Properties

Label 21.40.c.a
Level $21$
Weight $40$
Character orbit 21.c
Analytic conductor $202.313$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 40 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(202.313057918\)
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.

\(f(q)\) \(=\) \( q + ( 1162261467 - 2324522934 \zeta_{6} ) q^{3} + 549755813888 q^{4} + ( -34699901162609611 + 14804106004606482 \zeta_{6} ) q^{7} -4052555153018976267 q^{9} +O(q^{10})\) \( q +(1162261467 - 2324522934 \zeta_{6}) q^{3} +549755813888 q^{4} +(-34699901162609611 + 14804106004606482 \zeta_{6}) q^{7} -4052555153018976267 q^{9} +(\)\(63\!\cdots\!96\)\( - \)\(12\!\cdots\!92\)\( \zeta_{6}) q^{12} +(\)\(21\!\cdots\!56\)\( - \)\(42\!\cdots\!12\)\( \zeta_{6}) q^{13} +\)\(30\!\cdots\!44\)\( q^{16} +(-\)\(22\!\cdots\!30\)\( + \)\(45\!\cdots\!60\)\( \zeta_{6}) q^{19} +(-\)\(59\!\cdots\!49\)\( + \)\(63\!\cdots\!80\)\( \zeta_{6}) q^{21} -\)\(18\!\cdots\!25\)\( q^{25} +(-\)\(47\!\cdots\!89\)\( + \)\(94\!\cdots\!78\)\( \zeta_{6}) q^{27} +(-\)\(19\!\cdots\!68\)\( + \)\(81\!\cdots\!16\)\( \zeta_{6}) q^{28} +(-\)\(26\!\cdots\!30\)\( + \)\(53\!\cdots\!60\)\( \zeta_{6}) q^{31} -\)\(22\!\cdots\!96\)\( q^{36} +\)\(51\!\cdots\!30\)\( q^{37} -\)\(74\!\cdots\!56\)\( q^{39} +\)\(62\!\cdots\!40\)\( q^{43} +(\)\(35\!\cdots\!48\)\( - \)\(70\!\cdots\!96\)\( \zeta_{6}) q^{48} +(\)\(98\!\cdots\!97\)\( - \)\(80\!\cdots\!80\)\( \zeta_{6}) q^{49} +(\)\(11\!\cdots\!28\)\( - \)\(23\!\cdots\!56\)\( \zeta_{6}) q^{52} +\)\(78\!\cdots\!30\)\( q^{57} +(\)\(60\!\cdots\!80\)\( - \)\(12\!\cdots\!60\)\( \zeta_{6}) q^{61} +(\)\(14\!\cdots\!37\)\( - \)\(59\!\cdots\!94\)\( \zeta_{6}) q^{63} +\)\(16\!\cdots\!72\)\( q^{64} +\)\(36\!\cdots\!40\)\( q^{67} +(-\)\(18\!\cdots\!96\)\( + \)\(36\!\cdots\!92\)\( \zeta_{6}) q^{73} +(-\)\(21\!\cdots\!75\)\( + \)\(42\!\cdots\!50\)\( \zeta_{6}) q^{75} +(-\)\(12\!\cdots\!40\)\( + \)\(24\!\cdots\!80\)\( \zeta_{6}) q^{76} +\)\(10\!\cdots\!24\)\( q^{79} +\)\(16\!\cdots\!89\)\( q^{81} +(-\)\(32\!\cdots\!12\)\( + \)\(34\!\cdots\!40\)\( \zeta_{6}) q^{84} +(-\)\(10\!\cdots\!32\)\( + \)\(11\!\cdots\!40\)\( \zeta_{6}) q^{91} +\)\(92\!\cdots\!30\)\( q^{93} +(\)\(53\!\cdots\!12\)\( - \)\(10\!\cdots\!24\)\( \zeta_{6}) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 1099511627776q^{4} - 54595696320612740q^{7} - 8105110306037952534q^{9} + O(q^{10}) \) \( 2q + 1099511627776q^{4} - 54595696320612740q^{7} - 8105110306037952534q^{9} + \)\(60\!\cdots\!88\)\(q^{16} + \)\(51\!\cdots\!82\)\(q^{21} - \)\(36\!\cdots\!50\)\(q^{25} - \)\(30\!\cdots\!20\)\(q^{28} - \)\(44\!\cdots\!92\)\(q^{36} + \)\(10\!\cdots\!60\)\(q^{37} - \)\(14\!\cdots\!12\)\(q^{39} + \)\(12\!\cdots\!80\)\(q^{43} + \)\(11\!\cdots\!14\)\(q^{49} + \)\(15\!\cdots\!60\)\(q^{57} + \)\(22\!\cdots\!80\)\(q^{63} + \)\(33\!\cdots\!44\)\(q^{64} + \)\(72\!\cdots\!80\)\(q^{67} + \)\(21\!\cdots\!48\)\(q^{79} + \)\(32\!\cdots\!78\)\(q^{81} + \)\(28\!\cdots\!16\)\(q^{84} + \)\(95\!\cdots\!76\)\(q^{91} + \)\(18\!\cdots\!60\)\(q^{93} + O(q^{100}) \)

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 2.01310e9i 5.49756e11 0 0 −2.72978e16 + 1.28207e16i 0 −4.05256e18 0
20.2 0 2.01310e9i 5.49756e11 0 0 −2.72978e16 1.28207e16i 0 −4.05256e18 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 21.40.c.a 2
3.b odd 2 1 CM 21.40.c.a 2
7.b odd 2 1 inner 21.40.c.a 2
21.c even 2 1 inner 21.40.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.40.c.a 2 1.a even 1 1 trivial
21.40.c.a 2 3.b odd 2 1 CM
21.40.c.a 2 7.b odd 2 1 inner
21.40.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 \(S_{40}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4052555153018976267 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(90\!\cdots\!43\)\( + 54595696320612740 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(13\!\cdots\!08\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(15\!\cdots\!00\)\( + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(21\!\cdots\!00\)\( + T^{2} \)
$37$ \( ( -\)\(51\!\cdots\!30\)\( + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -\)\(62\!\cdots\!40\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(10\!\cdots\!00\)\( + T^{2} \)
$67$ \( ( -\)\(36\!\cdots\!40\)\( + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(99\!\cdots\!48\)\( + T^{2} \)
$79$ \( ( -\)\(10\!\cdots\!24\)\( + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(84\!\cdots\!32\)\( + T^{2} \)
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