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Properties

Label 21.5.b.a
Level $21$
Weight $5$
Character orbit 21.b
Analytic conductor $2.171$
Analytic rank $0$
Dimension $8$
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] = [21,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.8");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(2.17076922476\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 82x^{6} + 2017x^{4} + 13020x^{2} + 756 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3^{3}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} - 5) q^{4} - \beta_{6} q^{5} + (\beta_{7} + \beta_{6} - \beta_{4} + \cdots - 5) q^{6}+ \cdots + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} - 5) q^{4} - \beta_{6} q^{5} + (\beta_{7} + \beta_{6} - \beta_{4} + \cdots - 5) q^{6}+ \cdots + ( - 187 \beta_{7} - 202 \beta_{6} + \cdots - 3747) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} - 36 q^{4} - 34 q^{6} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} - 36 q^{4} - 34 q^{6} + 64 q^{9} - 4 q^{10} + 98 q^{12} + 420 q^{13} + 76 q^{15} - 444 q^{16} - 712 q^{18} - 372 q^{19} + 98 q^{21} - 16 q^{22} + 1146 q^{24} + 1056 q^{25} - 1862 q^{27} + 392 q^{28} + 2348 q^{30} - 2776 q^{31} + 1396 q^{33} + 2928 q^{34} - 3268 q^{36} - 2560 q^{37} - 2540 q^{39} - 1980 q^{40} - 2450 q^{42} + 4720 q^{43} + 9700 q^{45} + 7536 q^{46} - 2962 q^{48} + 2744 q^{49} + 4764 q^{51} - 20252 q^{52} + 4886 q^{54} + 184 q^{55} - 14144 q^{57} - 7504 q^{58} - 13828 q^{60} + 972 q^{61} - 6076 q^{63} + 22772 q^{64} + 36020 q^{66} + 10200 q^{67} - 5760 q^{69} + 10780 q^{70} + 14304 q^{72} - 32008 q^{73} + 2114 q^{75} + 17332 q^{76} - 29668 q^{78} - 23168 q^{79} - 17216 q^{81} - 31976 q^{82} - 14798 q^{84} + 32016 q^{85} + 50764 q^{87} + 29208 q^{88} - 24352 q^{90} + 11956 q^{91} + 31848 q^{93} - 64992 q^{94} + 28630 q^{96} + 28112 q^{97} - 32432 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 82x^{6} + 2017x^{4} + 13020x^{2} + 756 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} - 44\nu^{4} + 188\nu^{3} - 285\nu^{2} + 1560\nu + 198 ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 48\nu^{4} + 441\nu^{2} + 322 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 4\nu^{5} - 44\nu^{4} - 188\nu^{3} - 285\nu^{2} - 1576\nu + 182 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 64\nu^{5} + 1177\nu^{3} + 6114\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} - 20\nu^{5} - 44\nu^{4} - 892\nu^{3} - 285\nu^{2} - 6264\nu + 198 ) / 48 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 21 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{5} + 2\beta_{3} - 33\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 4\beta_{4} + 6\beta_{3} - 39\beta_{2} + 2\beta _1 + 691 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -47\beta_{7} + 45\beta_{5} - 88\beta_{3} + 1159\beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -96\beta_{5} - 176\beta_{4} - 288\beta_{3} + 1431\beta_{2} - 96\beta _1 - 24229 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1831\beta_{7} + 48\beta_{6} - 1703\beta_{5} + 3278\beta_{3} - 41449\beta _1 - 1703 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
− 6.02741i
− 5.97075i
− 3.15624i
− 0.242064i
0.242064i
3.15624i
5.97075i
6.02741i
− 6.02741i −8.92230 + 1.18010i −20.3296 − 15.7540i 7.11292 + 53.7783i −18.5203 26.0965i 78.2147 − 21.0583i −94.9556
8.2 − 5.97075i 5.29264 − 7.27928i −19.6499 22.9681i −43.4628 − 31.6011i 18.5203 21.7927i −24.9759 − 77.0533i 137.137
8.3 − 3.15624i 7.09942 + 5.53157i 6.03813 − 16.3361i 17.4590 − 22.4075i −18.5203 − 69.5577i 19.8035 + 78.5418i −51.5608
8.4 − 0.242064i −4.46977 + 7.81161i 15.9414 30.4863i 1.89091 + 1.08197i 18.5203 − 7.73187i −41.0424 − 69.8321i 7.37963
8.5 0.242064i −4.46977 − 7.81161i 15.9414 − 30.4863i 1.89091 − 1.08197i 18.5203 7.73187i −41.0424 + 69.8321i 7.37963
8.6 3.15624i 7.09942 − 5.53157i 6.03813 16.3361i 17.4590 + 22.4075i −18.5203 69.5577i 19.8035 − 78.5418i −51.5608
8.7 5.97075i 5.29264 + 7.27928i −19.6499 − 22.9681i −43.4628 + 31.6011i 18.5203 − 21.7927i −24.9759 + 77.0533i 137.137
8.8 6.02741i −8.92230 − 1.18010i −20.3296 15.7540i 7.11292 − 53.7783i −18.5203 − 26.0965i 78.2147 + 21.0583i −94.9556
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.8
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 21.5.b.a ✓ 8
3.b odd 2 1 inner 21.5.b.a ✓ 8
4.b odd 2 1 336.5.d.b 8
7.b odd 2 1 147.5.b.e 8
7.c even 3 2 147.5.h.e 16
7.d odd 6 2 147.5.h.c 16
12.b even 2 1 336.5.d.b 8
21.c even 2 1 147.5.b.e 8
21.g even 6 2 147.5.h.c 16
21.h odd 6 2 147.5.h.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.b.a ✓ 8 1.a even 1 1 trivial
21.5.b.a ✓ 8 3.b odd 2 1 inner
147.5.b.e 8 7.b odd 2 1
147.5.b.e 8 21.c even 2 1
147.5.h.c 16 7.d odd 6 2
147.5.h.c 16 21.g even 6 2
147.5.h.e 16 7.c even 3 2
147.5.h.e 16 21.h odd 6 2
336.5.d.b 8 4.b odd 2 1
336.5.d.b 8 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 82 T^{6} + \cdots + 756 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 32473916496 \) Copy content Toggle raw display
$7$ \( (T^{2} - 343)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{4} - 210 T^{3} + \cdots + 137696008)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{4} + 186 T^{3} + \cdots + 5829791872)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{4} + 1388 T^{3} + \cdots + 74477967264)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1280 T^{3} + \cdots - 319722971328)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{4} - 2360 T^{3} + \cdots + 117096670336)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 86\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 13501614097288)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 16869420316288)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 737570944880496)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 301006877602048)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 66\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 23\!\cdots\!48)^{2} \) Copy content Toggle raw display
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