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Properties

Label 63.9.m.c
Level $63$
Weight $9$
Character orbit 63.m
Analytic conductor $25.665$
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] = [63,9,Mod(10,63)]
 
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, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.10");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 63.m (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: \(25.6648524339\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 944 x^{8} - 4352 x^{7} + 713897 x^{6} - 2977250 x^{5} + 173584456 x^{4} + \cdots + 565428802500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 5^{2}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 21)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{2} + ( - \beta_{7} + \beta_{4} + \cdots + 4 \beta_1) q^{4}+ \cdots + (2 \beta_{9} + 2 \beta_{8} + \cdots + 1587) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{2} + ( - \beta_{7} + \beta_{4} + \cdots + 4 \beta_1) q^{4}+ \cdots + ( - 24024 \beta_{9} + 15575 \beta_{8} + \cdots + 11747561) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} - 605 q^{4} - 285 q^{5} + 4305 q^{7} + 15630 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} - 605 q^{4} - 285 q^{5} + 4305 q^{7} + 15630 q^{8} - 9531 q^{10} + 33039 q^{11} - 51786 q^{14} + 40311 q^{16} - 156150 q^{17} - 167028 q^{19} - 1685518 q^{22} - 511548 q^{23} + 599356 q^{25} - 1411824 q^{26} + 1401113 q^{28} + 4932834 q^{29} + 2934003 q^{31} - 171663 q^{32} + 1940484 q^{35} - 2745444 q^{37} - 1144098 q^{38} + 3663387 q^{40} + 1660776 q^{43} - 16647 q^{44} + 4626716 q^{46} + 14413992 q^{47} - 17807237 q^{49} + 15648492 q^{50} + 45867216 q^{52} - 30933039 q^{53} - 20526345 q^{56} - 30907837 q^{58} + 1785417 q^{59} - 25818192 q^{61} - 42983070 q^{64} + 23541048 q^{65} + 27726746 q^{67} + 76816314 q^{68} - 24859611 q^{70} + 98344824 q^{71} + 81756642 q^{73} - 126174342 q^{74} - 137009187 q^{77} + 38780939 q^{79} - 305800053 q^{80} - 123611550 q^{82} - 432983556 q^{85} + 49336398 q^{86} + 245381263 q^{88} + 94239126 q^{89} - 19088034 q^{91} + 683309928 q^{92} + 904609428 q^{94} - 451326426 q^{95} + 395409273 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 944 x^{8} - 4352 x^{7} + 713897 x^{6} - 2977250 x^{5} + 173584456 x^{4} + \cdots + 565428802500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 60\!\cdots\!79 \nu^{9} + \cdots + 40\!\cdots\!50 ) / 93\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 36\!\cdots\!09 \nu^{9} + \cdots - 24\!\cdots\!00 ) / 46\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!18 \nu^{9} + \cdots + 34\!\cdots\!50 ) / 93\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 89\!\cdots\!27 \nu^{9} + \cdots - 64\!\cdots\!50 ) / 93\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!27 \nu^{9} + \cdots + 88\!\cdots\!00 ) / 93\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 76\!\cdots\!19 \nu^{9} + \cdots + 91\!\cdots\!50 ) / 26\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31\!\cdots\!38 \nu^{9} + \cdots - 88\!\cdots\!50 ) / 95\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!87 \nu^{9} + \cdots - 20\!\cdots\!50 ) / 46\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 378\beta_{3} + 2\beta_{2} - 2\beta _1 - 378 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + 2\beta_{8} + 2\beta_{6} - 2\beta_{5} - 2\beta_{4} - 559\beta_{2} + 964 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 28 \beta_{9} + 56 \beta_{8} - 751 \beta_{7} + 42 \beta_{6} + 84 \beta_{5} + 765 \beta_{4} + 211836 \beta_{3} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3872 \beta_{9} + 1936 \beta_{8} - 4026 \beta_{7} - 3592 \beta_{6} - 1796 \beta_{5} + 3732 \beta_{4} + \cdots - 906562 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 23960 \beta_{9} - 23960 \beta_{8} + 42120 \beta_{6} - 42120 \beta_{5} - 555681 \beta_{4} + \cdots + 133756598 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1428642 \beta_{9} - 2857284 \beta_{8} + 2179386 \beta_{7} + 1383002 \beta_{6} + 2766004 \beta_{5} + \cdots - 1428642 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 31849656 \beta_{9} - 15924828 \beta_{8} + 352397611 \beta_{7} - 67059924 \beta_{6} - 33529962 \beta_{5} + \cdots - 89008526688 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 980558216 \beta_{9} + 980558216 \beta_{8} + 1016646716 \beta_{6} - 1016646716 \beta_{5} + \cdots + 664706175178 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−13.4055 + 23.2190i
−8.17212 + 14.1545i
2.18896 − 3.79138i
7.75834 − 13.4378i
12.6303 − 21.8764i
−13.4055 − 23.2190i
−8.17212 − 14.1545i
2.18896 + 3.79138i
7.75834 + 13.4378i
12.6303 + 21.8764i
−13.9055 − 24.0850i 0 −258.726 + 448.127i 307.943 − 177.791i 0 −1475.14 − 1894.40i 7271.26 0 −8564.20 − 4944.54i
10.2 −8.67212 − 15.0206i 0 −22.4115 + 38.8178i −582.722 + 336.435i 0 1370.11 + 1971.70i −3662.71 0 10106.9 + 5835.21i
10.3 1.68896 + 2.92536i 0 122.295 − 211.821i 854.850 − 493.548i 0 2400.94 + 16.9613i 1690.95 0 2887.61 + 1667.16i
10.4 7.25834 + 12.5718i 0 22.6330 − 39.2015i −855.322 + 493.820i 0 185.351 − 2393.84i 4373.38 0 −12416.4 − 7168.63i
10.5 12.1303 + 21.0104i 0 −166.290 + 288.023i 132.752 − 76.6442i 0 −328.761 + 2378.39i −1857.89 0 3220.64 + 1859.44i
19.1 −13.9055 + 24.0850i 0 −258.726 − 448.127i 307.943 + 177.791i 0 −1475.14 + 1894.40i 7271.26 0 −8564.20 + 4944.54i
19.2 −8.67212 + 15.0206i 0 −22.4115 − 38.8178i −582.722 − 336.435i 0 1370.11 − 1971.70i −3662.71 0 10106.9 − 5835.21i
19.3 1.68896 − 2.92536i 0 122.295 + 211.821i 854.850 + 493.548i 0 2400.94 − 16.9613i 1690.95 0 2887.61 − 1667.16i
19.4 7.25834 − 12.5718i 0 22.6330 + 39.2015i −855.322 − 493.820i 0 185.351 + 2393.84i 4373.38 0 −12416.4 + 7168.63i
19.5 12.1303 − 21.0104i 0 −166.290 − 288.023i 132.752 + 76.6442i 0 −328.761 − 2378.39i −1857.89 0 3220.64 − 1859.44i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.5
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 63.9.m.c 10
3.b odd 2 1 21.9.f.a ✓ 10
7.d odd 6 1 inner 63.9.m.c 10
21.g even 6 1 21.9.f.a ✓ 10
21.g even 6 1 147.9.d.a 10
21.h odd 6 1 147.9.d.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.9.f.a ✓ 10 3.b odd 2 1
21.9.f.a ✓ 10 21.g even 6 1
63.9.m.c 10 1.a even 1 1 trivial
63.9.m.c 10 7.d odd 6 1 inner
147.9.d.a 10 21.g even 6 1
147.9.d.a 10 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 3 T_{2}^{9} + 947 T_{2}^{8} - 3402 T_{2}^{7} + 699618 T_{2}^{6} - 1925676 T_{2}^{5} + \cdots + 329292345600 \) acting on \(S_{9}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 329292345600 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 27\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 12\!\cdots\!03 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 43\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 42\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 84\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 47\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 20\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
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