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Properties

Label 147.2.g.b
Level $147$
Weight $2$
Character orbit 147.g
Analytic conductor $1.174$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,2,Mod(68,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.g (of order \(6\), degree \(2\), not 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: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{14} - 16 x^{13} + 50 x^{12} + 56 x^{11} - 80 x^{10} + 240 x^{9} + 381 x^{8} + 144 x^{7} + \cdots + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{13} q^{2} + ( - \beta_{9} + \beta_{7}) q^{3} + ( - \beta_{14} - \beta_{4} - \beta_{2} + 1) q^{4} - \beta_{6} q^{5} + (\beta_{15} - \beta_{12} + \cdots - \beta_{5}) q^{6}+ \cdots + ( - \beta_{13} + \beta_{10} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{2} + ( - \beta_{9} + \beta_{7}) q^{3} + ( - \beta_{14} - \beta_{4} - \beta_{2} + 1) q^{4} - \beta_{6} q^{5} + (\beta_{15} - \beta_{12} + \cdots - \beta_{5}) q^{6}+ \cdots + ( - 2 \beta_{14} + \beta_{13} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 8 q^{9} - 16 q^{15} + 24 q^{16} - 16 q^{18} - 32 q^{22} + 8 q^{25} - 24 q^{30} + 16 q^{36} - 16 q^{37} - 8 q^{39} + 32 q^{43} - 32 q^{46} + 24 q^{51} - 64 q^{57} + 24 q^{60} + 16 q^{64} + 48 q^{67} + 32 q^{72} + 80 q^{78} - 16 q^{79} + 24 q^{81} - 32 q^{85} - 32 q^{88} + 16 q^{93} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{14} - 16 x^{13} + 50 x^{12} + 56 x^{11} - 80 x^{10} + 240 x^{9} + 381 x^{8} + 144 x^{7} + \cdots + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 42446925387 \nu^{15} - 49013264501 \nu^{14} - 449607400448 \nu^{13} - 333372408543 \nu^{12} + \cdots - 4632569388482 ) / 13978548468689 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 51339936 \nu^{15} + 130479544 \nu^{14} - 435471136 \nu^{13} - 1918490970 \nu^{12} + \cdots + 16357527067 ) / 11429720743 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 73980670651 \nu^{15} + 8278142623 \nu^{14} - 798175713114 \nu^{13} - 1270426137574 \nu^{12} + \cdots - 4951355393348 ) / 13978548468689 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 74359318484 \nu^{15} + 29882924854 \nu^{14} + 682222853954 \nu^{13} + \cdots + 6555751074397 ) / 13978548468689 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 76631050152 \nu^{15} - 273021074832 \nu^{14} + 718595341691 \nu^{13} + \cdots - 26706856226232 ) / 13978548468689 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 110031997352 \nu^{15} - 46327718359 \nu^{14} + 1051024667793 \nu^{13} + \cdots + 2401649960549 ) / 13978548468689 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 125644314653 \nu^{15} + 383053072184 \nu^{14} - 1064373739340 \nu^{13} + \cdots + 14808207101506 ) / 13978548468689 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19538194237 \nu^{15} - 1977472632 \nu^{14} - 172511924256 \nu^{13} - 286037658883 \nu^{12} + \cdots - 1544708016012 ) / 1996935495527 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 162430924499 \nu^{15} + 333525823938 \nu^{14} - 1217259507928 \nu^{13} + \cdots + 16432309148276 ) / 13978548468689 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 864730507 \nu^{15} + 440655048 \nu^{14} + 7445537199 \nu^{13} + 9891311434 \nu^{12} + \cdots + 84942999792 ) / 72427712273 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 195336741373 \nu^{15} - 405754000168 \nu^{14} + 1502896893234 \nu^{13} + \cdots - 18815409676812 ) / 13978548468689 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 204773667542 \nu^{15} - 132481168448 \nu^{14} - 1820466750632 \nu^{13} + \cdots - 16219455778253 ) / 13978548468689 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 217473612570 \nu^{15} - 148451384586 \nu^{14} - 1924716082305 \nu^{13} + \cdots - 16875378000012 ) / 13978548468689 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 41755774 \nu^{15} - 7982308 \nu^{14} - 349231374 \nu^{13} - 600752791 \nu^{12} + \cdots - 1058175027 ) / 1478116577 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 432597557144 \nu^{15} + 135818205979 \nu^{14} + 3509377463622 \nu^{13} + \cdots + 6831569802120 ) / 13978548468689 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - \beta_{12} - \beta_{10} - \beta_{8} + \beta_{4} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} - \beta_{14} + \beta_{11} + 2\beta_{7} - \beta_{4} - 2\beta_{2} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{15} - 7 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} - 7 \beta_{11} - 3 \beta_{10} - 11 \beta_{9} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{13} - 10 \beta_{12} - 4 \beta_{10} - 8 \beta_{8} + 6 \beta_{6} - 8 \beta_{5} - 7 \beta_{4} + \cdots - 8 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 60 \beta_{15} - 69 \beta_{14} - 27 \beta_{11} - 71 \beta_{9} + 80 \beta_{7} - 69 \beta_{4} + \cdots + 90 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9 \beta_{15} + 11 \beta_{14} + 116 \beta_{13} - 128 \beta_{12} - 116 \beta_{11} - 47 \beta_{10} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 107 \beta_{13} + 237 \beta_{12} - 53 \beta_{10} - 421 \beta_{8} + 504 \beta_{6} - 714 \beta_{5} + \cdots - 421 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 292 \beta_{15} - 330 \beta_{14} - 880 \beta_{11} - 1212 \beta_{9} + 424 \beta_{7} - 330 \beta_{4} + \cdots + 485 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3336 \beta_{15} + 3787 \beta_{14} + 3401 \beta_{13} - 1489 \beta_{12} - 3401 \beta_{11} - 1421 \beta_{10} + \cdots - 5412 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4897 \beta_{13} + 8976 \beta_{12} + 2046 \beta_{10} - 1077 \beta_{8} + 4821 \beta_{6} + \cdots - 1077 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 14102 \beta_{15} + 16017 \beta_{14} - 42127 \beta_{11} - 37883 \beta_{9} - 20042 \beta_{7} + \cdots - 22814 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 50026 \beta_{15} + 56893 \beta_{14} - 10884 \beta_{13} + 47714 \beta_{12} + 10884 \beta_{11} + \cdots - 80361 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 380311 \beta_{13} + 449515 \beta_{12} + 157393 \beta_{10} + 161849 \beta_{8} + 24180 \beta_{6} + \cdots + 161849 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 397857 \beta_{15} + 452423 \beta_{14} - 178490 \beta_{11} + 78272 \beta_{9} - 562444 \beta_{7} + \cdots - 639476 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1325324 \beta_{15} + 1506897 \beta_{14} - 2650041 \beta_{13} + 3950633 \beta_{12} + 2650041 \beta_{11} + \cdots - 2131704 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.198474 − 0.701908i
−2.06456 − 2.00847i
−0.0510945 + 0.786997i
0.886285 + 1.32819i
−0.656012 + 0.437748i
−1.59339 − 0.103448i
2.77167 + 0.783726i
0.508633 − 0.522837i
0.198474 + 0.701908i
−2.06456 + 2.00847i
−0.0510945 − 0.786997i
0.886285 − 1.32819i
−0.656012 − 0.437748i
−1.59339 + 0.103448i
2.77167 − 0.783726i
0.508633 + 0.522837i
−1.81952 − 1.05050i −1.70004 − 0.331437i 1.20711 + 2.09077i −0.804019 + 1.39260i 2.74509 + 2.38896i 0 − 0.870264i 2.78030 + 1.12691i 2.92586 − 1.68925i
68.2 −1.81952 − 1.05050i 1.70004 + 0.331437i 1.20711 + 2.09077i 0.804019 − 1.39260i −2.74509 − 2.38896i 0 − 0.870264i 2.78030 + 1.12691i −2.92586 + 1.68925i
68.3 −1.09057 − 0.629640i −0.353975 − 1.69549i −0.207107 − 0.358719i 1.16342 − 2.01511i −0.681517 + 2.07193i 0 3.04017i −2.74940 + 1.20032i −2.53759 + 1.46508i
68.4 −1.09057 − 0.629640i 0.353975 + 1.69549i −0.207107 − 0.358719i −1.16342 + 2.01511i 0.681517 − 2.07193i 0 3.04017i −2.74940 + 1.20032i 2.53759 − 1.46508i
68.5 1.09057 + 0.629640i −1.29135 − 1.15430i −0.207107 − 0.358719i 1.16342 − 2.01511i −0.681517 − 2.07193i 0 − 3.04017i 0.335190 + 2.98122i 2.53759 − 1.46508i
68.6 1.09057 + 0.629640i 1.29135 + 1.15430i −0.207107 − 0.358719i −1.16342 + 2.01511i 0.681517 + 2.07193i 0 − 3.04017i 0.335190 + 2.98122i −2.53759 + 1.46508i
68.7 1.81952 + 1.05050i −0.562989 + 1.63800i 1.20711 + 2.09077i 0.804019 − 1.39260i −2.74509 + 2.38896i 0 0.870264i −2.36609 − 1.84435i 2.92586 − 1.68925i
68.8 1.81952 + 1.05050i 0.562989 − 1.63800i 1.20711 + 2.09077i −0.804019 + 1.39260i 2.74509 − 2.38896i 0 0.870264i −2.36609 − 1.84435i −2.92586 + 1.68925i
80.1 −1.81952 + 1.05050i −1.70004 + 0.331437i 1.20711 − 2.09077i −0.804019 − 1.39260i 2.74509 − 2.38896i 0 0.870264i 2.78030 − 1.12691i 2.92586 + 1.68925i
80.2 −1.81952 + 1.05050i 1.70004 − 0.331437i 1.20711 − 2.09077i 0.804019 + 1.39260i −2.74509 + 2.38896i 0 0.870264i 2.78030 − 1.12691i −2.92586 − 1.68925i
80.3 −1.09057 + 0.629640i −0.353975 + 1.69549i −0.207107 + 0.358719i 1.16342 + 2.01511i −0.681517 − 2.07193i 0 − 3.04017i −2.74940 − 1.20032i −2.53759 − 1.46508i
80.4 −1.09057 + 0.629640i 0.353975 − 1.69549i −0.207107 + 0.358719i −1.16342 − 2.01511i 0.681517 + 2.07193i 0 − 3.04017i −2.74940 − 1.20032i 2.53759 + 1.46508i
80.5 1.09057 − 0.629640i −1.29135 + 1.15430i −0.207107 + 0.358719i 1.16342 + 2.01511i −0.681517 + 2.07193i 0 3.04017i 0.335190 − 2.98122i 2.53759 + 1.46508i
80.6 1.09057 − 0.629640i 1.29135 − 1.15430i −0.207107 + 0.358719i −1.16342 − 2.01511i 0.681517 − 2.07193i 0 3.04017i 0.335190 − 2.98122i −2.53759 − 1.46508i
80.7 1.81952 − 1.05050i −0.562989 − 1.63800i 1.20711 − 2.09077i 0.804019 + 1.39260i −2.74509 − 2.38896i 0 − 0.870264i −2.36609 + 1.84435i 2.92586 + 1.68925i
80.8 1.81952 − 1.05050i 0.562989 + 1.63800i 1.20711 − 2.09077i −0.804019 − 1.39260i 2.74509 + 2.38896i 0 − 0.870264i −2.36609 + 1.84435i −2.92586 − 1.68925i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.2.g.b 16
3.b odd 2 1 inner 147.2.g.b 16
7.b odd 2 1 inner 147.2.g.b 16
7.c even 3 1 147.2.c.b ✓ 8
7.c even 3 1 inner 147.2.g.b 16
7.d odd 6 1 147.2.c.b ✓ 8
7.d odd 6 1 inner 147.2.g.b 16
21.c even 2 1 inner 147.2.g.b 16
21.g even 6 1 147.2.c.b ✓ 8
21.g even 6 1 inner 147.2.g.b 16
21.h odd 6 1 147.2.c.b ✓ 8
21.h odd 6 1 inner 147.2.g.b 16
28.f even 6 1 2352.2.k.h 8
28.g odd 6 1 2352.2.k.h 8
84.j odd 6 1 2352.2.k.h 8
84.n even 6 1 2352.2.k.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.c.b ✓ 8 7.c even 3 1
147.2.c.b ✓ 8 7.d odd 6 1
147.2.c.b ✓ 8 21.g even 6 1
147.2.c.b ✓ 8 21.h odd 6 1
147.2.g.b 16 1.a even 1 1 trivial
147.2.g.b 16 3.b odd 2 1 inner
147.2.g.b 16 7.b odd 2 1 inner
147.2.g.b 16 7.c even 3 1 inner
147.2.g.b 16 7.d odd 6 1 inner
147.2.g.b 16 21.c even 2 1 inner
147.2.g.b 16 21.g even 6 1 inner
147.2.g.b 16 21.h odd 6 1 inner
2352.2.k.h 8 28.f even 6 1
2352.2.k.h 8 28.g odd 6 1
2352.2.k.h 8 84.j odd 6 1
2352.2.k.h 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6T_{2}^{6} + 29T_{2}^{4} - 42T_{2}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 6 T^{6} + 29 T^{4} + \cdots + 49)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{8} + 8 T^{6} + \cdots + 196)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 12 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 20 T^{2} + 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 16 T^{6} + \cdots + 196)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 32 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 20 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 84 T^{2} + 1372)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 8 T^{6} + 56 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots + 2116)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 56 T^{2} + 686)^{4} \) Copy content Toggle raw display
$43$ \( (T - 2)^{16} \) Copy content Toggle raw display
$47$ \( (T^{8} + 176 T^{6} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 40 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 128 T^{6} + \cdots + 12845056)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 116 T^{6} + \cdots + 11303044)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 12 T^{3} + \cdots + 1296)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 84 T^{2} + 1372)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 148 T^{6} + \cdots + 19518724)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + 20 T^{2} + \cdots + 16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 224 T^{2} + 2744)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 176 T^{6} + \cdots + 54848836)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 244 T^{2} + 10082)^{4} \) Copy content Toggle raw display
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