Properties

Label 63.12.a.d
Level $63$
Weight $12$
Character orbit 63.a
Self dual yes
Analytic conductor $48.406$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,12,Mod(1,63)]
 
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, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

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: yes
Analytic conductor: \(48.4056203753\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 818x - 4704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 26) q^{2} + (16 \beta_{2} + 7 \beta_1 + 1834) q^{4} + ( - 61 \beta_{2} - 59 \beta_1 - 1676) q^{5} - 16807 q^{7} + (10 \beta_{2} - 539 \beta_1 - 41518) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 26) q^{2} + (16 \beta_{2} + 7 \beta_1 + 1834) q^{4} + ( - 61 \beta_{2} - 59 \beta_1 - 1676) q^{5} - 16807 q^{7} + (10 \beta_{2} - 539 \beta_1 - 41518) q^{8} + (4134 \beta_{2} + 4026 \beta_1 + 203624) q^{10} + ( - 5742 \beta_{2} + 1694 \beta_1 + 343872) q^{11} + ( - 8673 \beta_{2} + 777 \beta_1 - 630224) q^{13} + (16807 \beta_{2} + 436982) q^{14} + (36878 \beta_{2} + 18473 \beta_1 - 3033102) q^{16} + (47582 \beta_{2} - 13966 \beta_1 - 5245262) q^{17} + (16931 \beta_{2} - 16027 \beta_1 + 4124438) q^{19} + ( - 246708 \beta_{2} - 153692 \beta_1 - 12691728) q^{20} + ( - 489380 \beta_{2} - 63140 \beta_1 + 10487968) q^{22} + (692992 \beta_{2} + 184576 \beta_1 - 2256112) q^{23} + (1050506 \beta_{2} + 298214 \beta_1 + 23225691) q^{25} + (503090 \beta_{2} + 13314 \beta_1 + 44659216) q^{26} + ( - 268912 \beta_{2} - 117649 \beta_1 - 30824038) q^{28} + ( - 172382 \beta_{2} - 659442 \beta_1 + 42446822) q^{29} + (1756262 \beta_{2} - 528982 \beta_1 + 54823252) q^{31} + (2420806 \beta_{2} - 281127 \beta_1 + 56779394) q^{32} + (6447314 \beta_{2} + 518852 \beta_1 - 24578612) q^{34} + (1025227 \beta_{2} + 991613 \beta_1 + 28168532) q^{35} + (1789782 \beta_{2} + 1498938 \beta_1 + 189137394) q^{37} + ( - 3121724 \beta_{2} + 859130 \beta_1 - 171164428) q^{38} + (9750200 \beta_{2} + 2856920 \beta_1 + 611386240) q^{40} + (12213250 \beta_{2} + 3360686 \beta_1 - 294943390) q^{41} + (4578266 \beta_{2} - 1844458 \beta_1 + 155540184) q^{43} + ( - 338872 \beta_{2} + 3807888 \beta_1 + 554004976) q^{44} + ( - 411920 \beta_{2} - 16110080 \beta_1 - 2051958688) q^{46} + ( - 5433762 \beta_{2} + 5512434 \beta_1 - 911590620) q^{47} + 282475249 q^{49} + ( - 28227759 \beta_{2} - 25544596 \beta_1 - 3792265374) q^{50} + ( - 22558340 \beta_{2} - 5925080 \beta_1 - 1475332376) q^{52} + ( - 17774864 \beta_{2} - 11963504 \beta_1 - 33079294) q^{53} + ( - 29109548 \beta_{2} - 7337972 \beta_1 - 1365170448) q^{55} + ( - 168070 \beta_{2} + 9058973 \beta_1 + 697793026) q^{56} + ( - 9879658 \beta_{2} + 41432636 \beta_1 - 947944764) q^{58} + (32129429 \beta_{2} - 6211549 \beta_1 - 1953611150) q^{59} + ( - 32560579 \beta_{2} + 53985851 \beta_1 + 673420436) q^{61} + ( - 9753568 \beta_{2} + 19974068 \beta_1 - 7374427688) q^{62} + ( - 93478874 \beta_{2} - 37629599 \beta_1 - 3194813838) q^{64} + (50101142 \beta_{2} + 58173178 \beta_1 + 1661005192) q^{65} + ( - 37614064 \beta_{2} + 55756400 \beta_1 - 9040218604) q^{67} + ( - 35376488 \beta_{2} - 48178802 \beta_1 - 8976399292) q^{68} + ( - 69480138 \beta_{2} - 67664982 \beta_1 - 3422308568) q^{70} + (88813676 \beta_{2} + 45822644 \beta_1 - 5507019632) q^{71} + ( - 277425284 \beta_{2} - 7137500 \beta_1 + 1922172050) q^{73} + ( - 249184350 \beta_{2} - 103963692 \beta_1 - 9753252660) q^{74} + (60597740 \beta_{2} + 2268434 \beta_1 + 6528869508) q^{76} + (96505794 \beta_{2} - 28471058 \beta_1 - 5779456704) q^{77} + (185300396 \beta_{2} + 27714932 \beta_1 - 18245810424) q^{79} + ( - 157186096 \beta_{2} + 72237696 \beta_1 - 19442658656) q^{80} + (242320218 \beta_{2} - 290494596 \beta_1 - 29464018388) q^{82} + ( - 232352071 \beta_{2} - 59430049 \beta_1 - 41244708638) q^{83} + (386397154 \beta_{2} + 202096846 \beta_1 + 15255729464) q^{85} + ( - 13845708 \beta_{2} + 80464076 \beta_1 - 19832329296) q^{86} + (246846368 \beta_{2} - 100598344 \beta_1 - 32504715632) q^{88} + ( - 425615272 \beta_{2} - 257804312 \beta_1 + 30493533382) q^{89} + (145767111 \beta_{2} - 13059039 \beta_1 + 10592174768) q^{91} + (1466316032 \beta_{2} + 607586672 \beta_1 + 49593790624) q^{92} + (570606432 \beta_{2} - 298222140 \beta_1 + 44440482360) q^{94} + ( - 79360080 \beta_{2} - 272680240 \beta_1 + 6535577760) q^{95} + ( - 820367702 \beta_{2} - 220553018 \beta_1 - 23945643106) q^{97} + ( - 282475249 \beta_{2} - 7344356474) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 77 q^{2} + 5493 q^{4} - 5026 q^{5} - 50421 q^{7} - 125103 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 77 q^{2} + 5493 q^{4} - 5026 q^{5} - 50421 q^{7} - 125103 q^{8} + 610764 q^{10} + 1039052 q^{11} - 1881222 q^{13} + 1294139 q^{14} - 9117711 q^{16} - 15797334 q^{17} + 12340356 q^{19} - 37982168 q^{20} + 31890144 q^{22} - 7276752 q^{23} + 68924781 q^{25} + 133487872 q^{26} - 92320851 q^{28} + 126853406 q^{29} + 162184512 q^{31} + 167636249 q^{32} - 79664298 q^{34} + 84471982 q^{35} + 567121338 q^{37} - 509512430 q^{38} + 1827265440 q^{40} - 893682734 q^{41} + 460197828 q^{43} + 1666161688 q^{44} - 6171574224 q^{46} - 2723825664 q^{47} + 847425747 q^{49} - 11374112959 q^{50} - 4409363868 q^{52} - 93426522 q^{53} - 4073739768 q^{55} + 2102606121 q^{56} - 2792521998 q^{58} - 5899174428 q^{59} + 2106807738 q^{61} - 22093555428 q^{62} - 9528592239 q^{64} + 4991087612 q^{65} - 27027285348 q^{67} - 26942000190 q^{68} - 10265110548 q^{70} - 16564049928 q^{71} + 6036803934 q^{73} - 29114537322 q^{74} + 19528279218 q^{76} - 17463346964 q^{77} - 54895016736 q^{79} - 58098552176 q^{80} - 88924869978 q^{82} - 123561203892 q^{83} + 45582888084 q^{85} - 59402678104 q^{86} - 97861591608 q^{88} + 91648411106 q^{89} + 31617698154 q^{91} + 147922642512 q^{92} + 132452618508 q^{94} + 19413413120 q^{95} - 71237114634 q^{97} - 21750594173 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 818x - 4704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{2} + 41\nu + 534 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 5\nu - 546 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 41\beta_{2} + 5\beta _1 + 3286 ) / 6 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
31.5985
−24.5296
−6.06890
−75.0789 0 3588.85 −12842.0 0 −16807.0 −115685. 0 964166.
1.2 −55.7251 0 1057.28 7066.04 0 −16807.0 55207.8 0 −393755.
1.3 53.8040 0 846.870 749.998 0 −16807.0 −64625.6 0 40352.9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.12.a.d 3
3.b odd 2 1 7.12.a.b 3
12.b even 2 1 112.12.a.h 3
15.d odd 2 1 175.12.a.b 3
15.e even 4 2 175.12.b.b 6
21.c even 2 1 49.12.a.d 3
21.g even 6 2 49.12.c.f 6
21.h odd 6 2 49.12.c.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.12.a.b 3 3.b odd 2 1
49.12.a.d 3 21.c even 2 1
49.12.c.f 6 21.g even 6 2
49.12.c.g 6 21.h odd 6 2
63.12.a.d 3 1.a even 1 1 trivial
112.12.a.h 3 12.b even 2 1
175.12.a.b 3 15.d odd 2 1
175.12.b.b 6 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 77T_{2}^{2} - 2854T_{2} - 225104 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(63))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 77 T^{2} - 2854 T - 225104 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 5026 T^{2} + \cdots + 68056486400 \) Copy content Toggle raw display
$7$ \( (T + 16807)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 1039052 T^{2} + \cdots + 33\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{3} + 1881222 T^{2} + \cdots - 91\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{3} + 15797334 T^{2} + \cdots + 62\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{3} - 12340356 T^{2} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + 7276752 T^{2} + \cdots - 42\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{3} - 126853406 T^{2} + \cdots + 25\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{3} - 162184512 T^{2} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} - 567121338 T^{2} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{3} + 893682734 T^{2} + \cdots - 46\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{3} - 460197828 T^{2} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{3} + 2723825664 T^{2} + \cdots + 21\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{3} + 93426522 T^{2} + \cdots + 36\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{3} + 5899174428 T^{2} + \cdots - 66\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{3} - 2106807738 T^{2} + \cdots + 35\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{3} + 27027285348 T^{2} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{3} + 16564049928 T^{2} + \cdots - 61\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{3} - 6036803934 T^{2} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{3} + 54895016736 T^{2} + \cdots + 29\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{3} + 123561203892 T^{2} + \cdots + 57\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{3} - 91648411106 T^{2} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + 71237114634 T^{2} + \cdots - 27\!\cdots\!84 \) Copy content Toggle raw display
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