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Properties

Label 21.7.f.b
Level $21$
Weight $7$
Character orbit 21.f
Analytic conductor $4.831$
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] = [21,7,Mod(10,21)]
 
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, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.10");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 21.f (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: \(4.83113575602\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 212x^{6} - 787x^{5} + 38792x^{4} - 92833x^{3} + 1563109x^{2} + 3107772x + 38787984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{2} + (9 \beta_{2} + 9) q^{3} + (\beta_{6} - \beta_{5} + 43 \beta_{2} - 43) q^{4} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots - 6) q^{5}+ \cdots + 243 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{2} + (9 \beta_{2} + 9) q^{3} + (\beta_{6} - \beta_{5} + 43 \beta_{2} - 43) q^{4} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots - 6) q^{5}+ \cdots + ( - 486 \beta_{7} - 2187 \beta_{5} + \cdots - 64395) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{2} + 108 q^{3} - 173 q^{4} - 42 q^{5} + 748 q^{7} - 454 q^{8} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{2} + 108 q^{3} - 173 q^{4} - 42 q^{5} + 748 q^{7} - 454 q^{8} + 972 q^{9} + 261 q^{10} - 1070 q^{11} - 4671 q^{12} + 6070 q^{14} - 756 q^{15} + 3911 q^{16} + 7212 q^{17} + 1215 q^{18} - 24606 q^{19} + 8154 q^{21} - 78 q^{22} - 15224 q^{23} - 6129 q^{24} + 22274 q^{25} - 19044 q^{26} - 3415 q^{28} + 32524 q^{29} + 2349 q^{30} + 40200 q^{31} + 70203 q^{32} - 28890 q^{33} - 242436 q^{35} - 84078 q^{36} - 45670 q^{37} + 503310 q^{38} + 93366 q^{39} - 94941 q^{40} + 55161 q^{42} - 445660 q^{43} - 188829 q^{44} - 10206 q^{45} + 525804 q^{46} + 82884 q^{47} + 24116 q^{49} - 1218884 q^{50} + 64908 q^{51} + 722856 q^{52} - 13034 q^{53} + 32805 q^{54} + 127061 q^{56} - 442908 q^{57} - 159501 q^{58} + 1810362 q^{59} + 429705 q^{60} - 392856 q^{61} + 38394 q^{63} - 1410446 q^{64} - 389004 q^{65} - 1053 q^{66} + 384094 q^{67} - 1616346 q^{68} + 406005 q^{70} + 225688 q^{71} - 55161 q^{72} + 903078 q^{73} + 1185530 q^{74} + 601398 q^{75} - 327674 q^{77} - 342792 q^{78} - 559592 q^{79} + 847713 q^{80} - 236196 q^{81} + 347634 q^{82} - 879444 q^{84} + 1953576 q^{85} - 2302402 q^{86} + 439074 q^{87} + 304887 q^{88} - 1770036 q^{89} - 2960718 q^{91} - 113064 q^{92} + 361800 q^{93} - 1837620 q^{94} + 1160112 q^{95} + 1895481 q^{96} + 5732467 q^{98} - 520020 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 212x^{6} - 787x^{5} + 38792x^{4} - 92833x^{3} + 1563109x^{2} + 3107772x + 38787984 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1585013359 \nu^{7} + 18232571539 \nu^{6} - 303603349712 \nu^{5} + 4245938445433 \nu^{4} + \cdots + 22\!\cdots\!68 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8019055 \nu^{7} - 15616321 \nu^{6} - 1444380025 \nu^{5} + 2053828205 \nu^{4} + \cdots - 29614389599556 ) / 11465622241311 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27322708193 \nu^{7} + 1000103862583 \nu^{6} + 22563530233273 \nu^{5} + \cdots + 69\!\cdots\!80 ) / 11\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39673486 \nu^{7} + 224426993 \nu^{6} - 7145928130 \nu^{5} + 10161113066 \nu^{4} + \cdots - 963541759742958 ) / 11465622241311 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 59472187739 \nu^{7} - 700951590437 \nu^{6} + 11672037067696 \nu^{5} - 218751249599021 \nu^{4} + \cdots - 85\!\cdots\!44 ) / 11\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 226304020214 \nu^{7} + 436984015237 \nu^{6} + 31940434175641 \nu^{5} + \cdots + 14\!\cdots\!28 ) / 11\!\cdots\!18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + 2\beta_{3} + 106\beta_{2} - 2\beta _1 - 106 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} + 6\beta_{5} - 2\beta_{4} - 145\beta_{3} - 2\beta _1 + 252 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{7} - 205\beta_{6} + 2\beta_{4} - 15886\beta_{2} + 772\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 424 \beta_{7} + 1560 \beta_{6} - 1560 \beta_{5} + 848 \beta_{4} + 24589 \beta_{3} + 90180 \beta_{2} + \cdots - 90180 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -304\beta_{7} + 38461\beta_{5} - 152\beta_{4} - 199046\beta_{3} - 152\beta _1 + 2727346 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 75858\beta_{7} - 355626\beta_{6} - 75858\beta_{4} - 22658292\beta_{2} + 4625113\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
5.73828 + 9.93899i
4.15432 + 7.19549i
−2.30325 − 3.98935i
−7.08935 − 12.2791i
5.73828 − 9.93899i
4.15432 − 7.19549i
−2.30325 + 3.98935i
−7.08935 + 12.2791i
−6.23828 − 10.8050i 13.5000 + 7.79423i −45.8323 + 79.3839i −175.367 + 101.248i − 194.490i 284.280 + 191.921i 345.159 121.500 + 210.444i 2187.98 + 1263.23i
10.2 −4.65432 − 8.06151i 13.5000 + 7.79423i −11.3253 + 19.6160i 151.343 − 87.3778i − 145.107i −271.614 − 209.463i −384.906 121.500 + 210.444i −1408.79 − 813.368i
10.3 1.80325 + 3.12332i 13.5000 + 7.79423i 25.4966 − 44.1614i 71.9311 − 41.5295i 56.2198i 77.0894 + 334.225i 414.723 121.500 + 210.444i 259.420 + 149.776i
10.4 6.58935 + 11.4131i 13.5000 + 7.79423i −54.8390 + 94.9839i −68.9069 + 39.7834i 205.435i 284.244 − 191.975i −601.976 121.500 + 210.444i −908.103 − 524.293i
19.1 −6.23828 + 10.8050i 13.5000 − 7.79423i −45.8323 − 79.3839i −175.367 − 101.248i 194.490i 284.280 − 191.921i 345.159 121.500 − 210.444i 2187.98 − 1263.23i
19.2 −4.65432 + 8.06151i 13.5000 − 7.79423i −11.3253 − 19.6160i 151.343 + 87.3778i 145.107i −271.614 + 209.463i −384.906 121.500 − 210.444i −1408.79 + 813.368i
19.3 1.80325 − 3.12332i 13.5000 − 7.79423i 25.4966 + 44.1614i 71.9311 + 41.5295i − 56.2198i 77.0894 − 334.225i 414.723 121.500 − 210.444i 259.420 − 149.776i
19.4 6.58935 − 11.4131i 13.5000 − 7.79423i −54.8390 − 94.9839i −68.9069 − 39.7834i − 205.435i 284.244 + 191.975i −601.976 121.500 − 210.444i −908.103 + 524.293i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.7.f.b ✓ 8
3.b odd 2 1 63.7.m.c 8
4.b odd 2 1 336.7.bh.b 8
7.b odd 2 1 147.7.f.a 8
7.c even 3 1 147.7.d.a 8
7.c even 3 1 147.7.f.a 8
7.d odd 6 1 inner 21.7.f.b ✓ 8
7.d odd 6 1 147.7.d.a 8
21.g even 6 1 63.7.m.c 8
21.g even 6 1 441.7.d.d 8
21.h odd 6 1 441.7.d.d 8
28.f even 6 1 336.7.bh.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.b ✓ 8 1.a even 1 1 trivial
21.7.f.b ✓ 8 7.d odd 6 1 inner
63.7.m.c 8 3.b odd 2 1
63.7.m.c 8 21.g even 6 1
147.7.d.a 8 7.c even 3 1
147.7.d.a 8 7.d odd 6 1
147.7.f.a 8 7.b odd 2 1
147.7.f.a 8 7.c even 3 1
336.7.bh.b 8 4.b odd 2 1
336.7.bh.b 8 28.f even 6 1
441.7.d.d 8 21.g even 6 1
441.7.d.d 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 5 T_{2}^{7} + 227 T_{2}^{6} + 818 T_{2}^{5} + 39854 T_{2}^{4} + 129428 T_{2}^{3} + \cdots + 30470400 \) acting on \(S_{7}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{7} + \cdots + 30470400 \) Copy content Toggle raw display
$3$ \( (T^{2} - 27 T + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 42\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 53\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 14\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 25\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 58\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
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