Properties

Label 81.7.d.a
Level 81
Weight 7
Character orbit 81.d
Analytic conductor 18.634
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 81.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6343807732\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -64 \zeta_{6} q^{4} + ( 286 - 286 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -64 \zeta_{6} q^{4} + ( 286 - 286 \zeta_{6} ) q^{7} -506 \zeta_{6} q^{13} + ( -4096 + 4096 \zeta_{6} ) q^{16} -10582 q^{19} + ( -15625 + 15625 \zeta_{6} ) q^{25} -18304 q^{28} -35282 \zeta_{6} q^{31} -89206 q^{37} + ( -111386 + 111386 \zeta_{6} ) q^{43} + 35853 \zeta_{6} q^{49} + ( -32384 + 32384 \zeta_{6} ) q^{52} + ( 420838 - 420838 \zeta_{6} ) q^{61} + 262144 q^{64} -172874 \zeta_{6} q^{67} + 638066 q^{73} + 677248 \zeta_{6} q^{76} + ( 204622 - 204622 \zeta_{6} ) q^{79} -144716 q^{91} + ( 56446 - 56446 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 64q^{4} + 286q^{7} + O(q^{10}) \) \( 2q - 64q^{4} + 286q^{7} - 506q^{13} - 4096q^{16} - 21164q^{19} - 15625q^{25} - 36608q^{28} - 35282q^{31} - 178412q^{37} - 111386q^{43} + 35853q^{49} - 32384q^{52} + 420838q^{61} + 524288q^{64} - 172874q^{67} + 1276132q^{73} + 677248q^{76} + 204622q^{79} - 289432q^{91} + 56446q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 −32.0000 55.4256i 0 0 143.000 247.683i 0 0 0
53.1 0 0 −32.0000 + 55.4256i 0 0 143.000 + 247.683i 0 0 0
\(n\): 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 CM by \(\Q(\sqrt{-3}) \)
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 81.7.d.a 2
3.b odd 2 1 CM 81.7.d.a 2
9.c even 3 1 3.7.b.a 1
9.c even 3 1 inner 81.7.d.a 2
9.d odd 6 1 3.7.b.a 1
9.d odd 6 1 inner 81.7.d.a 2
36.f odd 6 1 48.7.e.a 1
36.h even 6 1 48.7.e.a 1
45.h odd 6 1 75.7.c.a 1
45.j even 6 1 75.7.c.a 1
45.k odd 12 2 75.7.d.a 2
45.l even 12 2 75.7.d.a 2
63.l odd 6 1 147.7.b.a 1
63.o even 6 1 147.7.b.a 1
72.j odd 6 1 192.7.e.b 1
72.l even 6 1 192.7.e.a 1
72.n even 6 1 192.7.e.b 1
72.p odd 6 1 192.7.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.7.b.a 1 9.c even 3 1
3.7.b.a 1 9.d odd 6 1
48.7.e.a 1 36.f odd 6 1
48.7.e.a 1 36.h even 6 1
75.7.c.a 1 45.h odd 6 1
75.7.c.a 1 45.j even 6 1
75.7.d.a 2 45.k odd 12 2
75.7.d.a 2 45.l even 12 2
81.7.d.a 2 1.a even 1 1 trivial
81.7.d.a 2 3.b odd 2 1 CM
81.7.d.a 2 9.c even 3 1 inner
81.7.d.a 2 9.d odd 6 1 inner
147.7.b.a 1 63.l odd 6 1
147.7.b.a 1 63.o even 6 1
192.7.e.a 1 72.l even 6 1
192.7.e.a 1 72.p odd 6 1
192.7.e.b 1 72.j odd 6 1
192.7.e.b 1 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{7}^{\mathrm{new}}(81, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8 T + 64 T^{2} )( 1 + 8 T + 64 T^{2} ) \)
$3$ 1
$5$ \( ( 1 - 125 T + 15625 T^{2} )( 1 + 125 T + 15625 T^{2} ) \)
$7$ \( ( 1 - 683 T + 117649 T^{2} )( 1 + 397 T + 117649 T^{2} ) \)
$11$ \( ( 1 - 1331 T + 1771561 T^{2} )( 1 + 1331 T + 1771561 T^{2} ) \)
$13$ \( ( 1 - 3527 T + 4826809 T^{2} )( 1 + 4033 T + 4826809 T^{2} ) \)
$17$ \( ( 1 - 4913 T )^{2}( 1 + 4913 T )^{2} \)
$19$ \( ( 1 + 10582 T + 47045881 T^{2} )^{2} \)
$23$ \( ( 1 - 12167 T + 148035889 T^{2} )( 1 + 12167 T + 148035889 T^{2} ) \)
$29$ \( ( 1 - 24389 T + 594823321 T^{2} )( 1 + 24389 T + 594823321 T^{2} ) \)
$31$ \( ( 1 - 23939 T + 887503681 T^{2} )( 1 + 59221 T + 887503681 T^{2} ) \)
$37$ \( ( 1 + 89206 T + 2565726409 T^{2} )^{2} \)
$41$ \( ( 1 - 68921 T + 4750104241 T^{2} )( 1 + 68921 T + 4750104241 T^{2} ) \)
$43$ \( ( 1 - 42587 T + 6321363049 T^{2} )( 1 + 153973 T + 6321363049 T^{2} ) \)
$47$ \( ( 1 - 103823 T + 10779215329 T^{2} )( 1 + 103823 T + 10779215329 T^{2} ) \)
$53$ \( ( 1 - 148877 T )^{2}( 1 + 148877 T )^{2} \)
$59$ \( ( 1 - 205379 T + 42180533641 T^{2} )( 1 + 205379 T + 42180533641 T^{2} ) \)
$61$ \( ( 1 - 357839 T + 51520374361 T^{2} )( 1 - 62999 T + 51520374361 T^{2} ) \)
$67$ \( ( 1 - 412523 T + 90458382169 T^{2} )( 1 + 585397 T + 90458382169 T^{2} ) \)
$71$ \( ( 1 - 357911 T )^{2}( 1 + 357911 T )^{2} \)
$73$ \( ( 1 - 638066 T + 151334226289 T^{2} )^{2} \)
$79$ \( ( 1 - 937691 T + 243087455521 T^{2} )( 1 + 733069 T + 243087455521 T^{2} ) \)
$83$ \( ( 1 - 571787 T + 326940373369 T^{2} )( 1 + 571787 T + 326940373369 T^{2} ) \)
$89$ \( ( 1 - 704969 T )^{2}( 1 + 704969 T )^{2} \)
$97$ \( ( 1 - 1608263 T + 832972004929 T^{2} )( 1 + 1551817 T + 832972004929 T^{2} ) \)
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