Properties

Label 21.34.c.a
Level $21$
Weight $34$
Character orbit 21.c
Analytic conductor $144.864$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(144.863940650\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -43046721 + 86093442 \zeta_{6} ) q^{3} + 8589934592 q^{4} + ( 49967189809217 - 101524131070974 \zeta_{6} ) q^{7} -5559060566555523 q^{9} +O(q^{10})\) \( q +(-43046721 + 86093442 \zeta_{6}) q^{3} +8589934592 q^{4} +(49967189809217 - 101524131070974 \zeta_{6}) q^{7} -5559060566555523 q^{9} +(-369768517790072832 + 739537035580145664 \zeta_{6}) q^{12} +(-2738982428485658748 + 5477964856971317496 \zeta_{6}) q^{13} +73786976294838206464 q^{16} +(\)\(11\!\cdots\!10\)\( - \)\(23\!\cdots\!20\)\( \zeta_{6}) q^{19} +(\)\(65\!\cdots\!51\)\( - 68433587236834121340 \zeta_{6}) q^{21} -\)\(11\!\cdots\!25\)\( q^{25} +(\)\(23\!\cdots\!83\)\( - \)\(47\!\cdots\!66\)\( \zeta_{6}) q^{27} +(\)\(42\!\cdots\!64\)\( - \)\(87\!\cdots\!08\)\( \zeta_{6}) q^{28} +(-\)\(42\!\cdots\!10\)\( + \)\(84\!\cdots\!20\)\( \zeta_{6}) q^{31} -\)\(47\!\cdots\!16\)\( q^{36} +\)\(62\!\cdots\!30\)\( q^{37} -\)\(35\!\cdots\!24\)\( q^{39} -\)\(59\!\cdots\!40\)\( q^{43} +(-\)\(31\!\cdots\!44\)\( + \)\(63\!\cdots\!88\)\( \zeta_{6}) q^{48} +(-\)\(78\!\cdots\!87\)\( + \)\(16\!\cdots\!60\)\( \zeta_{6}) q^{49} +(-\)\(23\!\cdots\!16\)\( + \)\(47\!\cdots\!32\)\( \zeta_{6}) q^{52} +\)\(14\!\cdots\!30\)\( q^{57} +(-\)\(17\!\cdots\!40\)\( + \)\(34\!\cdots\!80\)\( \zeta_{6}) q^{61} +(-\)\(27\!\cdots\!91\)\( + \)\(56\!\cdots\!02\)\( \zeta_{6}) q^{63} +\)\(63\!\cdots\!88\)\( q^{64} -\)\(19\!\cdots\!60\)\( q^{67} +(-\)\(13\!\cdots\!12\)\( + \)\(26\!\cdots\!24\)\( \zeta_{6}) q^{73} +(\)\(50\!\cdots\!25\)\( - \)\(10\!\cdots\!50\)\( \zeta_{6}) q^{75} +(\)\(99\!\cdots\!20\)\( - \)\(19\!\cdots\!40\)\( \zeta_{6}) q^{76} -\)\(38\!\cdots\!84\)\( q^{79} +\)\(30\!\cdots\!29\)\( q^{81} +(\)\(56\!\cdots\!92\)\( - \)\(58\!\cdots\!80\)\( \zeta_{6}) q^{84} +(\)\(41\!\cdots\!88\)\( - \)\(43\!\cdots\!20\)\( \zeta_{6}) q^{91} -\)\(54\!\cdots\!30\)\( q^{93} +(-\)\(40\!\cdots\!44\)\( + \)\(80\!\cdots\!88\)\( \zeta_{6}) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 17179869184q^{4} - 1589751452540q^{7} - 11118121133111046q^{9} + O(q^{10}) \) \( 2q + 17179869184q^{4} - 1589751452540q^{7} - 11118121133111046q^{9} + \)\(14\!\cdots\!28\)\(q^{16} + \)\(13\!\cdots\!62\)\(q^{21} - \)\(23\!\cdots\!50\)\(q^{25} - \)\(13\!\cdots\!80\)\(q^{28} - \)\(95\!\cdots\!32\)\(q^{36} + \)\(12\!\cdots\!60\)\(q^{37} - \)\(70\!\cdots\!48\)\(q^{39} - \)\(11\!\cdots\!80\)\(q^{43} - \)\(15\!\cdots\!14\)\(q^{49} + \)\(29\!\cdots\!60\)\(q^{57} + \)\(88\!\cdots\!20\)\(q^{63} + \)\(12\!\cdots\!76\)\(q^{64} - \)\(39\!\cdots\!20\)\(q^{67} - \)\(76\!\cdots\!68\)\(q^{79} + \)\(61\!\cdots\!58\)\(q^{81} + \)\(11\!\cdots\!04\)\(q^{84} + \)\(83\!\cdots\!56\)\(q^{91} - \)\(10\!\cdots\!60\)\(q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1
0.500000 0.866025i
0.500000 + 0.866025i
0 7.45591e7i 8.58993e9 0 0 −7.94876e11 + 8.79225e13i 0 −5.55906e15 0
20.2 0 7.45591e7i 8.58993e9 0 0 −7.94876e11 8.79225e13i 0 −5.55906e15 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 \(\Q(\sqrt{-3}) \)
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.34.c.a 2
3.b odd 2 1 CM 21.34.c.a 2
7.b odd 2 1 inner 21.34.c.a 2
21.c even 2 1 inner 21.34.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.34.c.a 2 1.a even 1 1 trivial
21.34.c.a 2 3.b odd 2 1 CM
21.34.c.a 2 7.b odd 2 1 inner
21.34.c.a 2 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{34}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 5559060566555523 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(77\!\cdots\!07\)\( + 1589751452540 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(22\!\cdots\!12\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(39\!\cdots\!00\)\( + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(53\!\cdots\!00\)\( + T^{2} \)
$37$ \( ( -\)\(62\!\cdots\!30\)\( + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( \)\(59\!\cdots\!40\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(88\!\cdots\!00\)\( + T^{2} \)
$67$ \( ( \)\(19\!\cdots\!60\)\( + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(53\!\cdots\!32\)\( + T^{2} \)
$79$ \( ( \)\(38\!\cdots\!84\)\( + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(48\!\cdots\!08\)\( + T^{2} \)
show more
show less