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Properties

Label 21.6.g.c
Level $21$
Weight $6$
Character orbit 21.g
Analytic conductor $3.368$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,6,Mod(5,21)]
 
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([3, 5]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.5");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), 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: \(3.36806021607\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 171 x^{14} + 21495 x^{12} - 1128902 x^{10} + 42970860 x^{8} - 655075344 x^{6} + \cdots + 90230547456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{6} + \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{12} + \beta_{10} - \beta_{8} + \cdots + 12) q^{4}+ \cdots + ( - 5 \beta_{14} + 5 \beta_{13} + \cdots - 7 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{6} + \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{12} + \beta_{10} - \beta_{8} + \cdots + 12) q^{4}+ \cdots + (84 \beta_{15} - 580 \beta_{14} + \cdots + 15285) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 18 q^{3} + 86 q^{4} - 616 q^{7} - 606 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 18 q^{3} + 86 q^{4} - 616 q^{7} - 606 q^{9} - 546 q^{10} + 4128 q^{12} + 1188 q^{15} - 2858 q^{16} - 438 q^{18} + 3258 q^{19} + 882 q^{21} - 21900 q^{22} + 10374 q^{24} - 6578 q^{25} + 34006 q^{28} + 450 q^{30} - 9792 q^{31} + 17556 q^{33} - 15132 q^{36} - 10466 q^{37} - 17418 q^{39} - 60774 q^{40} - 56994 q^{42} + 111052 q^{43} - 31374 q^{45} + 81696 q^{46} - 51548 q^{49} + 31596 q^{51} + 126960 q^{52} + 7434 q^{54} - 31368 q^{57} - 41706 q^{58} + 60066 q^{60} - 128580 q^{61} - 123606 q^{63} - 158884 q^{64} - 160314 q^{66} - 102202 q^{67} + 234318 q^{70} + 330864 q^{72} + 345150 q^{73} + 428202 q^{75} - 277512 q^{78} - 251308 q^{79} + 179262 q^{81} - 534576 q^{82} + 106176 q^{84} + 205152 q^{85} - 551922 q^{87} - 359514 q^{88} + 65814 q^{91} + 17676 q^{93} + 276612 q^{94} + 1043910 q^{96} + 215472 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 171 x^{14} + 21495 x^{12} - 1128902 x^{10} + 42970860 x^{8} - 655075344 x^{6} + \cdots + 90230547456 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13230933495217 \nu^{14} + \cdots + 35\!\cdots\!44 ) / 25\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13230933495217 \nu^{15} + \cdots + 93\!\cdots\!12 \nu ) / 25\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 56\!\cdots\!45 \nu^{14} + \cdots - 31\!\cdots\!92 ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 58\!\cdots\!09 \nu^{14} + \cdots - 18\!\cdots\!84 ) / 80\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 65\!\cdots\!81 \nu^{15} + \cdots + 10\!\cdots\!16 ) / 51\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!65 \nu^{14} + \cdots - 19\!\cdots\!80 ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\!\cdots\!40 \nu^{15} + \cdots - 19\!\cdots\!96 ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!59 \nu^{15} + \cdots - 66\!\cdots\!60 ) / 51\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 988408623043793 \nu^{15} + \cdots + 62\!\cdots\!00 ) / 17\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!87 \nu^{14} + \cdots - 93\!\cdots\!48 ) / 64\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 65\!\cdots\!81 \nu^{15} + \cdots - 14\!\cdots\!00 ) / 51\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 22\!\cdots\!35 \nu^{15} + \cdots + 66\!\cdots\!60 ) / 51\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 51\!\cdots\!05 \nu^{15} + \cdots - 44\!\cdots\!88 ) / 51\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12\!\cdots\!29 \nu^{15} + \cdots + 44\!\cdots\!88 ) / 51\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{10} - \beta_{8} - \beta_{7} + \beta_{4} - 43\beta_{2} + \beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} + 10 \beta_{14} + 5 \beta_{13} + 8 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} - \beta_{9} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 159 \beta_{11} + 93 \beta_{10} - 93 \beta_{8} - 159 \beta_{7} + 93 \beta_{6} + 139 \beta_{5} + \cdots + 93 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 139 \beta_{15} + 643 \beta_{14} + 1286 \beta_{13} + 1166 \beta_{12} + 391 \beta_{11} + 384 \beta_{10} + \cdots - 1166 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8921\beta_{12} - 15927\beta_{11} - 3656\beta_{7} + 8921\beta_{6} + 19583\beta_{5} - 3656\beta_{4} - 346408 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15927 \beta_{15} - 72935 \beta_{14} + 72935 \beta_{13} - 44431 \beta_{11} - 39034 \beta_{10} + \cdots - 880102 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 902945 \beta_{12} + 470256 \beta_{11} - 902945 \beta_{10} + 902945 \beta_{8} + 1744655 \beta_{7} + \cdots - 35447344 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3489310 \beta_{15} - 15960734 \beta_{14} - 7980367 \beta_{13} - 15466810 \beta_{12} - 9725022 \beta_{11} + \cdots + 15466810 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 242314767 \beta_{11} - 94099521 \beta_{10} + 94099521 \beta_{8} + 242314767 \beta_{7} + \cdots - 94099521 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 188263727 \beta_{15} - 861092303 \beta_{14} - 1722184606 \beta_{13} - 1682200810 \beta_{12} + \cdots + 1682200810 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 9941629681 \beta_{12} + 20195416575 \beta_{11} + 5951790976 \beta_{7} - 9941629681 \beta_{6} + \cdots + 393676948736 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 20195416575 \beta_{15} + 92373091039 \beta_{14} - 92373091039 \beta_{13} + 56284253807 \beta_{11} + \cdots + 1049359678214 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1056780975361 \beta_{12} - 643984782144 \beta_{11} + 1056780975361 \beta_{10} - 1056780975361 \beta_{8} + \cdots + 41898267202496 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 4322048553950 \beta_{15} + 19769208691870 \beta_{14} + 9884604345935 \beta_{13} + \cdots - 19403874824906 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−8.95007 − 5.16733i
−5.89199 − 3.40174i
−3.16536 − 1.82752i
−1.84692 − 1.06632i
1.84692 + 1.06632i
3.16536 + 1.82752i
5.89199 + 3.40174i
8.95007 + 5.16733i
−8.95007 + 5.16733i
−5.89199 + 3.40174i
−3.16536 + 1.82752i
−1.84692 + 1.06632i
1.84692 − 1.06632i
3.16536 − 1.82752i
5.89199 − 3.40174i
8.95007 − 5.16733i
−8.95007 − 5.16733i 14.3022 + 6.20057i 37.4025 + 64.7831i 15.8704 − 27.4884i −95.9654 − 129.400i 20.8412 − 127.956i − 442.375i 166.106 + 177.364i −284.083 + 164.015i
5.2 −5.89199 − 3.40174i 1.92849 − 15.4687i 7.14372 + 12.3733i −7.45706 + 12.9160i −63.9832 + 84.5813i −125.096 + 34.0278i 120.507i −235.562 − 59.6625i 87.8738 − 50.7340i
5.3 −3.16536 − 1.82752i 7.01741 + 13.9196i −9.32032 − 16.1433i −37.1622 + 64.3668i 3.22580 − 56.8851i 45.3051 + 121.468i 185.094i −144.512 + 195.359i 235.264 − 135.830i
5.4 −1.84692 − 1.06632i −6.34074 + 14.2406i −13.7259 − 23.7740i 47.5262 − 82.3179i 26.8959 − 19.5400i −95.0499 − 88.1619i 126.790i −162.590 − 180.592i −175.555 + 101.356i
5.5 1.84692 + 1.06632i −15.5031 − 1.62906i −13.7259 − 23.7740i −47.5262 + 82.3179i −26.8959 − 19.5400i −95.0499 − 88.1619i − 126.790i 237.692 + 50.5111i −175.555 + 101.356i
5.6 3.16536 + 1.82752i −8.54605 − 13.0371i −9.32032 − 16.1433i 37.1622 − 64.3668i −3.22580 − 56.8851i 45.3051 + 121.468i − 185.094i −96.9302 + 222.831i 235.264 − 135.830i
5.7 5.89199 + 3.40174i 14.3605 + 6.06423i 7.14372 + 12.3733i 7.45706 − 12.9160i 63.9832 + 84.5813i −125.096 + 34.0278i − 120.507i 169.450 + 174.171i 87.8738 − 50.7340i
5.8 8.95007 + 5.16733i 1.78125 − 15.4864i 37.4025 + 64.7831i −15.8704 + 27.4884i 95.9654 − 129.400i 20.8412 − 127.956i 442.375i −236.654 − 55.1702i −284.083 + 164.015i
17.1 −8.95007 + 5.16733i 14.3022 − 6.20057i 37.4025 − 64.7831i 15.8704 + 27.4884i −95.9654 + 129.400i 20.8412 + 127.956i 442.375i 166.106 − 177.364i −284.083 − 164.015i
17.2 −5.89199 + 3.40174i 1.92849 + 15.4687i 7.14372 − 12.3733i −7.45706 − 12.9160i −63.9832 − 84.5813i −125.096 − 34.0278i − 120.507i −235.562 + 59.6625i 87.8738 + 50.7340i
17.3 −3.16536 + 1.82752i 7.01741 − 13.9196i −9.32032 + 16.1433i −37.1622 − 64.3668i 3.22580 + 56.8851i 45.3051 − 121.468i − 185.094i −144.512 − 195.359i 235.264 + 135.830i
17.4 −1.84692 + 1.06632i −6.34074 − 14.2406i −13.7259 + 23.7740i 47.5262 + 82.3179i 26.8959 + 19.5400i −95.0499 + 88.1619i − 126.790i −162.590 + 180.592i −175.555 − 101.356i
17.5 1.84692 − 1.06632i −15.5031 + 1.62906i −13.7259 + 23.7740i −47.5262 − 82.3179i −26.8959 + 19.5400i −95.0499 + 88.1619i 126.790i 237.692 − 50.5111i −175.555 − 101.356i
17.6 3.16536 − 1.82752i −8.54605 + 13.0371i −9.32032 + 16.1433i 37.1622 + 64.3668i −3.22580 + 56.8851i 45.3051 − 121.468i 185.094i −96.9302 − 222.831i 235.264 + 135.830i
17.7 5.89199 − 3.40174i 14.3605 − 6.06423i 7.14372 − 12.3733i 7.45706 + 12.9160i 63.9832 − 84.5813i −125.096 − 34.0278i 120.507i 169.450 − 174.171i 87.8738 + 50.7340i
17.8 8.95007 − 5.16733i 1.78125 + 15.4864i 37.4025 − 64.7831i −15.8704 − 27.4884i 95.9654 + 129.400i 20.8412 + 127.956i − 442.375i −236.654 + 55.1702i −284.083 − 164.015i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.6.g.c ✓ 16
3.b odd 2 1 inner 21.6.g.c ✓ 16
7.c even 3 1 147.6.c.c 16
7.d odd 6 1 inner 21.6.g.c ✓ 16
7.d odd 6 1 147.6.c.c 16
21.g even 6 1 inner 21.6.g.c ✓ 16
21.g even 6 1 147.6.c.c 16
21.h odd 6 1 147.6.c.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.g.c ✓ 16 1.a even 1 1 trivial
21.6.g.c ✓ 16 3.b odd 2 1 inner
21.6.g.c ✓ 16 7.d odd 6 1 inner
21.6.g.c ✓ 16 21.g even 6 1 inner
147.6.c.c 16 7.c even 3 1
147.6.c.c 16 7.d odd 6 1
147.6.c.c 16 21.g even 6 1
147.6.c.c 16 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 171 T_{2}^{14} + 21495 T_{2}^{12} - 1128902 T_{2}^{10} + 42970860 T_{2}^{8} + \cdots + 90230547456 \) acting on \(S_{6}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 90230547456 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 79\!\cdots\!01)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 39\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 62\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 29\!\cdots\!24)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 66\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 48\!\cdots\!04)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 62\!\cdots\!61)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 59\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 52\!\cdots\!64)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 24\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 25\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 83\!\cdots\!09)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 56\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 60\!\cdots\!24)^{2} \) Copy content Toggle raw display
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