Properties

Label 21.17.h.a
Level $21$
Weight $17$
Character orbit 21.h
Analytic conductor $34.088$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.0881542099\)
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.

\(f(q)\) \(=\) \( q -6561 \zeta_{6} q^{3} -65536 \zeta_{6} q^{4} + ( -5405345 - 661791 \zeta_{6} ) q^{7} + ( -43046721 + 43046721 \zeta_{6} ) q^{9} +O(q^{10})\) \( q -6561 \zeta_{6} q^{3} -65536 \zeta_{6} q^{4} + ( -5405345 - 661791 \zeta_{6} ) q^{7} + ( -43046721 + 43046721 \zeta_{6} ) q^{9} + ( -429981696 + 429981696 \zeta_{6} ) q^{12} -1205410561 q^{13} + ( -4294967296 + 4294967296 \zeta_{6} ) q^{16} + ( 23855013601 - 23855013601 \zeta_{6} ) q^{19} + ( -4342010751 + 39806479296 \zeta_{6} ) q^{21} -152587890625 \zeta_{6} q^{25} + 282429536481 q^{27} + ( -43371134976 + 397615824896 \zeta_{6} ) q^{28} + 202786989793 \zeta_{6} q^{31} + 2821109907456 q^{36} + ( -1766022963839 + 1766022963839 \zeta_{6} ) q^{37} + 7908698690721 \zeta_{6} q^{39} + 23369166652127 q^{43} + 28179280429056 q^{48} + ( 28779787241344 + 7592384673471 \zeta_{6} ) q^{49} + 78997786525696 \zeta_{6} q^{52} -156512744236161 q^{57} + ( -184288715234114 + 184288715234114 \zeta_{6} ) q^{61} + ( 261170310661056 - 232682378123745 \zeta_{6} ) q^{63} + 281474976710656 q^{64} -578874088431839 \zeta_{6} q^{67} -1438148649475199 \zeta_{6} q^{73} + ( -1001129150390625 + 1001129150390625 \zeta_{6} ) q^{75} -1563362171355136 q^{76} + ( -2574917432504159 + 2574917432504159 \zeta_{6} ) q^{79} -1853020188851841 \zeta_{6} q^{81} + ( 2608757427142656 - 2324199410565120 \zeta_{6} ) q^{84} + ( 6515659948848545 + 797729860574751 \zeta_{6} ) q^{91} + ( 1330485440031873 - 1330485440031873 \zeta_{6} ) q^{93} + 15621585304991234 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6561q^{3} - 65536q^{4} - 11472481q^{7} - 43046721q^{9} + O(q^{10}) \) \( 2q - 6561q^{3} - 65536q^{4} - 11472481q^{7} - 43046721q^{9} - 429981696q^{12} - 2410821122q^{13} - 4294967296q^{16} + 23855013601q^{19} + 31122457794q^{21} - 152587890625q^{25} + 564859072962q^{27} + 310873554944q^{28} + 202786989793q^{31} + 5642219814912q^{36} - 1766022963839q^{37} + 7908698690721q^{39} + 46738333304254q^{43} + 56358560858112q^{48} + 65151959156159q^{49} + 78997786525696q^{52} - 313025488472322q^{57} - 184288715234114q^{61} + 289658243198367q^{63} + 562949953421312q^{64} - 578874088431839q^{67} - 1438148649475199q^{73} - 1001129150390625q^{75} - 3126724342710272q^{76} - 2574917432504159q^{79} - 1853020188851841q^{81} + 2893315443720192q^{84} + 13829049758271841q^{91} + 1330485440031873q^{93} + 31243170609982468q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −3280.50 5681.99i −32768.0 56755.8i 0 0 −5.73624e6 573128.i 0 −2.15234e7 + 3.72796e7i 0
11.1 0 −3280.50 + 5681.99i −32768.0 + 56755.8i 0 0 −5.73624e6 + 573128.i 0 −2.15234e7 3.72796e7i 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.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.17.h.a 2
3.b odd 2 1 CM 21.17.h.a 2
7.c even 3 1 inner 21.17.h.a 2
21.h odd 6 1 inner 21.17.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.17.h.a 2 1.a even 1 1 trivial
21.17.h.a 2 3.b odd 2 1 CM
21.17.h.a 2 7.c even 3 1 inner
21.17.h.a 2 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{17}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 43046721 + 6561 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 33232930569601 + 11472481 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 1205410561 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(56\!\cdots\!01\)\( - 23855013601 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(41\!\cdots\!49\)\( - 202786989793 T + T^{2} \)
$37$ \( \)\(31\!\cdots\!21\)\( + 1766022963839 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -23369166652127 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(33\!\cdots\!96\)\( + 184288715234114 T + T^{2} \)
$67$ \( \)\(33\!\cdots\!21\)\( + 578874088431839 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(20\!\cdots\!01\)\( + 1438148649475199 T + T^{2} \)
$79$ \( \)\(66\!\cdots\!81\)\( + 2574917432504159 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -15621585304991234 + T )^{2} \)
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