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Properties

Label 63.7.d.f
Level $63$
Weight $7$
Character orbit 63.d
Analytic conductor $14.493$
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] = [63,7,Mod(55,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(2))
 
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("63.55");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 63.d (of order \(2\), degree \(1\), 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: \(14.4934072681\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 212x^{6} - 1123x^{5} + 44168x^{4} - 138697x^{3} + 660109x^{2} + 680340x + 1040400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{8}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + ( - \beta_{2} + \beta_1 + 43) q^{4} - \beta_{3} q^{5} + (\beta_{6} - \beta_{3} - 14 \beta_1) q^{7} + (\beta_{7} + \beta_{6} + \beta_{2} + \cdots + 199) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + ( - \beta_{2} + \beta_1 + 43) q^{4} - \beta_{3} q^{5} + (\beta_{6} - \beta_{3} - 14 \beta_1) q^{7} + (\beta_{7} + \beta_{6} + \beta_{2} + \cdots + 199) q^{8}+ \cdots + ( - 98 \beta_{7} - 2282 \beta_{6} + \cdots + 46991) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{2} + 346 q^{4} + 28 q^{7} + 1462 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{2} + 346 q^{4} + 28 q^{7} + 1462 q^{8} + 6848 q^{11} - 12082 q^{14} + 28466 q^{16} - 7764 q^{22} + 24320 q^{23} - 77056 q^{25} - 30142 q^{28} - 60496 q^{29} - 5082 q^{32} - 103656 q^{35} - 39112 q^{37} - 29272 q^{43} + 577884 q^{44} + 564972 q^{46} - 94864 q^{49} - 240154 q^{50} - 232288 q^{53} - 225722 q^{56} + 987684 q^{58} - 690734 q^{64} + 836304 q^{65} - 2163848 q^{67} - 366744 q^{70} + 506288 q^{71} - 512324 q^{74} - 1536304 q^{77} - 93272 q^{79} + 3740760 q^{85} - 3846452 q^{86} - 2077548 q^{88} + 1890336 q^{91} + 9701580 q^{92} - 4154832 q^{95} + 507542 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 212x^{6} - 1123x^{5} + 44168x^{4} - 138697x^{3} + 660109x^{2} + 680340x + 1040400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9082495 \nu^{7} + 20968969 \nu^{6} + 1852872025 \nu^{5} - 4069000805 \nu^{4} + \cdots + 6424259322060 ) / 6112279547031 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 57298949 \nu^{7} - 9710008 \nu^{6} + 11689257155 \nu^{5} - 25670200711 \nu^{4} + \cdots + 657724982153466 ) / 6112279547031 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 179401323979 \nu^{7} - 55492139374 \nu^{6} + 40306643472493 \nu^{5} - 173727333213157 \nu^{4} + \cdots + 74\!\cdots\!80 ) / 15\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 37789760718 \nu^{7} + 93374630118 \nu^{6} - 7883099181936 \nu^{5} + \cdots - 65\!\cdots\!60 ) / 173181253832545 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 83487828014 \nu^{7} + 255156934781 \nu^{6} - 17147785189814 \nu^{5} + \cdots - 10\!\cdots\!60 ) / 311726256898581 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 83872230808 \nu^{7} - 286834199911 \nu^{6} + 18384931162084 \nu^{5} + \cdots - 22\!\cdots\!79 ) / 311726256898581 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 169279565275 \nu^{7} + 75168547552 \nu^{6} - 35808432167449 \nu^{5} + \cdots - 96\!\cdots\!99 ) / 311726256898581 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{7} + 3\beta_{6} + 3\beta_{5} + 2\beta_{4} - 9\beta_{3} - 108\beta_1 ) / 216 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -9\beta_{7} + 9\beta_{6} - 63\beta_{5} + 118\beta_{4} + 9\beta_{3} + 108\beta_{2} - 324\beta _1 - 11448 ) / 216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - 2\beta_{2} + 197\beta _1 + 390 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 513 \beta_{7} - 297 \beta_{6} + 14679 \beta_{5} - 22582 \beta_{4} - 5841 \beta_{3} + 22572 \beta_{2} + \cdots - 2263248 ) / 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 75423 \beta_{7} - 121215 \beta_{6} - 154047 \beta_{5} + 50146 \beta_{4} + 425853 \beta_{3} + \cdots - 14259888 ) / 216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 244\beta_{7} + 244\beta_{6} - 43533\beta_{2} + 319627\beta _1 + 4441690 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 24416967 \beta_{7} + 15171951 \beta_{6} + 38378343 \beta_{5} - 21112810 \beta_{4} + \cdots - 3995757864 ) / 216 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
6.44130 − 11.1567i
6.44130 + 11.1567i
2.28617 + 3.95977i
2.28617 − 3.95977i
−0.564972 + 0.978561i
−0.564972 − 0.978561i
−7.66250 − 13.2718i
−7.66250 + 13.2718i
−13.8826 0 128.727 − 109.244i 0 86.6767 − 331.868i −898.572 0 1516.59i
55.2 −13.8826 0 128.727 109.244i 0 86.6767 + 331.868i −898.572 0 − 1516.59i
55.3 −5.57235 0 −32.9490 − 205.672i 0 342.970 − 4.53729i 540.233 0 1146.08i
55.4 −5.57235 0 −32.9490 205.672i 0 342.970 + 4.53729i 540.233 0 − 1146.08i
55.5 0.129945 0 −63.9831 − 126.447i 0 −213.284 + 268.624i −16.6307 0 − 16.4311i
55.6 0.129945 0 −63.9831 126.447i 0 −213.284 − 268.624i −16.6307 0 16.4311i
55.7 14.3250 0 141.206 − 175.511i 0 −202.362 − 276.945i 1105.97 0 − 2514.19i
55.8 14.3250 0 141.206 175.511i 0 −202.362 + 276.945i 1105.97 0 2514.19i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.7.d.f 8
3.b odd 2 1 21.7.d.a ✓ 8
7.b odd 2 1 inner 63.7.d.f 8
12.b even 2 1 336.7.f.a 8
21.c even 2 1 21.7.d.a ✓ 8
21.g even 6 1 147.7.f.b 8
21.g even 6 1 147.7.f.c 8
21.h odd 6 1 147.7.f.b 8
21.h odd 6 1 147.7.f.c 8
84.h odd 2 1 336.7.f.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.d.a ✓ 8 3.b odd 2 1
21.7.d.a ✓ 8 21.c even 2 1
63.7.d.f 8 1.a even 1 1 trivial
63.7.d.f 8 7.b odd 2 1 inner
147.7.f.b 8 21.g even 6 1
147.7.f.b 8 21.h odd 6 1
147.7.f.c 8 21.g even 6 1
147.7.f.c 8 21.h odd 6 1
336.7.f.a 8 12.b even 2 1
336.7.f.a 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 5T_{2}^{3} - 202T_{2}^{2} - 1082T_{2} + 144 \) acting on \(S_{7}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 5 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 3045338564760)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 32\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 94\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 14\!\cdots\!32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 55\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 14\!\cdots\!72)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 66\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 76\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 64\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 36\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 16\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 39\!\cdots\!44 \) Copy content Toggle raw display
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