Properties

Label 21.35.h.a
Level $21$
Weight $35$
Character orbit 21.h
Analytic conductor $153.774$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 35 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(153.773873490\)
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.

\(f(q)\) \(=\) \( q + 129140163 \zeta_{6} q^{3} -17179869184 \zeta_{6} q^{4} + ( -253015451951165 + 48377438356677 \zeta_{6} ) q^{7} + ( -16677181699666569 + 16677181699666569 \zeta_{6} ) q^{9} +O(q^{10})\) \( q +129140163 \zeta_{6} q^{3} -17179869184 \zeta_{6} q^{4} +(-253015451951165 + 48377438356677 \zeta_{6}) q^{7} +(-16677181699666569 + 16677181699666569 \zeta_{6}) q^{9} +(2218611106740436992 - 2218611106740436992 \zeta_{6}) q^{12} -6628589056805300737 q^{13} +(-\)\(29\!\cdots\!56\)\( + \)\(29\!\cdots\!56\)\( \zeta_{6}) q^{16} +(\)\(10\!\cdots\!49\)\( - \)\(10\!\cdots\!49\)\( \zeta_{6}) q^{19} +(-\)\(62\!\cdots\!51\)\( - \)\(26\!\cdots\!44\)\( \zeta_{6}) q^{21} -\)\(58\!\cdots\!25\)\( \zeta_{6} q^{25} -\)\(21\!\cdots\!47\)\( q^{27} +(\)\(83\!\cdots\!68\)\( + \)\(35\!\cdots\!92\)\( \zeta_{6}) q^{28} -\)\(19\!\cdots\!47\)\( \zeta_{6} q^{31} +\)\(28\!\cdots\!96\)\( q^{36} +(-\)\(85\!\cdots\!87\)\( + \)\(85\!\cdots\!87\)\( \zeta_{6}) q^{37} -\)\(85\!\cdots\!31\)\( \zeta_{6} q^{39} -\)\(87\!\cdots\!61\)\( q^{43} -\)\(38\!\cdots\!28\)\( q^{48} +(\)\(61\!\cdots\!96\)\( - \)\(22\!\cdots\!81\)\( \zeta_{6}) q^{49} +\)\(11\!\cdots\!08\)\( \zeta_{6} q^{52} +\)\(13\!\cdots\!87\)\( q^{57} +(\)\(44\!\cdots\!66\)\( - \)\(44\!\cdots\!66\)\( \zeta_{6}) q^{61} +(\)\(34\!\cdots\!72\)\( - \)\(42\!\cdots\!85\)\( \zeta_{6}) q^{63} +\)\(50\!\cdots\!04\)\( q^{64} +\)\(22\!\cdots\!13\)\( \zeta_{6} q^{67} -\)\(20\!\cdots\!43\)\( \zeta_{6} q^{73} +(\)\(75\!\cdots\!75\)\( - \)\(75\!\cdots\!75\)\( \zeta_{6}) q^{75} -\)\(18\!\cdots\!16\)\( q^{76} +(-\)\(33\!\cdots\!11\)\( + \)\(33\!\cdots\!11\)\( \zeta_{6}) q^{79} -\)\(27\!\cdots\!61\)\( \zeta_{6} q^{81} +(-\)\(45\!\cdots\!96\)\( + \)\(56\!\cdots\!80\)\( \zeta_{6}) q^{84} +(\)\(16\!\cdots\!05\)\( - \)\(32\!\cdots\!49\)\( \zeta_{6}) q^{91} +(\)\(25\!\cdots\!61\)\( - \)\(25\!\cdots\!61\)\( \zeta_{6}) q^{93} +\)\(20\!\cdots\!02\)\( q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 129140163q^{3} - 17179869184q^{4} - 457653465545653q^{7} - 16677181699666569q^{9} + O(q^{10}) \) \( 2q + 129140163q^{3} - 17179869184q^{4} - 457653465545653q^{7} - 16677181699666569q^{9} + 2218611106740436992q^{12} - 13257178113610601474q^{13} - \)\(29\!\cdots\!56\)\(q^{16} + \)\(10\!\cdots\!49\)\(q^{19} - \)\(38\!\cdots\!46\)\(q^{21} - \)\(58\!\cdots\!25\)\(q^{25} - \)\(43\!\cdots\!94\)\(q^{27} + \)\(51\!\cdots\!28\)\(q^{28} - \)\(19\!\cdots\!47\)\(q^{31} + \)\(57\!\cdots\!92\)\(q^{36} - \)\(85\!\cdots\!87\)\(q^{37} - \)\(85\!\cdots\!31\)\(q^{39} - \)\(17\!\cdots\!22\)\(q^{43} - \)\(76\!\cdots\!56\)\(q^{48} + \)\(10\!\cdots\!11\)\(q^{49} + \)\(11\!\cdots\!08\)\(q^{52} + \)\(27\!\cdots\!74\)\(q^{57} + \)\(44\!\cdots\!66\)\(q^{61} + \)\(26\!\cdots\!59\)\(q^{63} + \)\(10\!\cdots\!08\)\(q^{64} + \)\(22\!\cdots\!13\)\(q^{67} - \)\(20\!\cdots\!43\)\(q^{73} + \)\(75\!\cdots\!75\)\(q^{75} - \)\(36\!\cdots\!32\)\(q^{76} - \)\(33\!\cdots\!11\)\(q^{79} - \)\(27\!\cdots\!61\)\(q^{81} - \)\(34\!\cdots\!12\)\(q^{84} + \)\(30\!\cdots\!61\)\(q^{91} + \)\(25\!\cdots\!61\)\(q^{93} + \)\(41\!\cdots\!04\)\(q^{97} + O(q^{100}) \)

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 6.45701e7 + 1.11839e8i −8.58993e9 1.48782e10i 0 0 −2.28827e14 + 4.18961e13i 0 −8.33859e15 + 1.44429e16i 0
11.1 0 6.45701e7 1.11839e8i −8.58993e9 + 1.48782e10i 0 0 −2.28827e14 4.18961e13i 0 −8.33859e15 1.44429e16i 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 21.35.h.a 2
3.b odd 2 1 CM 21.35.h.a 2
7.c even 3 1 inner 21.35.h.a 2
21.h odd 6 1 inner 21.35.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.35.h.a 2 1.a even 1 1 trivial
21.35.h.a 2 3.b odd 2 1 CM
21.35.h.a 2 7.c even 3 1 inner
21.35.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 \(S_{35}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 16677181699666569 - 129140163 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(54\!\cdots\!49\)\( + 457653465545653 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 6628589056805300737 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(11\!\cdots\!01\)\( - \)\(10\!\cdots\!49\)\( T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(38\!\cdots\!09\)\( + \)\(19\!\cdots\!47\)\( T + T^{2} \)
$37$ \( \)\(73\!\cdots\!69\)\( + \)\(85\!\cdots\!87\)\( T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( \)\(87\!\cdots\!61\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(19\!\cdots\!56\)\( - \)\(44\!\cdots\!66\)\( T + T^{2} \)
$67$ \( \)\(48\!\cdots\!69\)\( - \)\(22\!\cdots\!13\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(40\!\cdots\!49\)\( + \)\(20\!\cdots\!43\)\( T + T^{2} \)
$79$ \( \)\(11\!\cdots\!21\)\( + \)\(33\!\cdots\!11\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -\)\(20\!\cdots\!02\)\( + T )^{2} \)
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