Properties

Label 63.4.e.c
Level $63$
Weight $4$
Character orbit 63.e
Analytic conductor $3.717$
Analytic rank $0$
Dimension $6$
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] = [63,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.9924270768.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 25x^{4} + 12x^{3} + 582x^{2} - 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} + 8 \beta_{4} + \beta_{2} + \beta_1 - 8) q^{4} + (\beta_{5} + 4 \beta_{4} - \beta_{3} - \beta_1) q^{5} + (3 \beta_{4} - \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 4) q^{7} + (\beta_{3} + 9 \beta_{2} - 10) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} + 8 \beta_{4} + \beta_{2} + \beta_1 - 8) q^{4} + (\beta_{5} + 4 \beta_{4} - \beta_{3} - \beta_1) q^{5} + (3 \beta_{4} - \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 4) q^{7} + (\beta_{3} + 9 \beta_{2} - 10) q^{8} + ( - \beta_{5} - 22 \beta_{4} + 11 \beta_{2} + 11 \beta_1 + 22) q^{10} + (3 \beta_{5} - 12 \beta_{4} + \beta_{2} + \beta_1 + 12) q^{11} + (\beta_{3} - 5 \beta_{2} + 19) q^{13} + ( - \beta_{5} - 22 \beta_{4} - 3 \beta_{3} - \beta_{2} + 6 \beta_1 + 64) q^{14} + ( - \beta_{5} - 74 \beta_{4} + \beta_{3} - 19 \beta_1) q^{16} + ( - 4 \beta_{5} - 16 \beta_{4} + 16) q^{17} + (\beta_{5} + 65 \beta_{4} - \beta_{3} + 7 \beta_1) q^{19} + (3 \beta_{3} - 11 \beta_{2} - 150) q^{20} + (\beta_{3} + 13 \beta_{2} + 2) q^{22} + ( - 4 \beta_{5} + 80 \beta_{4} + 4 \beta_{3} - 24 \beta_1) q^{23} + (\beta_{5} + 53 \beta_{4} - 29 \beta_{2} - 29 \beta_1 - 53) q^{25} + (5 \beta_{5} + 86 \beta_{4} - 5 \beta_{3} + 16 \beta_1) q^{26} + ( - \beta_{5} + 126 \beta_{4} - \beta_{3} - 16 \beta_{2} + 49 \beta_1 - 110) q^{28} + (5 \beta_{3} - 25 \beta_{2} - 26) q^{29} + ( - 2 \beta_{5} - 39 \beta_{4} + 22 \beta_{2} + 22 \beta_1 + 39) q^{31} + ( - 11 \beta_{5} - 218 \beta_{4} - 29 \beta_{2} - 29 \beta_1 + 218) q^{32} + ( - 48 \beta_{2} - 24) q^{34} + ( - 4 \beta_{5} + 158 \beta_{4} + 11 \beta_{3} - 47 \beta_{2} - 24 \beta_1 - 100) q^{35} + ( - \beta_{5} - 81 \beta_{4} + \beta_{3} - 19 \beta_1) q^{37} + (7 \beta_{5} + 106 \beta_{4} + 80 \beta_{2} + 80 \beta_1 - 106) q^{38} + (3 \beta_{5} + 18 \beta_{4} - 3 \beta_{3} - 75 \beta_1) q^{40} + ( - 14 \beta_{3} - 2 \beta_{2} - 82) q^{41} + (3 \beta_{3} + 69 \beta_{2} + 143) q^{43} + (11 \beta_{5} - 298 \beta_{4} - 11 \beta_{3} - 11 \beta_1) q^{44} + ( - 24 \beta_{5} - 360 \beta_{4} + 24 \beta_{2} + 24 \beta_1 + 360) q^{46} + ( - 28 \beta_{5} - 46 \beta_{4} + 28 \beta_{3} - 72 \beta_1) q^{47} + ( - 2 \beta_{5} - 95 \beta_{4} + 25 \beta_{3} - 35 \beta_{2} - 46 \beta_1 - 7) q^{49} + ( - 29 \beta_{3} + 32 \beta_{2} + 470) q^{50} + (24 \beta_{5} + 74 \beta_{4} + 102 \beta_{2} + 102 \beta_1 - 74) q^{52} + (11 \beta_{5} - 154 \beta_{4} + 69 \beta_{2} + 69 \beta_1 + 154) q^{53} + ( - 25 \beta_{3} - 19 \beta_{2} - 350) q^{55} + (33 \beta_{5} + 522 \beta_{4} - 8 \beta_{3} + 111 \beta_{2} + 81 \beta_1 - 454) q^{56} + (25 \beta_{5} + 430 \beta_{4} - 25 \beta_{3} - 41 \beta_1) q^{58} + (29 \beta_{5} - 358 \beta_{4} - 69 \beta_{2} - 69 \beta_1 + 358) q^{59} + ( - 20 \beta_{5} - 10 \beta_{4} + 20 \beta_{3} + 100 \beta_1) q^{61} + (22 \beta_{3} - 33 \beta_{2} - 364) q^{62} + ( - 21 \beta_{3} - 183 \beta_{2} - 194) q^{64} + (18 \beta_{5} - 174 \beta_{4} - 18 \beta_{3} + 50 \beta_1) q^{65} + (47 \beta_{5} - 215 \beta_{4} + 17 \beta_{2} + 17 \beta_1 + 215) q^{67} + (16 \beta_{5} + 640 \beta_{4} - 16 \beta_{3} + 24 \beta_1) q^{68} + (23 \beta_{5} + 434 \beta_{4} - 47 \beta_{3} + 102 \beta_{2} - 7 \beta_1 + 360) q^{70} + (12 \beta_{3} + 120 \beta_{2} - 66) q^{71} + ( - 23 \beta_{5} + 363 \beta_{4} - 101 \beta_{2} - 101 \beta_1 - 363) q^{73} + ( - 19 \beta_{5} - 298 \beta_{4} - 108 \beta_{2} - 108 \beta_1 + 298) q^{74} + (72 \beta_{3} + 186 \beta_{2} - 718) q^{76} + ( - 45 \beta_{5} + 438 \beta_{4} + 5 \beta_{3} - 24 \beta_{2} + 25 \beta_1 - 410) q^{77} + ( - 48 \beta_{5} - 299 \beta_{4} + 48 \beta_{3} + 36 \beta_1) q^{79} + ( - 51 \beta_{5} - 18 \beta_{4} - 121 \beta_{2} - 121 \beta_1 + 18) q^{80} + (2 \beta_{5} - 52 \beta_{4} - 2 \beta_{3} + 32 \beta_1) q^{82} + ( - 27 \beta_{3} + 51 \beta_{2} - 156) q^{83} + (72 \beta_{2} + 624) q^{85} + ( - 69 \beta_{5} - 1086 \beta_{4} + 69 \beta_{3} + 50 \beta_1) q^{86} + ( - 3 \beta_{5} - 258 \beta_{4} - 117 \beta_{2} - 117 \beta_1 + 258) q^{88} + (22 \beta_{5} + 532 \beta_{4} - 22 \beta_{3} + 170 \beta_1) q^{89} + ( - 23 \beta_{5} - 287 \beta_{4} - 23 \beta_{3} - 74 \beta_{2} - 49 \beta_1 + 18) q^{91} + (56 \beta_{3} - 336 \beta_{2} + 112) q^{92} + ( - 72 \beta_{5} - 984 \beta_{4} - 342 \beta_{2} - 342 \beta_1 + 984) q^{94} + (54 \beta_{5} + 246 \beta_{4} - 2 \beta_{2} - 2 \beta_1 - 246) q^{95} + ( - 49 \beta_{3} + 53 \beta_{2} + 24) q^{97} + ( - 11 \beta_{5} - 26 \beta_{4} - 35 \beta_{3} - 157 \beta_{2} + \cdots + 724) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 25 q^{4} + 11 q^{5} - 13 q^{7} - 78 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - 25 q^{4} + 11 q^{5} - 13 q^{7} - 78 q^{8} + 55 q^{10} + 35 q^{11} + 124 q^{13} + 326 q^{14} - 241 q^{16} + 48 q^{17} + 202 q^{19} - 878 q^{20} - 14 q^{22} + 216 q^{23} - 130 q^{25} + 274 q^{26} - 201 q^{28} - 106 q^{29} + 95 q^{31} + 683 q^{32} - 48 q^{34} - 56 q^{35} - 262 q^{37} - 398 q^{38} - 21 q^{40} - 488 q^{41} + 720 q^{43} - 905 q^{44} + 1056 q^{46} - 210 q^{47} - 303 q^{49} + 2756 q^{50} - 324 q^{52} + 393 q^{53} - 2062 q^{55} - 1299 q^{56} + 1249 q^{58} + 1143 q^{59} + 70 q^{61} - 2118 q^{62} - 798 q^{64} - 472 q^{65} + 628 q^{67} + 1944 q^{68} + 3251 q^{70} - 636 q^{71} - 988 q^{73} + 1002 q^{74} - 4680 q^{76} - 1073 q^{77} - 861 q^{79} + 175 q^{80} - 124 q^{82} - 1038 q^{83} + 3600 q^{85} - 3208 q^{86} + 891 q^{88} + 1766 q^{89} - 654 q^{91} + 1344 q^{92} + 3294 q^{94} - 736 q^{95} + 38 q^{97} + 4267 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 25x^{4} + 12x^{3} + 582x^{2} - 144x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 25\nu^{4} + 625\nu^{3} - 582\nu^{2} + 144\nu - 3600 ) / 14406 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -25\nu^{5} + 625\nu^{4} - 1219\nu^{3} + 14550\nu^{2} - 3600\nu + 234060 ) / 14406 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 100\nu^{5} - 99\nu^{4} + 2475\nu^{3} + 1825\nu^{2} + 57618\nu + 150 ) / 14406 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -1601\nu^{5} + 1609\nu^{4} - 40225\nu^{3} - 14212\nu^{2} - 936438\nu + 231696 ) / 14406 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 16\beta_{4} + \beta_{2} + \beta _1 - 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 25\beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -25\beta_{5} - 394\beta_{4} + 25\beta_{3} - 43\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -43\beta_{5} - 538\beta_{4} - 637\beta_{2} - 637\beta _1 + 538 \) Copy content Toggle raw display

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-\beta_{4}\) \(1\)

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
−2.27818 + 3.94593i
0.124036 0.214837i
2.65415 4.59712i
−2.27818 3.94593i
0.124036 + 0.214837i
2.65415 + 4.59712i
−2.27818 + 3.94593i 0 −6.38024 11.0509i 8.93660 15.4786i 0 2.26047 18.3818i 21.6905 0 40.7184 + 70.5264i
37.2 0.124036 0.214837i 0 3.96923 + 6.87491i −6.21730 + 10.7687i 0 −18.4385 1.73873i 3.95388 0 1.54234 + 2.67141i
37.3 2.65415 4.59712i 0 −10.0890 17.4746i 2.78070 4.81631i 0 9.67799 + 15.7904i −64.6443 0 −14.7608 25.5664i
46.1 −2.27818 3.94593i 0 −6.38024 + 11.0509i 8.93660 + 15.4786i 0 2.26047 + 18.3818i 21.6905 0 40.7184 70.5264i
46.2 0.124036 + 0.214837i 0 3.96923 6.87491i −6.21730 10.7687i 0 −18.4385 + 1.73873i 3.95388 0 1.54234 2.67141i
46.3 2.65415 + 4.59712i 0 −10.0890 + 17.4746i 2.78070 + 4.81631i 0 9.67799 15.7904i −64.6443 0 −14.7608 + 25.5664i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.3
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 63.4.e.c 6
3.b odd 2 1 21.4.e.b 6
7.b odd 2 1 441.4.e.w 6
7.c even 3 1 inner 63.4.e.c 6
7.c even 3 1 441.4.a.s 3
7.d odd 6 1 441.4.a.t 3
7.d odd 6 1 441.4.e.w 6
12.b even 2 1 336.4.q.k 6
21.c even 2 1 147.4.e.n 6
21.g even 6 1 147.4.a.m 3
21.g even 6 1 147.4.e.n 6
21.h odd 6 1 21.4.e.b 6
21.h odd 6 1 147.4.a.l 3
84.j odd 6 1 2352.4.a.cg 3
84.n even 6 1 336.4.q.k 6
84.n even 6 1 2352.4.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.b 6 3.b odd 2 1
21.4.e.b 6 21.h odd 6 1
63.4.e.c 6 1.a even 1 1 trivial
63.4.e.c 6 7.c even 3 1 inner
147.4.a.l 3 21.h odd 6 1
147.4.a.m 3 21.g even 6 1
147.4.e.n 6 21.c even 2 1
147.4.e.n 6 21.g even 6 1
336.4.q.k 6 12.b even 2 1
336.4.q.k 6 84.n even 6 1
441.4.a.s 3 7.c even 3 1
441.4.a.t 3 7.d odd 6 1
441.4.e.w 6 7.b odd 2 1
441.4.e.w 6 7.d odd 6 1
2352.4.a.cg 3 84.j odd 6 1
2352.4.a.ci 3 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 25T_{2}^{4} + 12T_{2}^{3} + 582T_{2}^{2} - 144T_{2} + 36 \) acting on \(S_{4}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + 25 T^{4} + 12 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 11 T^{5} + 313 T^{4} + \cdots + 1527696 \) Copy content Toggle raw display
$7$ \( T^{6} + 13 T^{5} + 236 T^{4} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{6} - 35 T^{5} + 2593 T^{4} + \cdots + 91470096 \) Copy content Toggle raw display
$13$ \( (T^{3} - 62 T^{2} + 425 T + 18452)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 48 T^{5} + \cdots + 12745506816 \) Copy content Toggle raw display
$19$ \( T^{6} - 202 T^{5} + \cdots + 54664310416 \) Copy content Toggle raw display
$23$ \( T^{6} - 216 T^{5} + \cdots + 2498119335936 \) Copy content Toggle raw display
$29$ \( (T^{3} + 53 T^{2} - 20472 T + 824976)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 95 T^{5} + \cdots + 139783329 \) Copy content Toggle raw display
$37$ \( T^{6} + 262 T^{5} + \cdots + 2415919104 \) Copy content Toggle raw display
$41$ \( (T^{3} + 244 T^{2} - 18780 T + 300384)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 360 T^{2} - 72363 T + 18269746)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 210 T^{5} + \cdots + 26205471480384 \) Copy content Toggle raw display
$53$ \( T^{6} - 393 T^{5} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{6} - 1143 T^{5} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{6} - 70 T^{5} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 783608160972004 \) Copy content Toggle raw display
$71$ \( (T^{3} + 318 T^{2} - 330804 T + 28535976)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 988 T^{5} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{6} + 861 T^{5} + \cdots + 37\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{3} + 519 T^{2} - 131616 T - 47916036)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 169118164647936 \) Copy content Toggle raw display
$97$ \( (T^{3} - 19 T^{2} - 569600 T + 44776452)^{2} \) Copy content Toggle raw display
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