Properties

Label 63.10.e.b
Level $63$
Weight $10$
Character orbit 63.e
Analytic conductor $32.447$
Analytic rank $0$
Dimension $10$
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] = [63,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.37");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(32.4472576783\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + 1399072384 x^{2} + 1034227200 x + 761760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 7)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 4 \beta_{3} - \beta_1 + 4) q^{2} + (\beta_{8} - \beta_{7} - 191 \beta_{3} + 7 \beta_{2} - 7 \beta_1) q^{4} + ( - \beta_{9} + \beta_{6} + 307 \beta_{3} - 307) q^{5} + (\beta_{9} + 10 \beta_{8} + 10 \beta_{7} + 2 \beta_{6} + \beta_{5} - 8 \beta_{4} + \cdots + 633) q^{7}+ \cdots + (66 \beta_{5} + 10 \beta_{4} + 26 \beta_{2} - 3462) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \beta_{3} - \beta_1 + 4) q^{2} + (\beta_{8} - \beta_{7} - 191 \beta_{3} + 7 \beta_{2} - 7 \beta_1) q^{4} + ( - \beta_{9} + \beta_{6} + 307 \beta_{3} - 307) q^{5} + (\beta_{9} + 10 \beta_{8} + 10 \beta_{7} + 2 \beta_{6} + \beta_{5} - 8 \beta_{4} + \cdots + 633) q^{7}+ \cdots + ( - 330232 \beta_{9} + 1446564 \beta_{8} + \cdots + 279003144) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 18 q^{2} - 940 q^{4} - 1533 q^{5} - 1036 q^{7} - 34272 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 18 q^{2} - 940 q^{4} - 1533 q^{5} - 1036 q^{7} - 34272 q^{8} + 4298 q^{10} - 42213 q^{11} - 319676 q^{13} + 39522 q^{14} + 322064 q^{16} - 324681 q^{17} - 16121 q^{19} + 350616 q^{20} - 62692 q^{22} - 2638863 q^{23} - 1304092 q^{25} - 4179252 q^{26} - 22156316 q^{28} - 15292500 q^{29} + 19179237 q^{31} + 6263520 q^{32} - 62909700 q^{34} + 43746759 q^{35} + 39566985 q^{37} - 67365270 q^{38} + 5721744 q^{40} + 53436852 q^{41} + 101835992 q^{43} - 99704916 q^{44} - 14489202 q^{46} - 32509659 q^{47} - 49024598 q^{49} - 3328464 q^{50} + 103893272 q^{52} + 25714707 q^{53} - 144695222 q^{55} - 115352832 q^{56} - 46645516 q^{58} - 46776513 q^{59} - 113075039 q^{61} - 467465628 q^{62} - 192008960 q^{64} + 338113566 q^{65} - 126707879 q^{67} - 32262636 q^{68} + 697712470 q^{70} + 1188736032 q^{71} - 859257651 q^{73} - 591757530 q^{74} + 1101475592 q^{76} - 1911891891 q^{77} - 527065417 q^{79} + 1257352656 q^{80} - 1341703076 q^{82} + 144863208 q^{83} - 1197360222 q^{85} + 678648216 q^{86} + 903700608 q^{88} - 1661554797 q^{89} + 726641384 q^{91} + 1301840952 q^{92} - 272580882 q^{94} + 1197123495 q^{95} + 869770188 q^{97} + 2404833858 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + 1399072384 x^{2} + 1034227200 x + 761760000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 174976544647 \nu^{9} + 543280294119 \nu^{8} - 2080323777608 \nu^{7} + 306747495137243 \nu^{6} - 527825025841115 \nu^{5} + \cdots - 72\!\cdots\!00 ) / 49\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 439915122044593 \nu^{9} - 359425911506973 \nu^{8} + \cdots + 45\!\cdots\!00 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9108600924689 \nu^{9} + 744234087958953 \nu^{8} + \cdots + 14\!\cdots\!20 ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 44987322634457 \nu^{9} - 60129676436289 \nu^{8} + 230247989821048 \nu^{7} + \cdots + 26\!\cdots\!80 ) / 69\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10228081391315 \nu^{9} + \cdots + 13\!\cdots\!84 ) / 69\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24\!\cdots\!57 \nu^{9} + \cdots - 60\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!03 \nu^{9} + \cdots + 20\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!87 \nu^{9} + \cdots + 10\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} - 687\beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -27\beta_{5} + \beta_{4} - 507\beta_{2} - 375 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14 \beta_{9} - 169 \beta_{8} + 155 \beta_{7} - 14 \beta_{6} + 155 \beta_{5} + 169 \beta_{4} + 87639 \beta_{3} - 134 \beta _1 - 87639 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 28\beta_{9} - 583\beta_{8} - 11457\beta_{7} + 99345\beta_{3} + 142643\beta_{2} - 142643\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12040\beta_{6} - 89029\beta_{5} - 107929\beta_{4} + 71655\beta_{2} + 49481343 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 9450 \beta_{9} + 115386 \beta_{8} + 1949460 \beta_{7} + 9450 \beta_{6} + 1949460 \beta_{5} - 115386 \beta_{4} - 13417998 \beta_{3} + 21095143 \beta _1 + 13417998 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4129692 \beta_{9} + 33731905 \beta_{8} - 25531489 \beta_{7} - 14657685447 \beta_{3} - 22648273 \beta_{2} + 22648273 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -8200416\beta_{6} - 1242725823\beta_{5} + 81699181\beta_{4} - 12756581151\beta_{2} - 8505431979 ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(1\)

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
8.80924 15.2580i
5.89912 10.2176i
−0.371984 + 0.644295i
−5.11725 + 8.86334i
−8.71912 + 15.1020i
8.80924 + 15.2580i
5.89912 + 10.2176i
−0.371984 0.644295i
−5.11725 8.86334i
−8.71912 15.1020i
−15.6185 + 27.0520i 0 −231.874 401.617i 239.755 415.269i 0 1428.40 6189.77i −1507.26 0 7489.23 + 12971.7i
37.2 −9.79824 + 16.9710i 0 63.9892 + 110.832i −983.791 + 1703.98i 0 −5768.52 + 2660.41i −12541.3 0 −19278.8 33391.9i
37.3 2.74397 4.75269i 0 240.941 + 417.323i 828.924 1435.74i 0 2822.68 + 5690.88i 5454.36 0 −4549.08 7879.24i
37.4 12.2345 21.1908i 0 −43.3662 75.1124i −1014.15 + 1756.56i 0 −4235.51 4734.35i 10405.9 0 24815.2 + 42981.2i
37.5 19.4382 33.6680i 0 −499.691 865.489i 162.760 281.909i 0 5234.95 3598.46i −18947.7 0 −6327.54 10959.6i
46.1 −15.6185 27.0520i 0 −231.874 + 401.617i 239.755 + 415.269i 0 1428.40 + 6189.77i −1507.26 0 7489.23 12971.7i
46.2 −9.79824 16.9710i 0 63.9892 110.832i −983.791 1703.98i 0 −5768.52 2660.41i −12541.3 0 −19278.8 + 33391.9i
46.3 2.74397 + 4.75269i 0 240.941 417.323i 828.924 + 1435.74i 0 2822.68 5690.88i 5454.36 0 −4549.08 + 7879.24i
46.4 12.2345 + 21.1908i 0 −43.3662 + 75.1124i −1014.15 1756.56i 0 −4235.51 + 4734.35i 10405.9 0 24815.2 42981.2i
46.5 19.4382 + 33.6680i 0 −499.691 + 865.489i 162.760 + 281.909i 0 5234.95 + 3598.46i −18947.7 0 −6327.54 + 10959.6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
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 63.10.e.b 10
3.b odd 2 1 7.10.c.a 10
7.c even 3 1 inner 63.10.e.b 10
12.b even 2 1 112.10.i.c 10
21.c even 2 1 49.10.c.g 10
21.g even 6 1 49.10.a.f 5
21.g even 6 1 49.10.c.g 10
21.h odd 6 1 7.10.c.a 10
21.h odd 6 1 49.10.a.e 5
84.n even 6 1 112.10.i.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.c.a 10 3.b odd 2 1
7.10.c.a 10 21.h odd 6 1
49.10.a.e 5 21.h odd 6 1
49.10.a.f 5 21.g even 6 1
49.10.c.g 10 21.c even 2 1
49.10.c.g 10 21.g even 6 1
63.10.e.b 10 1.a even 1 1 trivial
63.10.e.b 10 7.c even 3 1 inner
112.10.i.c 10 12.b even 2 1
112.10.i.c 10 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 18 T_{2}^{9} + 1912 T_{2}^{8} - 8760 T_{2}^{7} + 2328112 T_{2}^{6} - 13776576 T_{2}^{5} + 1132329984 T_{2}^{4} - 258296832 T_{2}^{3} + 340280893440 T_{2}^{2} + \cdots + 10212166139904 \) acting on \(S_{10}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 18 T^{9} + \cdots + 10212166139904 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 1533 T^{9} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + 1036 T^{9} + \cdots + 10\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{10} + 42213 T^{9} + \cdots + 91\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{5} + 159838 T^{4} + \cdots + 79\!\cdots\!92)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 324681 T^{9} + \cdots + 29\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{10} + 16121 T^{9} + \cdots + 39\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{10} + 2638863 T^{9} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{5} + 7646250 T^{4} + \cdots - 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 19179237 T^{9} + \cdots + 14\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{10} - 39566985 T^{9} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{5} - 26718426 T^{4} + \cdots + 20\!\cdots\!40)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 50917996 T^{4} + \cdots + 72\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 32509659 T^{9} + \cdots + 34\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{10} - 25714707 T^{9} + \cdots + 47\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{10} + 46776513 T^{9} + \cdots + 79\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{10} + 113075039 T^{9} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{10} + 126707879 T^{9} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{5} - 594368016 T^{4} + \cdots - 25\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 859257651 T^{9} + \cdots + 87\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( T^{10} + 527065417 T^{9} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( (T^{5} - 72431604 T^{4} + \cdots - 39\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 1661554797 T^{9} + \cdots + 21\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( (T^{5} - 434885094 T^{4} + \cdots - 84\!\cdots\!56)^{2} \) Copy content Toggle raw display
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