Properties

Label 21.20.g.a
Level $21$
Weight $20$
Character orbit 21.g
Analytic conductor $48.052$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0515062768\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( 19683 + 19683 \zeta_{6} ) q^{3} + ( -524288 + 524288 \zeta_{6} ) q^{4} + ( 103363346 - 109870239 \zeta_{6} ) q^{7} + 1162261467 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 19683 + 19683 \zeta_{6} ) q^{3} + ( -524288 + 524288 \zeta_{6} ) q^{4} + ( 103363346 - 109870239 \zeta_{6} ) q^{7} + 1162261467 \zeta_{6} q^{9} + ( -20639121408 + 10319560704 \zeta_{6} ) q^{12} + ( 43912988783 - 87825977566 \zeta_{6} ) q^{13} -274877906944 \zeta_{6} q^{16} + ( 1365151658466 - 682575829233 \zeta_{6} ) q^{19} + ( 4197076653555 - 2290651089156 \zeta_{6} ) q^{21} + ( 19073486328125 - 19073486328125 \zeta_{6} ) q^{25} + ( -22876792454961 + 45753584909922 \zeta_{6} ) q^{27} + ( 3411485917184 + 54192161947648 \zeta_{6} ) q^{28} + ( -159714263521255 - 159714263521255 \zeta_{6} ) q^{31} -609359740010496 q^{36} + 1581011344413647 \zeta_{6} q^{37} + ( 2593018074647367 - 2593018074647367 \zeta_{6} ) q^{39} + 6012529514786335 q^{43} + ( 5410421842378752 - 10820843684757504 \zeta_{6} ) q^{48} + ( -1387488121601405 - 10641641639802267 \zeta_{6} ) q^{49} + ( 23023053063061504 + 23023053063061504 \zeta_{6} ) q^{52} + 40305420140379417 q^{57} + ( 134845233947509688 - 67422616973754844 \zeta_{6} ) q^{61} + ( 127697945159780613 - 7562711003792031 \zeta_{6} ) q^{63} + 144115188075855872 q^{64} + ( 441174519368848177 - 441174519368848177 \zeta_{6} ) q^{67} + ( -95532016723177113 - 95532016723177113 \zeta_{6} ) q^{73} + ( 750846862792968750 - 375423431396484375 \zeta_{6} ) q^{75} + ( -357866316356911104 + 715732632713822208 \zeta_{6} ) q^{76} -262389124703851223 \zeta_{6} q^{79} + ( -1350851717672992089 + 1350851717672992089 \zeta_{6} ) q^{81} + ( -999516046307622912 + 2200476924539043840 \zeta_{6} ) q^{84} + ( -5110467692113710356 - 4253256334150166699 \zeta_{6} ) q^{91} -9430967546666586495 \zeta_{6} q^{93} + ( -6681938461866185624 + 13363876923732371248 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 59049q^{3} - 524288q^{4} + 96856453q^{7} + 1162261467q^{9} + O(q^{10}) \) \( 2q + 59049q^{3} - 524288q^{4} + 96856453q^{7} + 1162261467q^{9} - 30958682112q^{12} - 274877906944q^{16} + 2047727487699q^{19} + 6103502217954q^{21} + 19073486328125q^{25} + 61015133782016q^{28} - 479142790563765q^{31} - 1218719480020992q^{36} + 1581011344413647q^{37} + 2593018074647367q^{39} + 12025059029572670q^{43} - 13416617883005077q^{49} + 69069159189184512q^{52} + 80610840280758834q^{57} + 202267850921264532q^{61} + 247833179315769195q^{63} + 288230376151711744q^{64} + 441174519368848177q^{67} - 286596050169531339q^{73} + 1126270294189453125q^{75} - 262389124703851223q^{79} - 1350851717672992089q^{81} + 201444831923798016q^{84} - 14474191718377587411q^{91} - 9430967546666586495q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 0.866025i
0.500000 + 0.866025i
0 29524.5 17046.0i −262144. 454047.i 0 0 4.84282e7 + 9.51504e7i 0 5.81131e8 1.00655e9i 0
17.1 0 29524.5 + 17046.0i −262144. + 454047.i 0 0 4.84282e7 9.51504e7i 0 5.81131e8 + 1.00655e9i 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.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.20.g.a 2
3.b odd 2 1 CM 21.20.g.a 2
7.d odd 6 1 inner 21.20.g.a 2
21.g even 6 1 inner 21.20.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.20.g.a 2 1.a even 1 1 trivial
21.20.g.a 2 3.b odd 2 1 CM
21.20.g.a 2 7.d odd 6 1 inner
21.20.g.a 2 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{20}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1162261467 - 59049 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 11398895185373143 - 96856453 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(57\!\cdots\!67\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(13\!\cdots\!67\)\( - 2047727487699 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(76\!\cdots\!75\)\( + 479142790563765 T + T^{2} \)
$37$ \( \)\(24\!\cdots\!09\)\( - 1581011344413647 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -6012529514786335 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(13\!\cdots\!08\)\( - 202267850921264532 T + T^{2} \)
$67$ \( \)\(19\!\cdots\!29\)\( - 441174519368848177 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(27\!\cdots\!07\)\( + 286596050169531339 T + T^{2} \)
$79$ \( \)\(68\!\cdots\!29\)\( + 262389124703851223 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(13\!\cdots\!28\)\( + T^{2} \)
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