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Properties

Label 63.6.e.e
Level $63$
Weight $6$
Character orbit 63.e
Analytic conductor $10.104$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,6,Mod(37,63)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("63.37"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1 + 1) q^{2} + (\beta_{6} + \beta_{3} + 18 \beta_{2} - \beta_1) q^{4} + ( - \beta_{5} + \beta_{4} + \cdots - 2 \beta_1) q^{5} + (\beta_{7} + 6 \beta_{6} + \beta_{5} + \cdots + 53) q^{7}+ \cdots + (88 \beta_{7} + 8219 \beta_{6} + \cdots - 86434) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{2} - 69 q^{4} + 258 q^{7} - 246 q^{8} - 283 q^{10} + 402 q^{11} + 924 q^{13} - 1926 q^{14} - 3273 q^{16} + 276 q^{17} - 510 q^{19} - 9438 q^{20} + 2750 q^{22} + 6900 q^{23} - 2814 q^{25} - 15138 q^{26}+ \cdots - 382479 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8905 \nu^{7} + 366059 \nu^{6} - 1191820 \nu^{5} + 33153695 \nu^{4} - 47979268 \nu^{3} + \cdots + 307508418 ) / 9888988410 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1257 \nu^{7} - 279647 \nu^{6} + 388990 \nu^{5} - 26314019 \nu^{4} - 14452616 \nu^{3} + \cdots - 251390874 ) / 156968070 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22795 \nu^{7} + 56368 \nu^{6} + 2163175 \nu^{5} + 2122285 \nu^{4} + 222588334 \nu^{3} + \cdots - 31645054854 ) / 706356315 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 341623 \nu^{7} - 8468248 \nu^{6} + 27571040 \nu^{5} - 966332554 \nu^{4} + \cdots - 235881275616 ) / 9888988410 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25511 \nu^{7} + 22795 \nu^{6} - 2420915 \nu^{5} - 2375153 \nu^{4} - 232964210 \nu^{3} + \cdots - 54979470 ) / 706356315 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15589754 \nu^{7} - 23777815 \nu^{6} + 1539409445 \nu^{5} + 516927917 \nu^{4} + \cdots + 24613669710 ) / 9888988410 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{3} + 49\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 91\beta_{6} + 2\beta_{5} - 2\beta_{4} - 44 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 99\beta_{4} - 99\beta_{3} - 4505\beta_{2} - 181\beta _1 - 4406 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 196\beta_{7} + 8707\beta_{6} - 286\beta_{3} - 8624\beta_{2} - 8707\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 376\beta_{7} + 25333\beta_{6} - 376\beta_{5} + 9581\beta_{4} + 421300 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -18786\beta_{5} + 36042\beta_{4} + 36042\beta_{3} + 1222350\beta_{2} + 841891\beta _1 + 1186308 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
5.09061 − 8.81720i
0.895402 − 1.55088i
−0.874091 + 1.51397i
−4.61193 + 7.98809i
5.09061 + 8.81720i
0.895402 + 1.55088i
−0.874091 − 1.51397i
−4.61193 − 7.98809i
−4.59061 + 7.95118i 0 −26.1475 − 45.2888i 11.0358 − 19.1146i 0 126.882 − 26.6059i 186.333 0 101.322 + 175.495i
37.2 −0.395402 + 0.684857i 0 15.6873 + 27.1712i −52.0958 + 90.2327i 0 −7.12980 − 129.446i −50.1170 0 −41.1977 − 71.3564i
37.3 1.37409 − 2.37999i 0 12.2237 + 21.1722i 29.1836 − 50.5475i 0 −21.4366 + 127.857i 155.128 0 −80.2019 − 138.914i
37.4 5.11193 − 8.85412i 0 −36.2636 − 62.8104i 11.8764 − 20.5705i 0 30.6840 − 125.958i −414.344 0 −121.423 − 210.310i
46.1 −4.59061 − 7.95118i 0 −26.1475 + 45.2888i 11.0358 + 19.1146i 0 126.882 + 26.6059i 186.333 0 101.322 − 175.495i
46.2 −0.395402 − 0.684857i 0 15.6873 − 27.1712i −52.0958 − 90.2327i 0 −7.12980 + 129.446i −50.1170 0 −41.1977 + 71.3564i
46.3 1.37409 + 2.37999i 0 12.2237 − 21.1722i 29.1836 + 50.5475i 0 −21.4366 − 127.857i 155.128 0 −80.2019 + 138.914i
46.4 5.11193 + 8.85412i 0 −36.2636 + 62.8104i 11.8764 + 20.5705i 0 30.6840 + 125.958i −414.344 0 −121.423 + 210.310i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
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 63.6.e.e 8
3.b odd 2 1 21.6.e.c ✓ 8
7.c even 3 1 inner 63.6.e.e 8
7.c even 3 1 441.6.a.w 4
7.d odd 6 1 441.6.a.v 4
12.b even 2 1 336.6.q.j 8
21.c even 2 1 147.6.e.o 8
21.g even 6 1 147.6.a.l 4
21.g even 6 1 147.6.e.o 8
21.h odd 6 1 21.6.e.c ✓ 8
21.h odd 6 1 147.6.a.m 4
84.n even 6 1 336.6.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c ✓ 8 3.b odd 2 1
21.6.e.c ✓ 8 21.h odd 6 1
63.6.e.e 8 1.a even 1 1 trivial
63.6.e.e 8 7.c even 3 1 inner
147.6.a.l 4 21.g even 6 1
147.6.a.m 4 21.h odd 6 1
147.6.e.o 8 21.c even 2 1
147.6.e.o 8 21.g even 6 1
336.6.q.j 8 12.b even 2 1
336.6.q.j 8 84.n even 6 1
441.6.a.v 4 7.d odd 6 1
441.6.a.w 4 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 3T_{2}^{7} + 103T_{2}^{6} - 90T_{2}^{5} + 9190T_{2}^{4} - 16260T_{2}^{3} + 53772T_{2}^{2} + 37944T_{2} + 41616 \) acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 3 T^{7} + \cdots + 41616 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 10164899803536 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{4} - 462 T^{3} + \cdots + 149501563456)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 54\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 408027025117872)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 75\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 18\!\cdots\!52)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 991662745581932)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 73\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 72\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 66\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 21\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 27\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 41\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
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