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Properties

Label 63.6.e.f
Level $63$
Weight $6$
Character orbit 63.e
Analytic conductor $10.104$
Analytic rank $0$
Dimension $12$
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] = [63,6,Mod(37,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), 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: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 187x^{10} + 25399x^{8} + 1518438x^{6} + 66232188x^{4} + 1297462320x^{2} + 18380851776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{8} - \beta_{5} + 30 \beta_{2}) q^{4} + ( - \beta_{7} - 2 \beta_1) q^{5} + (3 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} - \beta_{5} + 6 \beta_{2} + 13) q^{7} + (\beta_{11} - \beta_{10} - 2 \beta_{7} + 2 \beta_{4} - 21 \beta_{3} - 21 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{8} - \beta_{5} + 30 \beta_{2}) q^{4} + ( - \beta_{7} - 2 \beta_1) q^{5} + (3 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} - \beta_{5} + 6 \beta_{2} + 13) q^{7} + (\beta_{11} - \beta_{10} - 2 \beta_{7} + 2 \beta_{4} - 21 \beta_{3} - 21 \beta_1) q^{8} + ( - 16 \beta_{9} + 7 \beta_{8} - 8 \beta_{6} + 7 \beta_{5} - 120 \beta_{2} + \cdots + 8) q^{10}+ \cdots + (54 \beta_{11} + 237 \beta_{10} - 992 \beta_{7} + 1600 \beta_{4} + \cdots + 5194 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 182 q^{4} + 142 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 182 q^{4} + 142 q^{7} + 686 q^{10} + 308 q^{13} - 1898 q^{16} + 9422 q^{19} - 18292 q^{22} - 7526 q^{25} + 37074 q^{28} + 23422 q^{31} - 55608 q^{34} - 18182 q^{37} + 69258 q^{40} - 87372 q^{43} + 25332 q^{46} + 30354 q^{49} + 34272 q^{52} - 96320 q^{55} - 89782 q^{58} - 16156 q^{61} + 380580 q^{64} + 144650 q^{67} - 187262 q^{70} - 100058 q^{73} - 685440 q^{76} + 101994 q^{79} + 75712 q^{82} + 602352 q^{85} + 752310 q^{88} - 282306 q^{91} - 120456 q^{94} - 866096 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 187x^{10} + 25399x^{8} + 1518438x^{6} + 66232188x^{4} + 1297462320x^{2} + 18380851776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 334805 \nu^{10} + 56078291 \nu^{8} + 7616751407 \nu^{6} + 387910489914 \nu^{4} + 19861967445084 \nu^{2} + \cdots + 120250595793888 ) / 268837476625872 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 334805 \nu^{11} - 56078291 \nu^{9} - 7616751407 \nu^{7} - 387910489914 \nu^{5} - 19861967445084 \nu^{3} + \cdots - 389088072419760 \nu ) / 268837476625872 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4837597 \nu^{11} + 1494767143 \nu^{9} + 203024549011 \nu^{7} + 16589437928838 \nu^{5} + 529420847226732 \nu^{3} + \cdots + 10\!\cdots\!80 \nu ) / 537674953251744 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 289\nu^{10} + 39253\nu^{8} + 5331481\nu^{6} + 102358836\nu^{4} + 2005169040\nu^{2} - 465300162504 ) / 11897569332 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8147771 \nu^{10} + 247743581 \nu^{8} - 579963031 \nu^{6} - 12528900245394 \nu^{4} - 474402432194820 \nu^{2} + \cdots - 15\!\cdots\!32 ) / 268837476625872 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 25381 \nu^{11} - 3447337 \nu^{9} - 404606113 \nu^{7} - 8989514244 \nu^{5} - 176101022160 \nu^{3} + 6931671291720 \nu ) / 999395823888 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7113833 \nu^{10} + 1294946627 \nu^{8} + 175884221279 \nu^{6} + 10868775058206 \nu^{4} + 458647852670748 \nu^{2} + \cdots + 89\!\cdots\!20 ) / 134418738312936 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 901281 \nu^{10} + 185645795 \nu^{8} + 23947312631 \nu^{6} + 1451072814470 \nu^{4} + 44932220248548 \nu^{2} + \cdots + 663388031086704 ) / 9956943578736 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 37519 \nu^{11} + 5095963 \nu^{9} + 628528315 \nu^{7} + 13288585356 \nu^{5} + 260318121840 \nu^{3} - 37967330091600 \nu ) / 499697911944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4460963 \nu^{11} - 895743515 \nu^{9} - 121663045655 \nu^{7} - 7874821552212 \nu^{5} - 317256967300860 \nu^{3} + \cdots - 62\!\cdots\!00 \nu ) / 44806246104312 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} - \beta_{5} + 62\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - 2\beta_{7} + 2\beta_{4} - 85\beta_{3} - 85\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -14\beta_{9} + 139\beta_{8} + 14\beta_{6} - 5230\beta_{2} - 5230 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -139\beta_{11} - 362\beta_{4} + 8497\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2618\beta_{9} - 5236\beta_{6} + 16423\beta_{5} - 2618\beta_{2} + 522864 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16423\beta_{10} + 48554\beta_{7} + 911065\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 711172\beta_{9} - 1876447\beta_{8} + 355586\beta_{6} - 1876447\beta_{5} + 55996200\beta_{2} - 355586 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1876447 \beta_{11} - 1876447 \beta_{10} - 5886410 \beta_{7} + 5886410 \beta_{4} - 100576825 \beta_{3} - 100576825 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -43338386\beta_{9} + 212572543\beta_{8} + 43338386\beta_{6} - 6135103078\beta_{2} - 6135103078 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -212572543\beta_{11} - 685175402\beta_{4} + 11240963497\beta_{3} \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−5.31117 + 9.19921i
−3.54467 + 6.13954i
−2.44476 + 4.23445i
2.44476 − 4.23445i
3.54467 − 6.13954i
5.31117 − 9.19921i
−5.31117 − 9.19921i
−3.54467 − 6.13954i
−2.44476 − 4.23445i
2.44476 + 4.23445i
3.54467 + 6.13954i
5.31117 + 9.19921i
−5.31117 + 9.19921i 0 −40.4170 − 70.0043i 33.7376 − 58.4352i 0 −120.281 + 48.3679i 518.731 0 358.372 + 620.718i
37.2 −3.54467 + 6.13954i 0 −9.12931 − 15.8124i −41.1020 + 71.1908i 0 112.556 + 64.3292i −97.4172 0 −291.386 − 504.695i
37.3 −2.44476 + 4.23445i 0 4.04630 + 7.00840i 21.3752 − 37.0229i 0 43.2256 − 122.223i −196.034 0 104.514 + 181.024i
37.4 2.44476 − 4.23445i 0 4.04630 + 7.00840i −21.3752 + 37.0229i 0 43.2256 − 122.223i 196.034 0 104.514 + 181.024i
37.5 3.54467 − 6.13954i 0 −9.12931 − 15.8124i 41.1020 − 71.1908i 0 112.556 + 64.3292i 97.4172 0 −291.386 − 504.695i
37.6 5.31117 − 9.19921i 0 −40.4170 − 70.0043i −33.7376 + 58.4352i 0 −120.281 + 48.3679i −518.731 0 358.372 + 620.718i
46.1 −5.31117 − 9.19921i 0 −40.4170 + 70.0043i 33.7376 + 58.4352i 0 −120.281 − 48.3679i 518.731 0 358.372 − 620.718i
46.2 −3.54467 − 6.13954i 0 −9.12931 + 15.8124i −41.1020 − 71.1908i 0 112.556 − 64.3292i −97.4172 0 −291.386 + 504.695i
46.3 −2.44476 − 4.23445i 0 4.04630 − 7.00840i 21.3752 + 37.0229i 0 43.2256 + 122.223i −196.034 0 104.514 − 181.024i
46.4 2.44476 + 4.23445i 0 4.04630 − 7.00840i −21.3752 − 37.0229i 0 43.2256 + 122.223i 196.034 0 104.514 − 181.024i
46.5 3.54467 + 6.13954i 0 −9.12931 + 15.8124i 41.1020 + 71.1908i 0 112.556 − 64.3292i 97.4172 0 −291.386 + 504.695i
46.6 5.31117 + 9.19921i 0 −40.4170 + 70.0043i −33.7376 − 58.4352i 0 −120.281 − 48.3679i −518.731 0 358.372 − 620.718i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.e.f ✓ 12
3.b odd 2 1 inner 63.6.e.f ✓ 12
7.c even 3 1 inner 63.6.e.f ✓ 12
7.c even 3 1 441.6.a.bc 6
7.d odd 6 1 441.6.a.bd 6
21.g even 6 1 441.6.a.bd 6
21.h odd 6 1 inner 63.6.e.f ✓ 12
21.h odd 6 1 441.6.a.bc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.e.f ✓ 12 1.a even 1 1 trivial
63.6.e.f ✓ 12 3.b odd 2 1 inner
63.6.e.f ✓ 12 7.c even 3 1 inner
63.6.e.f ✓ 12 21.h odd 6 1 inner
441.6.a.bc 6 7.c even 3 1
441.6.a.bc 6 21.h odd 6 1
441.6.a.bd 6 7.d odd 6 1
441.6.a.bd 6 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 187T_{2}^{10} + 25399T_{2}^{8} + 1518438T_{2}^{6} + 66232188T_{2}^{4} + 1297462320T_{2}^{2} + 18380851776 \) acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 187 T^{10} + \cdots + 18380851776 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 13138 T^{10} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( (T^{6} - 71 T^{5} + \cdots + 4747561509943)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 902578 T^{10} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{3} - 77 T^{2} - 783976 T + 60080636)^{4} \) Copy content Toggle raw display
$17$ \( T^{12} + 2284392 T^{10} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{6} - 4711 T^{5} + \cdots + 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 10191432 T^{10} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{6} - 66189634 T^{4} + \cdots - 75\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 11711 T^{5} + \cdots + 73\!\cdots\!01)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 9091 T^{5} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 727900480 T^{4} + \cdots - 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 21843 T^{2} + \cdots - 1643621929904)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + 613454880 T^{10} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + 666461298 T^{10} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{12} + 4695322386 T^{10} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{6} + 8078 T^{5} + \cdots + 28\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 72325 T^{5} + \cdots + 50\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 3481391808 T^{4} + \cdots - 30\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 50029 T^{5} + \cdots + 87\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 50997 T^{5} + \cdots + 12\!\cdots\!89)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 7727019426 T^{4} + \cdots - 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 2820380296 T^{10} + \cdots + 36\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{3} + 216524 T^{2} + \cdots - 546802680855102)^{4} \) Copy content Toggle raw display
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