Properties

Label 21.3.b.a
Level $21$
Weight $3$
Character orbit 21.b
Analytic conductor $0.572$
Analytic rank $0$
Dimension $4$
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] = [21,3,Mod(8,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.8");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 21.b (of order \(2\), degree \(1\), 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: \(0.572208555157\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.65856.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 14x^{2} + 21 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - \beta_{2} - \beta_1) q^{3} + (2 \beta_{2} - 3) q^{4} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{6} - \beta_{2} q^{7} + (4 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 2) q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - \beta_{2} - \beta_1) q^{3} + (2 \beta_{2} - 3) q^{4} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{6} - \beta_{2} q^{7} + (4 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 2) q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 6) q^{9} + (\beta_{2} + 7) q^{10} + 2 \beta_1 q^{11} + ( - \beta_{3} + \beta_{2} + 7 \beta_1 - 6) q^{12} + (\beta_{2} - 9) q^{13} + ( - 2 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{14} + (4 \beta_{3} + 2 \beta_{2} - \beta_1 + 9) q^{15} + ( - 4 \beta_{2} + 9) q^{16} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{17} + (2 \beta_{3} - 2 \beta_{2} - 8 \beta_1 + 15) q^{18} + ( - 5 \beta_{2} + 3) q^{19} + ( - 6 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 3) q^{20} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{21} + (4 \beta_{2} - 14) q^{22} + ( - 4 \beta_{3} + 2 \beta_{2} + 8 \beta_1 - 2) q^{23} + ( - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 33) q^{24} + ( - 10 \beta_{2} - 3) q^{25} + (2 \beta_{3} - \beta_{2} - 12 \beta_1 + 1) q^{26} + (\beta_{3} + 8 \beta_{2} + 5 \beta_1 + 3) q^{27} + (3 \beta_{2} - 14) q^{28} + ( - 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{29} + (8 \beta_{3} - 8 \beta_{2} - 5 \beta_1 - 3) q^{30} + (2 \beta_{2} + 34) q^{31} + (8 \beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{32} + ( - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 6) q^{33} - 6 \beta_{2} q^{34} + (6 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{35} + ( - 10 \beta_{3} - 8 \beta_{2} + 13 \beta_1 + 24) q^{36} + (14 \beta_{2} + 4) q^{37} + ( - 10 \beta_{3} + 5 \beta_{2} + 18 \beta_1 - 5) q^{38} + ( - 8 \beta_{3} + 8 \beta_{2} + 11 \beta_1 - 3) q^{39} + (15 \beta_{2} + 21) q^{40} + (16 \beta_{3} - 8 \beta_{2} - 22 \beta_1 + 8) q^{41} + (\beta_{3} - 4 \beta_{2} + 2 \beta_1 + 18) q^{42} + ( - 6 \beta_{2} - 40) q^{43} + (8 \beta_{3} - 4 \beta_{2} - 18 \beta_1 + 4) q^{44} + (4 \beta_{3} - 13 \beta_{2} + 2 \beta_1 - 33) q^{45} + (18 \beta_{2} - 42) q^{46} + (8 \beta_{3} - 4 \beta_{2} + 8 \beta_1 + 4) q^{47} + (5 \beta_{3} - 5 \beta_{2} - 17 \beta_1 + 12) q^{48} + 7 q^{49} + ( - 20 \beta_{3} + 10 \beta_{2} + 27 \beta_1 - 10) q^{50} + ( - 6 \beta_{3} - 24) q^{51} + ( - 21 \beta_{2} + 41) q^{52} + ( - 32 \beta_{3} + 16 \beta_{2} + 10 \beta_1 - 16) q^{53} + (17 \beta_{3} + \beta_{2} - 23 \beta_1 - 30) q^{54} + (2 \beta_{2} + 14) q^{55} + ( - 2 \beta_{3} + \beta_{2} - 11 \beta_1 - 1) q^{56} + ( - 2 \beta_{3} + 2 \beta_{2} - 13 \beta_1 + 15) q^{57} + ( - 2 \beta_{2} + 28) q^{58} + ( - 2 \beta_{3} + \beta_{2} - 26 \beta_1 - 1) q^{59} + (8 \beta_{3} - 2 \beta_{2} + \beta_1 + 39) q^{60} + (7 \beta_{2} - 39) q^{61} + (4 \beta_{3} - 2 \beta_{2} + 28 \beta_1 + 2) q^{62} + (8 \beta_{3} + \beta_{2} - 5 \beta_1 - 3) q^{63} + ( - 18 \beta_{2} + 1) q^{64} + (12 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 6) q^{65} + ( - 10 \beta_{3} + 10 \beta_{2} + 22 \beta_1 - 12) q^{66} + ( - 8 \beta_{2} - 6) q^{67} + (4 \beta_{3} - 2 \beta_{2} + 10 \beta_1 + 2) q^{68} + ( - 12 \beta_{2} + 6 \beta_1 + 42) q^{69} + ( - 7 \beta_{2} - 7) q^{70} + (12 \beta_{3} - 6 \beta_{2} + 18 \beta_1 + 6) q^{71} + ( - 18 \beta_{3} + 36 \beta_{2} + 36 \beta_1 - 9) q^{72} + (26 \beta_{2} - 8) q^{73} + (28 \beta_{3} - 14 \beta_{2} - 38 \beta_1 + 14) q^{74} + ( - 13 \beta_{3} + 13 \beta_{2} - 17 \beta_1 + 30) q^{75} + (21 \beta_{2} - 79) q^{76} + ( - 4 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 2) q^{77} + (8 \beta_{3} + 22 \beta_{2} - 11 \beta_1 - 45) q^{78} + ( - 36 \beta_{2} + 32) q^{79} + (6 \beta_{3} - 3 \beta_{2} - 8 \beta_1 + 3) q^{80} + (4 \beta_{3} - 22 \beta_{2} + 20 \beta_1 - 15) q^{81} + ( - 52 \beta_{2} + 98) q^{82} + ( - 18 \beta_{3} + 9 \beta_{2} - 18 \beta_1 - 9) q^{83} + ( - 11 \beta_{3} + 11 \beta_{2} + 20 \beta_1 - 9) q^{84} + (18 \beta_{2} + 42) q^{85} + ( - 12 \beta_{3} + 6 \beta_{2} - 22 \beta_1 - 6) q^{86} + (10 \beta_{3} + 8 \beta_{2} - 4 \beta_1 + 12) q^{87} + ( - 24 \beta_{2} + 42) q^{88} + ( - 32 \beta_{3} + 16 \beta_{2} + 42 \beta_1 - 16) q^{89} + ( - 22 \beta_{3} + 13 \beta_{2} - 2 \beta_1 - 39) q^{90} + (9 \beta_{2} - 7) q^{91} + (20 \beta_{3} - 10 \beta_{2} - 64 \beta_1 + 10) q^{92} + (36 \beta_{3} - 36 \beta_{2} - 30 \beta_1 - 6) q^{93} + (12 \beta_{2} - 84) q^{94} + (24 \beta_{3} - 12 \beta_{2} - 10 \beta_1 + 12) q^{95} + ( - 17 \beta_{3} - 10 \beta_{2} + 5 \beta_1 - 33) q^{96} + (8 \beta_{2} - 2) q^{97} + 7 \beta_1 q^{98} + (4 \beta_{3} - 4 \beta_{2} - 16 \beta_1 + 30) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 12 q^{4} + 14 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 12 q^{4} + 14 q^{6} - 20 q^{9} + 28 q^{10} - 22 q^{12} - 36 q^{13} + 28 q^{15} + 36 q^{16} + 56 q^{18} + 12 q^{19} + 14 q^{21} - 56 q^{22} - 126 q^{24} - 12 q^{25} + 10 q^{27} - 56 q^{28} - 28 q^{30} + 136 q^{31} + 28 q^{33} + 116 q^{36} + 16 q^{37} + 4 q^{39} + 84 q^{40} + 70 q^{42} - 160 q^{43} - 140 q^{45} - 168 q^{46} + 38 q^{48} + 28 q^{49} - 84 q^{51} + 164 q^{52} - 154 q^{54} + 56 q^{55} + 64 q^{57} + 112 q^{58} + 140 q^{60} - 156 q^{61} - 28 q^{63} + 4 q^{64} - 28 q^{66} - 24 q^{67} + 168 q^{69} - 28 q^{70} - 32 q^{73} + 146 q^{75} - 316 q^{76} - 196 q^{78} + 128 q^{79} - 68 q^{81} + 392 q^{82} - 14 q^{84} + 168 q^{85} + 28 q^{87} + 168 q^{88} - 112 q^{90} - 28 q^{91} - 96 q^{93} - 336 q^{94} - 98 q^{96} - 8 q^{97} + 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 14x^{2} + 21 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + \nu^{2} + 13\nu + 5 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 2\beta_{2} - 13\beta _1 + 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
3.50592i
1.30710i
1.30710i
3.50592i
3.50592i 0.822876 + 2.88494i −8.29150 1.24197i 10.1144 2.88494i 2.64575 15.0457i −7.64575 + 4.74789i 4.35425
8.2 1.30710i −1.82288 2.38267i 2.29150 7.37953i −3.11438 + 2.38267i −2.64575 8.22359i −2.35425 + 8.68663i 9.64575
8.3 1.30710i −1.82288 + 2.38267i 2.29150 7.37953i −3.11438 2.38267i −2.64575 8.22359i −2.35425 8.68663i 9.64575
8.4 3.50592i 0.822876 2.88494i −8.29150 1.24197i 10.1144 + 2.88494i 2.64575 15.0457i −7.64575 4.74789i 4.35425
\(n\): 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.3.b.a 4
3.b odd 2 1 inner 21.3.b.a 4
4.b odd 2 1 336.3.d.c 4
5.b even 2 1 525.3.c.a 4
5.c odd 4 2 525.3.f.a 8
7.b odd 2 1 147.3.b.f 4
7.c even 3 2 147.3.h.e 8
7.d odd 6 2 147.3.h.c 8
8.b even 2 1 1344.3.d.f 4
8.d odd 2 1 1344.3.d.b 4
9.c even 3 2 567.3.r.c 8
9.d odd 6 2 567.3.r.c 8
12.b even 2 1 336.3.d.c 4
15.d odd 2 1 525.3.c.a 4
15.e even 4 2 525.3.f.a 8
21.c even 2 1 147.3.b.f 4
21.g even 6 2 147.3.h.c 8
21.h odd 6 2 147.3.h.e 8
24.f even 2 1 1344.3.d.b 4
24.h odd 2 1 1344.3.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.b.a 4 1.a even 1 1 trivial
21.3.b.a 4 3.b odd 2 1 inner
147.3.b.f 4 7.b odd 2 1
147.3.b.f 4 21.c even 2 1
147.3.h.c 8 7.d odd 6 2
147.3.h.c 8 21.g even 6 2
147.3.h.e 8 7.c even 3 2
147.3.h.e 8 21.h odd 6 2
336.3.d.c 4 4.b odd 2 1
336.3.d.c 4 12.b even 2 1
525.3.c.a 4 5.b even 2 1
525.3.c.a 4 15.d odd 2 1
525.3.f.a 8 5.c odd 4 2
525.3.f.a 8 15.e even 4 2
567.3.r.c 8 9.c even 3 2
567.3.r.c 8 9.d odd 6 2
1344.3.d.b 4 8.d odd 2 1
1344.3.d.b 4 24.f even 2 1
1344.3.d.f 4 8.b even 2 1
1344.3.d.f 4 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 14T^{2} + 21 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 12 T^{2} + 18 T + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 56T^{2} + 84 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 56T^{2} + 336 \) Copy content Toggle raw display
$13$ \( (T^{2} + 18 T + 74)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 168T^{2} + 3024 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T - 166)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 672 T^{2} + 12096 \) Copy content Toggle raw display
$29$ \( T^{4} + 392 T^{2} + 27216 \) Copy content Toggle raw display
$31$ \( (T^{2} - 68 T + 1128)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T - 1356)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 5432 T^{2} + \cdots + 4139856 \) Copy content Toggle raw display
$43$ \( (T^{2} + 80 T + 1348)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2688 T^{2} + \cdots + 1741824 \) Copy content Toggle raw display
$53$ \( T^{4} + 11256 T^{2} + \cdots + 2543184 \) Copy content Toggle raw display
$59$ \( T^{4} + 10248 T^{2} + \cdots + 14606676 \) Copy content Toggle raw display
$61$ \( (T^{2} + 78 T + 1178)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12 T - 412)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 9576 T^{2} + \cdots + 22888656 \) Copy content Toggle raw display
$73$ \( (T^{2} + 16 T - 4668)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 64 T - 8048)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 13608 T^{2} + \cdots + 44641044 \) Copy content Toggle raw display
$89$ \( T^{4} + 20216 T^{2} + \cdots + 64754256 \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T - 444)^{2} \) Copy content Toggle raw display
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