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Properties

Label 147.4.g.d
Level $147$
Weight $4$
Character orbit 147.g
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(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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.68");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 21)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{2}) q^{2} + ( - \beta_{8} - \beta_{7}) q^{3} + ( - \beta_{11} + \beta_{10} + \cdots - 2 \beta_{4}) q^{4}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{9} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{2}) q^{2} + ( - \beta_{8} - \beta_{7}) q^{3} + ( - \beta_{11} + \beta_{10} + \cdots - 2 \beta_{4}) q^{4}+ \cdots + (75 \beta_{11} - 87 \beta_{10} + \cdots - 303) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} + 14 q^{4} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} + 14 q^{4} - 3 q^{9} - 30 q^{10} + 192 q^{12} + 6 q^{15} + 134 q^{16} + 66 q^{18} - 300 q^{19} - 268 q^{22} - 414 q^{24} - 42 q^{25} - 822 q^{30} + 930 q^{31} + 855 q^{33} + 852 q^{36} + 764 q^{37} - 426 q^{39} - 2298 q^{40} - 1012 q^{43} - 2367 q^{45} + 608 q^{46} - 1341 q^{51} + 3000 q^{52} + 4158 q^{54} + 270 q^{57} + 2870 q^{58} - 918 q^{60} - 2358 q^{61} - 548 q^{64} - 2934 q^{66} + 792 q^{67} - 2712 q^{72} + 2904 q^{73} + 2418 q^{75} + 4296 q^{78} + 1674 q^{79} + 837 q^{81} - 5040 q^{82} + 348 q^{85} - 1638 q^{87} - 554 q^{88} - 1479 q^{93} + 1356 q^{94} + 4410 q^{96} - 3354 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( - 538 \nu^{11} - 22601 \nu^{10} + 146502 \nu^{9} - 1327 \nu^{8} + 161148 \nu^{7} + \cdots + 142839531 ) / 489398112 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 661 \nu^{11} - 17114 \nu^{10} - 13815 \nu^{9} + 226448 \nu^{8} + 617943 \nu^{7} + \cdots + 575491554 ) / 489398112 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 109 \nu^{11} - 3322 \nu^{10} - 26109 \nu^{9} + 69172 \nu^{8} + 83625 \nu^{7} - 33772 \nu^{6} + \cdots - 271271106 ) / 69914016 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 857 \nu^{11} + 5426 \nu^{10} - 627 \nu^{9} + 6088 \nu^{8} - 24405 \nu^{7} + 289700 \nu^{6} + \cdots + 173722158 ) / 163132704 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4301 \nu^{11} + 145853 \nu^{10} - 231543 \nu^{9} - 1385435 \nu^{8} - 5417499 \nu^{7} + \cdots + 8693961417 ) / 489398112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4345 \nu^{11} + 19933 \nu^{10} + 136449 \nu^{9} + 345029 \nu^{8} - 1557771 \nu^{7} + \cdots - 1180448559 ) / 489398112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} + \cdots - 151814979 ) / 54377568 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} + \cdots - 151814979 ) / 54377568 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22376 \nu^{11} + 116845 \nu^{10} - 61380 \nu^{9} - 2138089 \nu^{8} - 3679350 \nu^{7} + \cdots + 5512873689 ) / 489398112 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 635 \nu^{11} - 221 \nu^{10} + 4041 \nu^{9} + 4103 \nu^{8} + 30441 \nu^{7} - 164387 \nu^{6} + \cdots - 66193929 ) / 13226976 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12970 \nu^{11} - 10169 \nu^{10} + 146502 \nu^{9} + 359201 \nu^{8} + 86556 \nu^{7} + \cdots + 876936699 ) / 244699056 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} + \beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} + 2\beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + 3\beta_{3} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{11} - 3 \beta_{10} - \beta_{9} - 6 \beta_{8} - 3 \beta_{7} - \beta_{5} + 3 \beta_{3} + \cdots + 21 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11 \beta_{11} + 27 \beta_{10} - 7 \beta_{9} + 2 \beta_{8} + 25 \beta_{7} - 24 \beta_{6} + \cdots - 19 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 33 \beta_{11} + 12 \beta_{9} - 10 \beta_{8} + 52 \beta_{7} - 6 \beta_{5} + 216 \beta_{4} + 72 \beta_{3} + \cdots + 216 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 134 \beta_{11} - 114 \beta_{10} + 8 \beta_{9} - 280 \beta_{8} - 140 \beta_{7} + 8 \beta_{5} + \cdots - 1629 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 260 \beta_{11} + 552 \beta_{10} - 10 \beta_{9} + 681 \beta_{8} - 129 \beta_{7} - 912 \beta_{6} + \cdots - 406 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1282 \beta_{11} - 1582 \beta_{9} - 493 \beta_{8} - 1013 \beta_{7} + 1176 \beta_{6} + 791 \beta_{5} + \cdots + 3648 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1824 \beta_{11} + 2709 \beta_{10} + 681 \beta_{9} + 974 \beta_{8} + 487 \beta_{7} + 681 \beta_{5} + \cdots - 29619 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 10351 \beta_{11} - 4467 \beta_{10} + 3905 \beta_{9} + 8606 \beta_{8} - 13073 \beta_{7} + \cdots + 7409 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 3799 \beta_{11} - 20608 \beta_{9} - 25200 \beta_{8} - 96240 \beta_{7} + 40080 \beta_{6} + \cdots - 144912 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(1 + \beta_{4}\)

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
2.70662 + 1.29391i
0.00299931 − 3.00000i
−2.23014 + 2.00661i
2.85284 − 0.928053i
−2.59957 − 1.49740i
−0.232749 + 2.99096i
2.70662 − 1.29391i
0.00299931 + 3.00000i
−2.23014 − 2.00661i
2.85284 + 0.928053i
−2.59957 + 1.49740i
−0.232749 − 2.99096i
−3.93653 − 2.27276i 2.24112 − 4.68800i 6.33084 + 10.9653i 5.80193 − 10.0492i −19.4769 + 13.3609i 0 − 21.1897i −16.9548 − 21.0128i −45.6790 + 26.3728i
68.2 −2.24076 − 1.29370i −5.19615 − 0.00519496i −0.652660 − 1.13044i −8.05907 + 13.9587i 11.6366 + 6.73392i 0 24.0767i 26.9999 + 0.0539876i 36.1169 − 20.8521i
68.3 −1.65310 − 0.954416i 3.47555 + 3.86271i −2.17818 − 3.77272i −0.623706 + 1.08029i −2.05878 − 9.70256i 0 23.5862i −2.84113 + 26.8501i 2.06209 − 1.19055i
68.4 1.65310 + 0.954416i −1.60743 − 4.94127i −2.17818 − 3.77272i 0.623706 − 1.08029i 2.05878 − 9.70256i 0 − 23.5862i −21.8323 + 15.8855i 2.06209 − 1.19055i
68.5 2.24076 + 1.29370i −2.59358 + 4.50260i −0.652660 − 1.13044i 8.05907 − 13.9587i −11.6366 + 6.73392i 0 − 24.0767i −13.5467 − 23.3556i 36.1169 − 20.8521i
68.6 3.93653 + 2.27276i 5.18049 + 0.403134i 6.33084 + 10.9653i −5.80193 + 10.0492i 19.4769 + 13.3609i 0 21.1897i 26.6750 + 4.17686i −45.6790 + 26.3728i
80.1 −3.93653 + 2.27276i 2.24112 + 4.68800i 6.33084 − 10.9653i 5.80193 + 10.0492i −19.4769 − 13.3609i 0 21.1897i −16.9548 + 21.0128i −45.6790 − 26.3728i
80.2 −2.24076 + 1.29370i −5.19615 + 0.00519496i −0.652660 + 1.13044i −8.05907 − 13.9587i 11.6366 − 6.73392i 0 − 24.0767i 26.9999 − 0.0539876i 36.1169 + 20.8521i
80.3 −1.65310 + 0.954416i 3.47555 − 3.86271i −2.17818 + 3.77272i −0.623706 − 1.08029i −2.05878 + 9.70256i 0 − 23.5862i −2.84113 − 26.8501i 2.06209 + 1.19055i
80.4 1.65310 − 0.954416i −1.60743 + 4.94127i −2.17818 + 3.77272i 0.623706 + 1.08029i 2.05878 + 9.70256i 0 23.5862i −21.8323 − 15.8855i 2.06209 + 1.19055i
80.5 2.24076 − 1.29370i −2.59358 − 4.50260i −0.652660 + 1.13044i 8.05907 + 13.9587i −11.6366 − 6.73392i 0 24.0767i −13.5467 + 23.3556i 36.1169 + 20.8521i
80.6 3.93653 − 2.27276i 5.18049 − 0.403134i 6.33084 − 10.9653i −5.80193 − 10.0492i 19.4769 − 13.3609i 0 − 21.1897i 26.6750 − 4.17686i −45.6790 − 26.3728i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.6
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{12} - 31T_{2}^{10} + 723T_{2}^{8} - 6370T_{2}^{6} + 41020T_{2}^{4} - 119952T_{2}^{2} + 254016 \) Copy content Toggle raw display
\( T_{19}^{6} + 150T_{19}^{5} + 9753T_{19}^{4} + 337950T_{19}^{3} + 6428709T_{19}^{2} + 60952662T_{19} + 243972972 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 31 T^{10} + \cdots + 254016 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 2962842624 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{6} + 4335 T^{4} + \cdots + 82121472)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{6} + 150 T^{5} + \cdots + 243972972)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 89\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 14683734245376)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 465 T^{5} + \cdots + 33414175107)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 3418867564324)^{2} \) 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
$47$ \( T^{12} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 64\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 96\!\cdots\!44)^{2} \) 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
$79$ \( (T^{6} + \cdots + 363201760969609)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 388952511994368)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 88\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 98\!\cdots\!48)^{2} \) Copy content Toggle raw display
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