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Properties

Label 147.4.c.a
Level $147$
Weight $4$
Character orbit 147.c
Analytic conductor $8.673$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,4,Mod(146,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.146");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.c (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: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 29x^{9} + 6x^{8} - 49x^{7} + 1564x^{6} - 441x^{5} + 486x^{4} - 21141x^{3} - 59049x + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{11} - 2) q^{4} - \beta_{10} q^{5} + ( - \beta_{10} - \beta_{5} + \cdots + 2 \beta_{2}) q^{6}+ \cdots + ( - \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{11} - 2) q^{4} - \beta_{10} q^{5} + ( - \beta_{10} - \beta_{5} + \cdots + 2 \beta_{2}) q^{6}+ \cdots + ( - 18 \beta_{11} - 93 \beta_{8} + \cdots - 303) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 28 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 28 q^{4} + 6 q^{9} + 6 q^{15} - 268 q^{16} - 132 q^{18} - 268 q^{22} + 84 q^{25} + 1644 q^{30} + 852 q^{36} - 1528 q^{37} + 852 q^{39} - 1012 q^{43} - 1216 q^{46} + 2682 q^{51} + 270 q^{57} - 5740 q^{58} + 1836 q^{60} - 548 q^{64} - 1584 q^{67} + 5424 q^{72} + 4296 q^{78} - 3348 q^{79} - 1674 q^{81} + 348 q^{85} + 1108 q^{88} + 2958 q^{93} - 3354 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 29x^{9} + 6x^{8} - 49x^{7} + 1564x^{6} - 441x^{5} + 486x^{4} - 21141x^{3} - 59049x + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 661 \nu^{11} - 17114 \nu^{10} - 13815 \nu^{9} + 226448 \nu^{8} + 617943 \nu^{7} + \cdots + 575491554 ) / 489398112 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 857 \nu^{11} + 5426 \nu^{10} - 627 \nu^{9} + 6088 \nu^{8} - 24405 \nu^{7} + 289700 \nu^{6} + \cdots + 255288510 ) / 81566352 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2476 \nu^{11} + 15515 \nu^{10} + 15948 \nu^{9} - 216551 \nu^{8} - 468246 \nu^{7} + \cdots - 1990167813 ) / 163132704 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31 \nu^{11} + 220 \nu^{10} + 1107 \nu^{9} + 1790 \nu^{8} - 14415 \nu^{7} - 28370 \nu^{6} + \cdots - 11337408 ) / 1889568 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3517 \nu^{11} - 9850 \nu^{10} + 7683 \nu^{9} + 231268 \nu^{8} + 322857 \nu^{7} + \cdots - 1904802642 ) / 163132704 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1523 \nu^{11} - 209 \nu^{10} - 6255 \nu^{9} - 6421 \nu^{8} + 82569 \nu^{7} + 227449 \nu^{6} + \cdots + 151814979 ) / 54377568 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6025 \nu^{11} + 87566 \nu^{10} - 97641 \nu^{9} - 1174508 \nu^{8} - 3032283 \nu^{7} + \cdots + 4735611702 ) / 163132704 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23393 \nu^{11} + 32191 \nu^{10} + 346095 \nu^{9} - 558841 \nu^{8} - 77001 \nu^{7} + \cdots - 1125769185 ) / 489398112 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25402 \nu^{11} + 2263 \nu^{10} + 146502 \nu^{9} + 719729 \nu^{8} + 11964 \nu^{7} + \cdots + 1611033867 ) / 489398112 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 103 \nu^{11} - 112 \nu^{10} + 657 \nu^{9} - 2906 \nu^{8} + 6711 \nu^{7} - 30454 \nu^{6} + \cdots - 12282192 ) / 1889568 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 95 \nu^{11} - 112 \nu^{10} + 207 \nu^{9} + 2026 \nu^{8} - 399 \nu^{7} + 9974 \nu^{6} + \cdots + 4723920 ) / 944784 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - 3\beta_{6} - \beta_{3} + \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{10} + 3\beta_{9} - 2\beta_{8} + \beta_{7} + 3\beta_{5} - \beta_{3} - \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} - \beta_{8} - \beta_{7} - 3\beta_{3} - 22\beta _1 + 21 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{11} - 21 \beta_{10} + 30 \beta_{9} + 19 \beta_{8} + 7 \beta_{7} - 21 \beta_{6} + 27 \beta_{5} + \cdots + 48 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 95 \beta_{11} + 18 \beta_{10} - 171 \beta_{9} + 33 \beta_{8} + 6 \beta_{7} - 42 \beta_{6} + 72 \beta_{5} + \cdots + 216 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -6\beta_{11} + 20\beta_{8} + 8\beta_{7} - 140\beta_{3} - 200\beta _1 - 1629 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 535 \beta_{11} - 30 \beta_{10} + 666 \beta_{9} + 406 \beta_{8} + 10 \beta_{7} + 1491 \beta_{6} + \cdots - 984 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 762 \beta_{11} - 2373 \beta_{10} - 3819 \beta_{9} + 1282 \beta_{8} - 791 \beta_{7} + 1506 \beta_{6} + \cdots + 3648 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2311\beta_{11} + 4533\beta_{8} + 681\beta_{7} + 487\beta_{3} + 12470\beta _1 - 29619 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5664 \beta_{11} + 11715 \beta_{10} - 17760 \beta_{9} - 7409 \beta_{8} - 3905 \beta_{7} + 30285 \beta_{6} + \cdots - 114192 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 67241 \beta_{11} - 30912 \beta_{10} + 34443 \beta_{9} + 3799 \beta_{8} - 10304 \beta_{7} + \cdots - 144912 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
−0.232749 − 2.99096i
2.70662 + 1.29391i
−2.59957 + 1.49740i
0.00299931 − 3.00000i
−2.23014 + 2.00661i
2.85284 + 0.928053i
−2.23014 − 2.00661i
2.85284 − 0.928053i
−2.59957 − 1.49740i
0.00299931 + 3.00000i
−0.232749 + 2.99096i
2.70662 − 1.29391i
− 4.54551i −2.93937 + 4.28487i −12.6617 11.6039 19.4769 + 13.3609i 0 21.1897i −9.72022 − 25.1896i − 52.7455i
146.2 − 4.54551i 2.93937 − 4.28487i −12.6617 −11.6039 −19.4769 − 13.3609i 0 21.1897i −9.72022 − 25.1896i 52.7455i
146.3 − 2.58741i −2.60257 − 4.49740i 1.30532 −16.1181 −11.6366 + 6.73392i 0 − 24.0767i −13.4532 + 23.4096i 41.7042i
146.4 − 2.58741i 2.60257 + 4.49740i 1.30532 16.1181 11.6366 − 6.73392i 0 − 24.0767i −13.4532 + 23.4096i − 41.7042i
146.5 − 1.90883i −5.08298 − 1.07856i 4.35636 1.24741 −2.05878 + 9.70256i 0 − 23.5862i 24.6734 + 10.9646i − 2.38110i
146.6 − 1.90883i 5.08298 + 1.07856i 4.35636 −1.24741 2.05878 − 9.70256i 0 − 23.5862i 24.6734 + 10.9646i 2.38110i
146.7 1.90883i −5.08298 + 1.07856i 4.35636 1.24741 −2.05878 − 9.70256i 0 23.5862i 24.6734 − 10.9646i 2.38110i
146.8 1.90883i 5.08298 − 1.07856i 4.35636 −1.24741 2.05878 + 9.70256i 0 23.5862i 24.6734 − 10.9646i − 2.38110i
146.9 2.58741i −2.60257 + 4.49740i 1.30532 −16.1181 −11.6366 − 6.73392i 0 24.0767i −13.4532 − 23.4096i − 41.7042i
146.10 2.58741i 2.60257 − 4.49740i 1.30532 16.1181 11.6366 + 6.73392i 0 24.0767i −13.4532 − 23.4096i 41.7042i
146.11 4.54551i −2.93937 − 4.28487i −12.6617 11.6039 19.4769 − 13.3609i 0 − 21.1897i −9.72022 + 25.1896i 52.7455i
146.12 4.54551i 2.93937 + 4.28487i −12.6617 −11.6039 −19.4769 + 13.3609i 0 − 21.1897i −9.72022 + 25.1896i − 52.7455i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.12
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
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.c.a 12
3.b odd 2 1 inner 147.4.c.a 12
7.b odd 2 1 inner 147.4.c.a 12
7.c even 3 1 21.4.g.a ✓ 12
7.c even 3 1 147.4.g.d 12
7.d odd 6 1 21.4.g.a ✓ 12
7.d odd 6 1 147.4.g.d 12
21.c even 2 1 inner 147.4.c.a 12
21.g even 6 1 21.4.g.a ✓ 12
21.g even 6 1 147.4.g.d 12
21.h odd 6 1 21.4.g.a ✓ 12
21.h odd 6 1 147.4.g.d 12
28.f even 6 1 336.4.bc.d 12
28.g odd 6 1 336.4.bc.d 12
84.j odd 6 1 336.4.bc.d 12
84.n even 6 1 336.4.bc.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.g.a ✓ 12 7.c even 3 1
21.4.g.a ✓ 12 7.d odd 6 1
21.4.g.a ✓ 12 21.g even 6 1
21.4.g.a ✓ 12 21.h odd 6 1
147.4.c.a 12 1.a even 1 1 trivial
147.4.c.a 12 3.b odd 2 1 inner
147.4.c.a 12 7.b odd 2 1 inner
147.4.c.a 12 21.c even 2 1 inner
147.4.g.d 12 7.c even 3 1
147.4.g.d 12 7.d odd 6 1
147.4.g.d 12 21.g even 6 1
147.4.g.d 12 21.h odd 6 1
336.4.bc.d 12 28.f even 6 1
336.4.bc.d 12 28.g odd 6 1
336.4.bc.d 12 84.j odd 6 1
336.4.bc.d 12 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 31T_{2}^{4} + 238T_{2}^{2} + 504 \) acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 31 T^{4} + \cdots + 504)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( (T^{6} - 396 T^{4} + \cdots - 54432)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + 3244 T^{4} + \cdots + 673513344)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 4335 T^{4} + \cdots + 82121472)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 12261 T^{4} + \cdots - 4206722688)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 2994 T^{4} + \cdots + 243972972)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 14311 T^{4} + \cdots + 94816842624)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 14683734245376)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 42033 T^{4} + \cdots + 33414175107)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 382 T^{2} + \cdots - 1849018)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 4591113633792)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 253 T^{2} + \cdots - 6662944)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 104940131240448)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 36\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 623203616502432)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 64\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 396 T^{2} + \cdots + 98311462)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 6720226523136)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 40\!\cdots\!92)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 837 T^{2} + \cdots - 19057853)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 388952511994368)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 94\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 98\!\cdots\!48)^{2} \) Copy content Toggle raw display
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