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Properties

Label 147.3.h.c
Level $147$
Weight $3$
Character orbit 147.h
Analytic conductor $4.005$
Analytic rank $0$
Dimension $8$
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] = [147,3,Mod(116,147)]
 
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, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.116");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 147.h (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: \(4.00545988610\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.39033114624.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 6x^{6} - 30x^{5} + 34x^{4} - 102x^{3} + 486x^{2} - 730x + 373 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_{6} + \beta_{2}) q^{3} + ( - 3 \beta_{4} + 2 \beta_{2}) q^{4} + ( - \beta_{7} + 2 \beta_{5} + \beta_{4} + \cdots + 1) q^{5}+ \cdots + (2 \beta_{7} - 3 \beta_{5} + 4 \beta_{4} + \cdots + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + ( - \beta_{6} + \beta_{2}) q^{3} + ( - 3 \beta_{4} + 2 \beta_{2}) q^{4} + ( - \beta_{7} + 2 \beta_{5} + \beta_{4} + \cdots + 1) q^{5}+ \cdots + (4 \beta_{7} - 16 \beta_{6} + \cdots + 26) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 12 q^{4} - 28 q^{6} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 12 q^{4} - 28 q^{6} + 20 q^{9} + 28 q^{10} - 22 q^{12} + 72 q^{13} + 56 q^{15} - 36 q^{16} - 56 q^{18} + 12 q^{19} - 112 q^{22} - 126 q^{24} + 12 q^{25} - 20 q^{27} + 28 q^{30} + 136 q^{31} + 28 q^{33} + 232 q^{36} - 16 q^{37} - 4 q^{39} + 84 q^{40} - 320 q^{43} - 140 q^{45} + 168 q^{46} - 76 q^{48} + 84 q^{51} + 164 q^{52} - 154 q^{54} - 112 q^{55} + 128 q^{57} - 112 q^{58} - 140 q^{60} - 156 q^{61} + 8 q^{64} - 28 q^{66} + 24 q^{67} - 336 q^{69} - 32 q^{73} + 146 q^{75} + 632 q^{76} - 392 q^{78} - 128 q^{79} + 68 q^{81} + 392 q^{82} + 336 q^{85} + 28 q^{87} - 168 q^{88} + 224 q^{90} + 96 q^{93} - 336 q^{94} - 98 q^{96} + 16 q^{97} + 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 6x^{6} - 30x^{5} + 34x^{4} - 102x^{3} + 486x^{2} - 730x + 373 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 50\nu^{7} + 15\nu^{6} - 1344\nu^{5} - 2577\nu^{4} - 16648\nu^{3} + 10993\nu^{2} + 13664\nu + 174191 ) / 46998 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 62\nu^{7} + 1809\nu^{6} + 3078\nu^{5} + 13500\nu^{4} - 11602\nu^{3} - 37037\nu^{2} - 174898\nu + 164120 ) / 46998 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 140\nu^{7} - 704\nu^{6} - 257\nu^{5} - 7514\nu^{4} + 3666\nu^{3} - 6072\nu^{2} + 120543\nu - 105932 ) / 15666 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -163\nu^{7} + 63\nu^{6} - 945\nu^{5} + 3276\nu^{4} - 469\nu^{3} + 15883\nu^{2} - 51751\nu + 36554 ) / 6714 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1223\nu^{7} + 255\nu^{6} + 7365\nu^{5} - 20310\nu^{4} - 655\nu^{3} - 114503\nu^{2} + 346799\nu - 202912 ) / 46998 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -616\nu^{7} + 39\nu^{6} - 2823\nu^{5} + 12099\nu^{4} + 7667\nu^{3} + 62674\nu^{2} - 160597\nu + 51101 ) / 23499 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -152\nu^{7} + 29\nu^{6} - 808\nu^{5} + 3373\nu^{4} + 986\nu^{3} + 16429\nu^{2} - 45746\nu + 19023 ) / 5222 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + \beta_{5} - 5\beta_{4} - 4\beta_{3} - 4\beta_{2} + 4\beta _1 - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} - 11\beta_{6} + 11\beta_{5} + 17\beta_{4} - 2\beta_{3} - 8\beta_{2} - 16\beta _1 + 38 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{7} - 2\beta_{6} - 2\beta_{5} - 6\beta_{4} + 8\beta_{3} + 10\beta_{2} + 2\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -47\beta_{7} + 136\beta_{6} - 91\beta_{5} - 199\beta_{4} - 44\beta_{3} + 10\beta_{2} + 107\beta _1 - 163 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -170\beta_{7} + 22\beta_{6} + 377\beta_{5} + 575\beta_{4} + 4\beta_{3} - 86\beta_{2} - 136\beta _1 - 100 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1190\beta_{7} - 1309\beta_{6} + 301\beta_{5} + 37\beta_{4} - 280\beta_{3} - 490\beta_{2} - 434\beta _1 + 1036 ) / 3 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−0.279898 − 3.02113i
−1.85391 + 1.90397i
1.03103 − 0.478705i
2.10277 − 0.136187i
−0.279898 + 3.02113i
−1.85391 − 1.90397i
1.03103 + 0.478705i
2.10277 + 0.136187i
−3.03622 + 1.75296i 2.90987 + 0.729839i 4.14575 − 7.18065i −1.07558 + 0.620984i −10.1144 + 2.88494i 0 15.0457i 7.93467 + 4.24747i 2.17712 − 3.77089i
116.2 −1.13198 + 0.653548i −2.97489 + 0.387321i −1.14575 + 1.98450i −6.39086 + 3.68977i 3.11438 − 2.38267i 0 − 8.22359i 8.69997 − 2.30448i 4.82288 − 8.35347i
116.3 1.13198 − 0.653548i 1.15202 + 2.76999i −1.14575 + 1.98450i 6.39086 − 3.68977i 3.11438 + 2.38267i 0 8.22359i −6.34572 + 6.38215i 4.82288 − 8.35347i
116.4 3.03622 − 1.75296i −2.08699 − 2.15510i 4.14575 − 7.18065i 1.07558 − 0.620984i −10.1144 − 2.88494i 0 − 15.0457i −0.288920 + 8.99536i 2.17712 − 3.77089i
128.1 −3.03622 − 1.75296i 2.90987 − 0.729839i 4.14575 + 7.18065i −1.07558 − 0.620984i −10.1144 − 2.88494i 0 − 15.0457i 7.93467 − 4.24747i 2.17712 + 3.77089i
128.2 −1.13198 − 0.653548i −2.97489 − 0.387321i −1.14575 − 1.98450i −6.39086 − 3.68977i 3.11438 + 2.38267i 0 8.22359i 8.69997 + 2.30448i 4.82288 + 8.35347i
128.3 1.13198 + 0.653548i 1.15202 − 2.76999i −1.14575 − 1.98450i 6.39086 + 3.68977i 3.11438 − 2.38267i 0 − 8.22359i −6.34572 − 6.38215i 4.82288 + 8.35347i
128.4 3.03622 + 1.75296i −2.08699 + 2.15510i 4.14575 + 7.18065i 1.07558 + 0.620984i −10.1144 + 2.88494i 0 15.0457i −0.288920 − 8.99536i 2.17712 + 3.77089i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 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 147.3.h.c 8
3.b odd 2 1 inner 147.3.h.c 8
7.b odd 2 1 147.3.h.e 8
7.c even 3 1 147.3.b.f 4
7.c even 3 1 inner 147.3.h.c 8
7.d odd 6 1 21.3.b.a ✓ 4
7.d odd 6 1 147.3.h.e 8
21.c even 2 1 147.3.h.e 8
21.g even 6 1 21.3.b.a ✓ 4
21.g even 6 1 147.3.h.e 8
21.h odd 6 1 147.3.b.f 4
21.h odd 6 1 inner 147.3.h.c 8
28.f even 6 1 336.3.d.c 4
35.i odd 6 1 525.3.c.a 4
35.k even 12 2 525.3.f.a 8
56.j odd 6 1 1344.3.d.f 4
56.m even 6 1 1344.3.d.b 4
63.i even 6 1 567.3.r.c 8
63.k odd 6 1 567.3.r.c 8
63.s even 6 1 567.3.r.c 8
63.t odd 6 1 567.3.r.c 8
84.j odd 6 1 336.3.d.c 4
105.p even 6 1 525.3.c.a 4
105.w odd 12 2 525.3.f.a 8
168.ba even 6 1 1344.3.d.f 4
168.be odd 6 1 1344.3.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.b.a ✓ 4 7.d odd 6 1
21.3.b.a ✓ 4 21.g even 6 1
147.3.b.f 4 7.c even 3 1
147.3.b.f 4 21.h odd 6 1
147.3.h.c 8 1.a even 1 1 trivial
147.3.h.c 8 3.b odd 2 1 inner
147.3.h.c 8 7.c even 3 1 inner
147.3.h.c 8 21.h odd 6 1 inner
147.3.h.e 8 7.b odd 2 1
147.3.h.e 8 7.d odd 6 1
147.3.h.e 8 21.c even 2 1
147.3.h.e 8 21.g even 6 1
336.3.d.c 4 28.f even 6 1
336.3.d.c 4 84.j odd 6 1
525.3.c.a 4 35.i odd 6 1
525.3.c.a 4 105.p even 6 1
525.3.f.a 8 35.k even 12 2
525.3.f.a 8 105.w odd 12 2
567.3.r.c 8 63.i even 6 1
567.3.r.c 8 63.k odd 6 1
567.3.r.c 8 63.s even 6 1
567.3.r.c 8 63.t odd 6 1
1344.3.d.b 4 56.m even 6 1
1344.3.d.b 4 168.be odd 6 1
1344.3.d.f 4 56.j odd 6 1
1344.3.d.f 4 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{8} - 14T_{2}^{6} + 175T_{2}^{4} - 294T_{2}^{2} + 441 \) Copy content Toggle raw display
\( T_{5}^{8} - 56T_{5}^{6} + 3052T_{5}^{4} - 4704T_{5}^{2} + 7056 \) Copy content Toggle raw display
\( T_{13}^{2} - 18T_{13} + 74 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 14 T^{6} + \cdots + 441 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} - 56 T^{6} + \cdots + 7056 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 56 T^{6} + \cdots + 112896 \) Copy content Toggle raw display
$13$ \( (T^{2} - 18 T + 74)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} - 168 T^{6} + \cdots + 9144576 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + \cdots + 27556)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 672 T^{6} + \cdots + 146313216 \) Copy content Toggle raw display
$29$ \( (T^{4} + 392 T^{2} + 27216)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 68 T^{3} + \cdots + 1272384)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 8 T^{3} + \cdots + 1838736)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 5432 T^{2} + 4139856)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 80 T + 1348)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 3033950846976 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 6467784857856 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 213354983768976 \) Copy content Toggle raw display
$61$ \( (T^{4} + 78 T^{3} + \cdots + 1387684)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 12 T^{3} + \cdots + 169744)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 9576 T^{2} + 22888656)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 16 T^{3} + \cdots + 21790224)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 64 T^{3} + \cdots + 64770304)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 13608 T^{2} + 44641044)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{2} - 4 T - 444)^{4} \) Copy content Toggle raw display
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