Properties

Label 21.4.e.a
Level $21$
Weight $4$
Character orbit 21.e
Analytic conductor $1.239$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,4,Mod(4,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 3) q^{2} - 3 \zeta_{6} q^{3} - \zeta_{6} q^{4} + ( - 3 \zeta_{6} + 3) q^{5} - 9 q^{6} + (21 \zeta_{6} - 14) q^{7} + 21 q^{8} + (9 \zeta_{6} - 9) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \zeta_{6} + 3) q^{2} - 3 \zeta_{6} q^{3} - \zeta_{6} q^{4} + ( - 3 \zeta_{6} + 3) q^{5} - 9 q^{6} + (21 \zeta_{6} - 14) q^{7} + 21 q^{8} + (9 \zeta_{6} - 9) q^{9} - 9 \zeta_{6} q^{10} + 15 \zeta_{6} q^{11} + (3 \zeta_{6} - 3) q^{12} - 64 q^{13} + (42 \zeta_{6} + 21) q^{14} - 9 q^{15} + ( - 71 \zeta_{6} + 71) q^{16} - 84 \zeta_{6} q^{17} + 27 \zeta_{6} q^{18} + ( - 16 \zeta_{6} + 16) q^{19} - 3 q^{20} + ( - 21 \zeta_{6} + 63) q^{21} + 45 q^{22} + ( - 84 \zeta_{6} + 84) q^{23} - 63 \zeta_{6} q^{24} + 116 \zeta_{6} q^{25} + (192 \zeta_{6} - 192) q^{26} + 27 q^{27} + ( - 7 \zeta_{6} + 21) q^{28} - 297 q^{29} + (27 \zeta_{6} - 27) q^{30} + 253 \zeta_{6} q^{31} - 45 \zeta_{6} q^{32} + ( - 45 \zeta_{6} + 45) q^{33} - 252 q^{34} + (42 \zeta_{6} + 21) q^{35} + 9 q^{36} + ( - 316 \zeta_{6} + 316) q^{37} - 48 \zeta_{6} q^{38} + 192 \zeta_{6} q^{39} + ( - 63 \zeta_{6} + 63) q^{40} + 360 q^{41} + ( - 189 \zeta_{6} + 126) q^{42} + 26 q^{43} + ( - 15 \zeta_{6} + 15) q^{44} + 27 \zeta_{6} q^{45} - 252 \zeta_{6} q^{46} + ( - 30 \zeta_{6} + 30) q^{47} - 213 q^{48} + ( - 147 \zeta_{6} - 245) q^{49} + 348 q^{50} + (252 \zeta_{6} - 252) q^{51} + 64 \zeta_{6} q^{52} - 363 \zeta_{6} q^{53} + ( - 81 \zeta_{6} + 81) q^{54} + 45 q^{55} + (441 \zeta_{6} - 294) q^{56} - 48 q^{57} + (891 \zeta_{6} - 891) q^{58} + 15 \zeta_{6} q^{59} + 9 \zeta_{6} q^{60} + ( - 118 \zeta_{6} + 118) q^{61} + 759 q^{62} + ( - 126 \zeta_{6} - 63) q^{63} + 433 q^{64} + (192 \zeta_{6} - 192) q^{65} - 135 \zeta_{6} q^{66} + 370 \zeta_{6} q^{67} + (84 \zeta_{6} - 84) q^{68} - 252 q^{69} + ( - 63 \zeta_{6} + 189) q^{70} - 342 q^{71} + (189 \zeta_{6} - 189) q^{72} - 362 \zeta_{6} q^{73} - 948 \zeta_{6} q^{74} + ( - 348 \zeta_{6} + 348) q^{75} - 16 q^{76} + (105 \zeta_{6} - 315) q^{77} + 576 q^{78} + (467 \zeta_{6} - 467) q^{79} - 213 \zeta_{6} q^{80} - 81 \zeta_{6} q^{81} + ( - 1080 \zeta_{6} + 1080) q^{82} + 477 q^{83} + ( - 42 \zeta_{6} - 21) q^{84} - 252 q^{85} + ( - 78 \zeta_{6} + 78) q^{86} + 891 \zeta_{6} q^{87} + 315 \zeta_{6} q^{88} + (906 \zeta_{6} - 906) q^{89} + 81 q^{90} + ( - 1344 \zeta_{6} + 896) q^{91} - 84 q^{92} + ( - 759 \zeta_{6} + 759) q^{93} - 90 \zeta_{6} q^{94} - 48 \zeta_{6} q^{95} + (135 \zeta_{6} - 135) q^{96} + 503 q^{97} + (735 \zeta_{6} - 1176) q^{98} - 135 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 3 q^{3} - q^{4} + 3 q^{5} - 18 q^{6} - 7 q^{7} + 42 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} - 3 q^{3} - q^{4} + 3 q^{5} - 18 q^{6} - 7 q^{7} + 42 q^{8} - 9 q^{9} - 9 q^{10} + 15 q^{11} - 3 q^{12} - 128 q^{13} + 84 q^{14} - 18 q^{15} + 71 q^{16} - 84 q^{17} + 27 q^{18} + 16 q^{19} - 6 q^{20} + 105 q^{21} + 90 q^{22} + 84 q^{23} - 63 q^{24} + 116 q^{25} - 192 q^{26} + 54 q^{27} + 35 q^{28} - 594 q^{29} - 27 q^{30} + 253 q^{31} - 45 q^{32} + 45 q^{33} - 504 q^{34} + 84 q^{35} + 18 q^{36} + 316 q^{37} - 48 q^{38} + 192 q^{39} + 63 q^{40} + 720 q^{41} + 63 q^{42} + 52 q^{43} + 15 q^{44} + 27 q^{45} - 252 q^{46} + 30 q^{47} - 426 q^{48} - 637 q^{49} + 696 q^{50} - 252 q^{51} + 64 q^{52} - 363 q^{53} + 81 q^{54} + 90 q^{55} - 147 q^{56} - 96 q^{57} - 891 q^{58} + 15 q^{59} + 9 q^{60} + 118 q^{61} + 1518 q^{62} - 252 q^{63} + 866 q^{64} - 192 q^{65} - 135 q^{66} + 370 q^{67} - 84 q^{68} - 504 q^{69} + 315 q^{70} - 684 q^{71} - 189 q^{72} - 362 q^{73} - 948 q^{74} + 348 q^{75} - 32 q^{76} - 525 q^{77} + 1152 q^{78} - 467 q^{79} - 213 q^{80} - 81 q^{81} + 1080 q^{82} + 954 q^{83} - 84 q^{84} - 504 q^{85} + 78 q^{86} + 891 q^{87} + 315 q^{88} - 906 q^{89} + 162 q^{90} + 448 q^{91} - 168 q^{92} + 759 q^{93} - 90 q^{94} - 48 q^{95} - 135 q^{96} + 1006 q^{97} - 1617 q^{98} - 270 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 0.866025i
0.500000 + 0.866025i
1.50000 + 2.59808i −1.50000 + 2.59808i −0.500000 + 0.866025i 1.50000 + 2.59808i −9.00000 −3.50000 18.1865i 21.0000 −4.50000 7.79423i −4.50000 + 7.79423i
16.1 1.50000 2.59808i −1.50000 2.59808i −0.500000 0.866025i 1.50000 2.59808i −9.00000 −3.50000 + 18.1865i 21.0000 −4.50000 + 7.79423i −4.50000 7.79423i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.4.e.a 2
3.b odd 2 1 63.4.e.a 2
4.b odd 2 1 336.4.q.e 2
7.b odd 2 1 147.4.e.h 2
7.c even 3 1 inner 21.4.e.a 2
7.c even 3 1 147.4.a.b 1
7.d odd 6 1 147.4.a.a 1
7.d odd 6 1 147.4.e.h 2
21.c even 2 1 441.4.e.c 2
21.g even 6 1 441.4.a.k 1
21.g even 6 1 441.4.e.c 2
21.h odd 6 1 63.4.e.a 2
21.h odd 6 1 441.4.a.l 1
28.f even 6 1 2352.4.a.bd 1
28.g odd 6 1 336.4.q.e 2
28.g odd 6 1 2352.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 1.a even 1 1 trivial
21.4.e.a 2 7.c even 3 1 inner
63.4.e.a 2 3.b odd 2 1
63.4.e.a 2 21.h odd 6 1
147.4.a.a 1 7.d odd 6 1
147.4.a.b 1 7.c even 3 1
147.4.e.h 2 7.b odd 2 1
147.4.e.h 2 7.d odd 6 1
336.4.q.e 2 4.b odd 2 1
336.4.q.e 2 28.g odd 6 1
441.4.a.k 1 21.g even 6 1
441.4.a.l 1 21.h odd 6 1
441.4.e.c 2 21.c even 2 1
441.4.e.c 2 21.g even 6 1
2352.4.a.i 1 28.g odd 6 1
2352.4.a.bd 1 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 7T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$13$ \( (T + 64)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 84T + 7056 \) Copy content Toggle raw display
$19$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$23$ \( T^{2} - 84T + 7056 \) Copy content Toggle raw display
$29$ \( (T + 297)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 253T + 64009 \) Copy content Toggle raw display
$37$ \( T^{2} - 316T + 99856 \) Copy content Toggle raw display
$41$ \( (T - 360)^{2} \) Copy content Toggle raw display
$43$ \( (T - 26)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$53$ \( T^{2} + 363T + 131769 \) Copy content Toggle raw display
$59$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$61$ \( T^{2} - 118T + 13924 \) Copy content Toggle raw display
$67$ \( T^{2} - 370T + 136900 \) Copy content Toggle raw display
$71$ \( (T + 342)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 362T + 131044 \) Copy content Toggle raw display
$79$ \( T^{2} + 467T + 218089 \) Copy content Toggle raw display
$83$ \( (T - 477)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 906T + 820836 \) Copy content Toggle raw display
$97$ \( (T - 503)^{2} \) Copy content Toggle raw display
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