beta_{13} - 78 \beta_{12} + \cdots + 57 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 36 \beta_{17} + 45 \beta_{16} + 27 \beta_{15} - 57 \beta_{14} - 33 \beta_{13} - 51 \beta_{12} + \cdots - 96 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 204 \beta_{17} - 81 \beta_{16} - 30 \beta_{15} + 126 \beta_{14} + 240 \beta_{13} + 123 \beta_{12} + \cdots - 279 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 102 \beta_{17} - 333 \beta_{16} - 51 \beta_{15} + 126 \beta_{14} - 303 \beta_{13} + 177 \beta_{12} + \cdots - 288 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 261 \beta_{17} - 219 \beta_{16} + 108 \beta_{15} - 240 \beta_{14} - 345 \beta_{13} + 177 \beta_{12} + \cdots - 144 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 387 \beta_{17} + 126 \beta_{16} + 297 \beta_{15} + 711 \beta_{14} + 540 \beta_{13} + 720 \beta_{12} + \cdots + 2385 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1980 \beta_{17} - 459 \beta_{16} + 1062 \beta_{15} + 1611 \beta_{14} + 1818 \beta_{13} + 891 \beta_{12} + \cdots - 630 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1377 \beta_{17} + 2556 \beta_{16} - 621 \beta_{15} - 63 \beta_{14} + 2142 \beta_{13} - 180 \beta_{12} + \cdots - 3411 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−1.29960 + 1.14501i
1.16555 + 1.28120i
0.960398 − 1.44140i
1.68668 + 0.393823i
0.472963 + 1.66622i
−0.219955 − 1.71803i
0.381933 + 1.68942i
1.20201 − 1.24706i
−1.34999 − 1.08514i
0.381933 − 1.68942i
1.20201 + 1.24706i
−1.34999 + 1.08514i
1.68668 − 0.393823i
0.472963 − 1.66622i
−0.219955 + 1.71803i
−1.29960 − 1.14501i
1.16555 − 1.28120i
0.960398 + 1.44140i
−0.173648 − 0.984808i 0 −0.939693 + 0.342020i 0.766044 + 0.642788i 0 −4.75088 − 1.72918i 0.500000 + 0.866025i 0 0.500000 − 0.866025i
91.2 −0.173648 − 0.984808i 0 −0.939693 + 0.342020i 0.766044 + 0.642788i 0 −1.17449 − 0.427479i 0.500000 + 0.866025i 0 0.500000 − 0.866025i
91.3 −0.173648 − 0.984808i 0 −0.939693 + 0.342020i 0.766044 + 0.642788i 0 3.54598 + 1.29063i 0.500000 + 0.866025i 0 0.500000 − 0.866025i
181.1 0.939693 − 0.342020i 0 0.766044 − 0.642788i 0.173648 + 0.984808i 0 −2.60035 − 2.18196i 0.500000 − 0.866025i 0 0.500000 + 0.866025i
181.2 0.939693 − 0.342020i 0 0.766044 − 0.642788i 0.173648 + 0.984808i 0 1.67330 + 1.40407i 0.500000 − 0.866025i 0 0.500000 + 0.866025i
181.3 0.939693 − 0.342020i 0 0.766044 − 0.642788i 0.173648 + 0.984808i 0 1.95914 + 1.64391i 0.500000 − 0.866025i 0 0.500000 + 0.866025i
361.1 −0.766044 + 0.642788i 0 0.173648 − 0.984808i −0.939693 + 0.342020i 0 −0.500571 − 2.83888i 0.500000 + 0.866025i 0 0.500000 − 0.866025i
361.2 −0.766044 + 0.642788i 0 0.173648 − 0.984808i −0.939693 + 0.342020i 0 0.0883297 + 0.500943i 0.500000 + 0.866025i 0 0.500000 − 0.866025i
361.3 −0.766044 + 0.642788i 0 0.173648 − 0.984808i −0.939693 + 0.342020i 0 0.259537 + 1.47191i 0.500000 + 0.866025i 0 0.500000 − 0.866025i
451.1 −0.766044 − 0.642788i 0 0.173648 + 0.984808i −0.939693 − 0.342020i 0 −0.500571 + 2.83888i 0.500000 − 0.866025i 0 0.500000 + 0.866025i
451.2 −0.766044 − 0.642788i 0 0.173648 + 0.984808i −0.939693 − 0.342020i 0 0.0883297 − 0.500943i 0.500000 − 0.866025i 0 0.500000 + 0.866025i
451.3 −0.766044 − 0.642788i 0 0.173648 + 0.984808i −0.939693 − 0.342020i 0 0.259537 − 1.47191i 0.500000 − 0.866025i 0 0.500000 + 0.866025i
631.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i 0.173648 − 0.984808i 0 −2.60035 + 2.18196i 0.500000 + 0.866025i 0 0.500000 − 0.866025i
631.2 0.939693 + 0.342020i 0 0.766044 + 0.642788i 0.173648 − 0.984808i 0 1.67330 − 1.40407i 0.500000 + 0.866025i 0 0.500000 − 0.866025i
631.3 0.939693 + 0.342020i 0 0.766044 + 0.642788i 0.173648 − 0.984808i 0 1.95914 − 1.64391i 0.500000 + 0.866025i 0 0.500000 − 0.866025i
721.1 −0.173648 + 0.984808i 0 −0.939693 − 0.342020i 0.766044 − 0.642788i 0 −4.75088 + 1.72918i 0.500000 − 0.866025i 0 0.500000 + 0.866025i
721.2 −0.173648 + 0.984808i 0 −0.939693 − 0.342020i 0.766044 − 0.642788i 0 −1.17449 + 0.427479i 0.500000 − 0.866025i 0 0.500000 + 0.866025i
721.3 −0.173648 + 0.984808i 0 −0.939693 − 0.342020i 0.766044 − 0.642788i 0 3.54598 − 1.29063i 0.500000 − 0.866025i 0 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.k.d 18
3.b odd 2 1 270.2.k.d ✓ 18
27.e even 9 1 inner 810.2.k.d 18
27.e even 9 1 7290.2.a.q 9
27.f odd 18 1 270.2.k.d ✓ 18
27.f odd 18 1 7290.2.a.r 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.k.d ✓ 18 3.b odd 2 1
270.2.k.d ✓ 18 27.f odd 18 1
810.2.k.d 18 1.a even 1 1 trivial
810.2.k.d 18 27.e even 9 1 inner
7290.2.a.q 9 27.e even 9 1
7290.2.a.r 9 27.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{18} + 3 T_{7}^{17} - 21 T_{7}^{16} - 57 T_{7}^{15} + 249 T_{7}^{14} + 291 T_{7}^{13} + \cdots + 982081 \) acting on \(S_{2}^{\mathrm{new}}(810, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{18} + 3 T^{17} + \cdots + 982081 \) Copy content Toggle raw display
$11$ \( T^{18} + 9 T^{17} + \cdots + 2985984 \) Copy content Toggle raw display
$13$ \( T^{18} + 15 T^{17} + \cdots + 87616 \) Copy content Toggle raw display
$17$ \( T^{18} - 3 T^{17} + \cdots + 3779136 \) Copy content Toggle raw display
$19$ \( T^{18} - 9 T^{17} + \cdots + 19855936 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 297666009 \) Copy content Toggle raw display
$29$ \( T^{18} - 9 T^{17} + \cdots + 3674889 \) Copy content Toggle raw display
$31$ \( T^{18} + 3 T^{17} + \cdots + 4700224 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 578992548805696 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 269887523049 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 669790017649 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
$53$ \( (T^{9} + 12 T^{8} + \cdots + 2346408)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 1719926784 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 19813208147209 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 5542008097609 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 68\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 3169339909696 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 51843191481 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 90\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 679384765504 \) Copy content Toggle raw display
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