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Properties

Label 147.5.h.b
Level $147$
Weight $5$
Character orbit 147.h
Analytic conductor $15.195$
Analytic rank $0$
Dimension $16$
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] = [147,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.116");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
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: \(15.1953845733\)
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: \( x^{16} - 103 x^{14} + 7227 x^{12} - 270898 x^{10} + 7374256 x^{8} - 115494792 x^{6} + 1245573504 x^{4} + \cdots + 5639409216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{6} - \beta_{2}) q^{3} + (\beta_{9} + \beta_{8} + \cdots + 10 \beta_{2}) q^{4}+ \cdots + (3 \beta_{11} + \beta_{10} + 4 \beta_{9} + \cdots + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{6} - \beta_{2}) q^{3} + (\beta_{9} + \beta_{8} + \cdots + 10 \beta_{2}) q^{4}+ \cdots + (285 \beta_{15} - 126 \beta_{14} + \cdots - 2930) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} + 78 q^{4} - 28 q^{6} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} + 78 q^{4} - 28 q^{6} + 62 q^{9} + 98 q^{10} - 436 q^{12} + 492 q^{13} - 856 q^{15} + 150 q^{16} - 884 q^{18} + 1326 q^{19} + 3244 q^{22} - 630 q^{24} + 2094 q^{25} - 1028 q^{27} + 340 q^{30} + 2504 q^{31} + 616 q^{33} + 7728 q^{34} - 9644 q^{36} + 1342 q^{37} - 2626 q^{39} - 5754 q^{40} + 1460 q^{43} + 1330 q^{45} + 8844 q^{46} - 25864 q^{48} + 7794 q^{51} - 9536 q^{52} + 18578 q^{54} + 35140 q^{55} + 11696 q^{57} - 12446 q^{58} - 18890 q^{60} - 15012 q^{61} + 7660 q^{64} + 17192 q^{66} + 8658 q^{67} - 53676 q^{69} + 21312 q^{72} - 12322 q^{73} + 19514 q^{75} + 84520 q^{76} - 32120 q^{78} - 40168 q^{79} - 23986 q^{81} - 47348 q^{82} + 22536 q^{85} + 1162 q^{87} + 56430 q^{88} - 139636 q^{90} + 8556 q^{93} - 6468 q^{94} + 40894 q^{96} + 28268 q^{97} - 47812 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 103 x^{14} + 7227 x^{12} - 270898 x^{10} + 7374256 x^{8} - 115494792 x^{6} + 1245573504 x^{4} + \cdots + 5639409216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 805319043514 \nu^{14} - 80361724322821 \nu^{12} + \cdots - 39\!\cdots\!52 ) / 17\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 805319043514 \nu^{15} - 80361724322821 \nu^{13} + \cdots - 39\!\cdots\!52 \nu ) / 17\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6321739747609 \nu^{15} - 94449672964612 \nu^{14} + 429246974607973 \nu^{13} + \cdots + 12\!\cdots\!44 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6321739747609 \nu^{15} - 104735181626664 \nu^{14} + 429246974607973 \nu^{13} + \cdots + 78\!\cdots\!16 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 77376912306302 \nu^{15} - 18998427328522 \nu^{14} + \cdots + 14\!\cdots\!12 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 77376912306302 \nu^{15} - 21044339567884 \nu^{14} + \cdots - 25\!\cdots\!68 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 161075564360213 \nu^{15} - 215202916713528 \nu^{14} + \cdots - 12\!\cdots\!76 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6321739747609 \nu^{15} - 492819382147164 \nu^{14} - 429246974607973 \nu^{13} + \cdots + 16\!\cdots\!64 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 59901524063219 \nu^{15} - 174880886372968 \nu^{14} + \cdots + 14\!\cdots\!20 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6321739747609 \nu^{15} + 793973218456484 \nu^{14} + 429246974607973 \nu^{13} + \cdots - 58\!\cdots\!24 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 341327704587575 \nu^{15} - 94449672964612 \nu^{14} + \cdots + 12\!\cdots\!44 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 55949115474713 \nu^{15} + 51619834308255 \nu^{14} + \cdots - 49\!\cdots\!16 ) / 92\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 199118570462318 \nu^{15} - 18998427328522 \nu^{14} + \cdots + 14\!\cdots\!12 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 161075564360213 \nu^{15} - 306271249789248 \nu^{14} + \cdots - 56\!\cdots\!36 ) / 27\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{6} + 26\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + 2\beta_{13} + \beta_{12} - \beta_{11} - \beta_{7} - 2\beta_{6} - 2\beta_{4} + 36\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{11} + 55\beta_{9} - 12\beta_{5} + 71\beta_{4} + 968\beta_{2} - 968 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20 \beta_{15} + 59 \beta_{14} + 142 \beta_{13} + 142 \beta_{10} - 20 \beta_{9} - 20 \beta_{8} + \cdots - 1494 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 940\beta_{15} + 664\beta_{11} - 2871\beta_{8} + 664\beta_{7} + 4087\beta_{6} + 4087\beta_{4} - 41116 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3259\beta_{12} + 4087\beta_{11} + 8174\beta_{10} - 1604\beta_{9} + 1604\beta_{5} + 12158\beta_{4} - 67722\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 56236 \beta_{15} - 148427 \beta_{9} - 148427 \beta_{8} + 41336 \beta_{7} + 219563 \beta_{6} + \cdots - 1892900 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 97572 \beta_{15} - 174863 \beta_{14} - 439126 \beta_{13} - 174863 \beta_{12} + \cdots - 3241626 \beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2321752\beta_{11} - 7632983\beta_{9} + 3077916\beta_{5} - 11467063\beta_{4} - 91446724\beta_{2} + 91446724 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5399668 \beta_{15} - 9198571 \beta_{14} - 22934126 \beta_{13} - 22934126 \beta_{10} + \cdots + 160001874 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 162334604 \beta_{15} - 124454072 \beta_{11} + 391263459 \beta_{8} - 124454072 \beta_{7} + \cdots + 4536707828 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 477836999 \beta_{12} - 591478595 \beta_{11} - 1182957190 \beta_{10} + 286788676 \beta_{9} + \cdots + 8027287194 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 8418044924 \beta_{15} + 20017474303 \beta_{9} + 20017474303 \beta_{8} - 6517435672 \beta_{7} + \cdots + 228221098180 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 14935480596 \beta_{15} + 24634300723 \beta_{14} + 60672256958 \beta_{13} + 24634300723 \beta_{12} + \cdots + 406127317026 \beta_{3} \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−6.18666 + 3.57187i
−4.46939 + 2.58040i
−4.14633 + 2.39389i
−1.34450 + 0.776250i
1.34450 − 0.776250i
4.14633 − 2.39389i
4.46939 − 2.58040i
6.18666 − 3.57187i
−6.18666 − 3.57187i
−4.46939 − 2.58040i
−4.14633 − 2.39389i
−1.34450 − 0.776250i
1.34450 + 0.776250i
4.14633 + 2.39389i
4.46939 + 2.58040i
6.18666 + 3.57187i
−6.18666 + 3.57187i 7.57871 − 4.85420i 17.5165 − 30.3395i −13.6392 + 7.87461i −29.5483 + 57.1014i 0 135.967i 33.8736 − 73.5771i 56.2542 − 97.4351i
116.2 −4.46939 + 2.58040i −8.54954 − 2.81164i 5.31698 − 9.20927i 31.7353 − 18.3224i 45.4664 − 9.49493i 0 − 27.6932i 65.1893 + 48.0765i −94.5584 + 163.780i
116.3 −4.14633 + 2.39389i −2.75140 + 8.56912i 3.46138 − 5.99529i −30.3128 + 17.5011i −9.10526 − 42.1170i 0 − 43.4597i −65.8596 − 47.1542i 83.7912 − 145.131i
116.4 −1.34450 + 0.776250i 7.02686 + 5.62345i −6.79487 + 11.7691i 23.4142 − 13.5182i −13.8128 − 2.10615i 0 − 45.9381i 17.7536 + 79.0304i −20.9870 + 36.3506i
116.5 1.34450 − 0.776250i −8.38348 − 3.27372i −6.79487 + 11.7691i −23.4142 + 13.5182i −13.8128 + 2.10615i 0 45.9381i 59.5656 + 54.8903i −20.9870 + 36.3506i
116.6 4.14633 − 2.39389i −6.04537 + 6.66734i 3.46138 − 5.99529i 30.3128 − 17.5011i −9.10526 + 42.1170i 0 43.4597i −7.90694 − 80.6132i 83.7912 − 145.131i
116.7 4.46939 − 2.58040i 6.70973 + 5.99830i 5.31698 − 9.20927i −31.7353 + 18.3224i 45.4664 + 9.49493i 0 27.6932i 9.04086 + 80.4939i −94.5584 + 163.780i
116.8 6.18666 − 3.57187i 0.414505 − 8.99045i 17.5165 − 30.3395i 13.6392 − 7.87461i −29.5483 − 57.1014i 0 − 135.967i −80.6564 − 7.45317i 56.2542 − 97.4351i
128.1 −6.18666 − 3.57187i 7.57871 + 4.85420i 17.5165 + 30.3395i −13.6392 − 7.87461i −29.5483 − 57.1014i 0 − 135.967i 33.8736 + 73.5771i 56.2542 + 97.4351i
128.2 −4.46939 − 2.58040i −8.54954 + 2.81164i 5.31698 + 9.20927i 31.7353 + 18.3224i 45.4664 + 9.49493i 0 27.6932i 65.1893 − 48.0765i −94.5584 − 163.780i
128.3 −4.14633 − 2.39389i −2.75140 − 8.56912i 3.46138 + 5.99529i −30.3128 − 17.5011i −9.10526 + 42.1170i 0 43.4597i −65.8596 + 47.1542i 83.7912 + 145.131i
128.4 −1.34450 − 0.776250i 7.02686 − 5.62345i −6.79487 − 11.7691i 23.4142 + 13.5182i −13.8128 + 2.10615i 0 45.9381i 17.7536 − 79.0304i −20.9870 − 36.3506i
128.5 1.34450 + 0.776250i −8.38348 + 3.27372i −6.79487 − 11.7691i −23.4142 − 13.5182i −13.8128 − 2.10615i 0 − 45.9381i 59.5656 − 54.8903i −20.9870 − 36.3506i
128.6 4.14633 + 2.39389i −6.04537 − 6.66734i 3.46138 + 5.99529i 30.3128 + 17.5011i −9.10526 − 42.1170i 0 − 43.4597i −7.90694 + 80.6132i 83.7912 + 145.131i
128.7 4.46939 + 2.58040i 6.70973 − 5.99830i 5.31698 + 9.20927i −31.7353 − 18.3224i 45.4664 − 9.49493i 0 − 27.6932i 9.04086 − 80.4939i −94.5584 − 163.780i
128.8 6.18666 + 3.57187i 0.414505 + 8.99045i 17.5165 + 30.3395i 13.6392 + 7.87461i −29.5483 + 57.1014i 0 135.967i −80.6564 + 7.45317i 56.2542 + 97.4351i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.8
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 147.5.h.b 16
3.b odd 2 1 inner 147.5.h.b 16
7.b odd 2 1 21.5.h.b ✓ 16
7.c even 3 1 147.5.b.f 8
7.c even 3 1 inner 147.5.h.b 16
7.d odd 6 1 21.5.h.b ✓ 16
7.d odd 6 1 147.5.b.c 8
21.c even 2 1 21.5.h.b ✓ 16
21.g even 6 1 21.5.h.b ✓ 16
21.g even 6 1 147.5.b.c 8
21.h odd 6 1 147.5.b.f 8
21.h odd 6 1 inner 147.5.h.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.h.b ✓ 16 7.b odd 2 1
21.5.h.b ✓ 16 7.d odd 6 1
21.5.h.b ✓ 16 21.c even 2 1
21.5.h.b ✓ 16 21.g even 6 1
147.5.b.c 8 7.d odd 6 1
147.5.b.c 8 21.g even 6 1
147.5.b.f 8 7.c even 3 1
147.5.b.f 8 21.h odd 6 1
147.5.h.b 16 1.a even 1 1 trivial
147.5.h.b 16 3.b odd 2 1 inner
147.5.h.b 16 7.c even 3 1 inner
147.5.h.b 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{16} - 103 T_{2}^{14} + 7227 T_{2}^{12} - 270898 T_{2}^{10} + 7374256 T_{2}^{8} + \cdots + 5639409216 \) Copy content Toggle raw display
\( T_{13}^{4} - 123T_{13}^{3} - 25046T_{13}^{2} + 1759704T_{13} + 114515728 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 5639409216 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{4} - 123 T^{3} + \cdots + 114515728)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 27\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 43\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 42\!\cdots\!69)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 66\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 27\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 365 T^{3} + \cdots + 885324127312)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 77\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 68\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 30\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 30\!\cdots\!84)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 45\!\cdots\!49)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 51\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 368225255164464)^{4} \) Copy content Toggle raw display
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