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Properties

Label 147.5.h.e
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: \( x^{16} - 82 x^{14} + 4707 x^{12} - 139354 x^{10} + 2999893 x^{8} - 26137356 x^{6} + 167995548 x^{4} + \cdots + 571536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{7}\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{11} - \beta_{7}) q^{3} + (\beta_{10} - 5 \beta_{6} + \beta_{2} - 1) q^{4} - \beta_{5} q^{5} + ( - \beta_{15} - \beta_{14} - \beta_{8} + \cdots - 4) q^{6}+ \cdots + ( - \beta_{13} - \beta_{12} + \beta_{10} + \cdots - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{11} - \beta_{7}) q^{3} + (\beta_{10} - 5 \beta_{6} + \beta_{2} - 1) q^{4} - \beta_{5} q^{5} + ( - \beta_{15} - \beta_{14} - \beta_{8} + \cdots - 4) q^{6}+ \cdots + (187 \beta_{15} + 202 \beta_{14} + \cdots - 4141) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{3} + 36 q^{4} - 68 q^{6} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{3} + 36 q^{4} - 68 q^{6} - 64 q^{9} + 4 q^{10} - 98 q^{12} + 840 q^{13} + 152 q^{15} + 444 q^{16} + 712 q^{18} + 372 q^{19} - 32 q^{22} - 1146 q^{24} - 1056 q^{25} - 3724 q^{27} - 2348 q^{30} + 2776 q^{31} - 1396 q^{33} + 5856 q^{34} - 6536 q^{36} + 2560 q^{37} + 2540 q^{39} + 1980 q^{40} + 9440 q^{43} - 9700 q^{45} - 7536 q^{46} - 5924 q^{48} - 4764 q^{51} + 20252 q^{52} - 4886 q^{54} + 368 q^{55} - 28288 q^{57} + 7504 q^{58} + 13828 q^{60} - 972 q^{61} + 45544 q^{64} - 36020 q^{66} - 10200 q^{67} - 11520 q^{69} - 14304 q^{72} + 32008 q^{73} - 2114 q^{75} + 34664 q^{76} - 59336 q^{78} + 23168 q^{79} + 17216 q^{81} + 31976 q^{82} + 64032 q^{85} - 50764 q^{87} - 29208 q^{88} - 48704 q^{90} - 31848 q^{93} + 64992 q^{94} - 28630 q^{96} + 56224 q^{97} - 64864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 82 x^{14} + 4707 x^{12} - 139354 x^{10} + 2999893 x^{8} - 26137356 x^{6} + 167995548 x^{4} + \cdots + 571536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5665843765 \nu^{14} + 423436825558 \nu^{12} - 23672201436075 \nu^{10} + \cdots - 13\!\cdots\!44 ) / 66\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 98833 \nu^{14} - 8106206 \nu^{12} + 412929615 \nu^{10} - 10772839630 \nu^{8} + \cdots + 37451109732904 ) / 2022292980928 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 619007382659 \nu^{15} - 50707612784153 \nu^{13} + \cdots - 60\!\cdots\!60 \nu ) / 60\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4034306868397 \nu^{15} - 306581933071846 \nu^{13} + \cdots + 40\!\cdots\!64 \nu ) / 32\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 619007382659 \nu^{14} + 50707612784153 \nu^{12} + \cdots + 54\!\cdots\!04 ) / 60\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 900529175132 \nu^{15} - 14854539277199 \nu^{14} + 67618422541784 \nu^{13} + \cdots + 14\!\cdots\!88 ) / 32\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4230685374940 \nu^{15} + 14854539277199 \nu^{14} + 317767145082136 \nu^{13} + \cdots - 14\!\cdots\!88 ) / 32\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13210266703915 \nu^{14} + \cdots + 11\!\cdots\!24 ) / 68\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4316054147318 \nu^{14} + 353682979012397 \nu^{12} + \cdots + 42\!\cdots\!12 ) / 20\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 321389890624616 \nu^{15} + 589135519629 \nu^{14} + \cdots + 43\!\cdots\!96 ) / 96\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 974073789996392 \nu^{15} + 1767406558887 \nu^{14} + \cdots + 11\!\cdots\!92 ) / 96\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 482984513710102 \nu^{15} - 62777070965567 \nu^{14} + \cdots + 55\!\cdots\!20 ) / 48\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 627384518986495 \nu^{15} + \cdots - 55\!\cdots\!24 \nu ) / 48\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 212887602846142 \nu^{15} - 7525458891871 \nu^{14} + \cdots + 66\!\cdots\!28 ) / 16\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 21\beta_{6} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{12} + 2\beta_{11} - \beta_{8} + 33\beta_{4} + 33\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{13} - 2\beta_{12} + 39\beta_{10} - 4\beta_{9} - 6\beta_{7} - 691\beta_{6} - 4\beta_{3} + 2\beta _1 - 691 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -47\beta_{15} + 45\beta_{13} + 88\beta_{11} - 88\beta_{7} + 45\beta_{6} + 1159\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -96\beta_{12} - 288\beta_{11} + 96\beta_{4} - 176\beta_{3} - 1431\beta_{2} + 96\beta _1 - 22798 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1703 \beta_{13} + 1703 \beta_{12} + 1831 \beta_{8} - 3278 \beta_{7} + 1703 \beta_{6} + 48 \beta_{5} + \cdots + 1703 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3838 \beta_{13} - 11514 \beta_{11} - 51699 \beta_{10} + 6364 \beta_{9} + 11514 \beta_{7} + 865695 \beta_{6} + \cdots + 51699 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 68363 \beta_{15} + 3936 \beta_{14} + 61901 \beta_{12} - 117340 \beta_{11} + 68363 \beta_{8} + \cdots + 61901 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 147124 \beta_{13} + 147124 \beta_{12} - 1860015 \beta_{10} + 218936 \beta_{9} + 441372 \beta_{7} + \cdots + 31098041 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2523823 \beta_{15} + 225936 \beta_{14} - 2226075 \beta_{13} - 4154402 \beta_{11} + \cdots - 53644157 \beta_{4} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5571330 \beta_{12} + 16713990 \beta_{11} - 5571330 \beta_{4} + 7405060 \beta_{3} + 66846483 \beta_{2} + \cdots + 1052025248 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 79822873 \beta_{13} - 79822873 \beta_{12} - 92869403 \beta_{8} + 146599216 \beta_{7} + \cdots - 79822873 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 209998136 \beta_{13} + 629994408 \beta_{11} + 2401800495 \beta_{10} - 248347232 \beta_{9} + \cdots - 2401800495 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3413442079 \beta_{15} - 514947120 \beta_{14} - 2860145863 \beta_{12} + 5166995510 \beta_{11} + \cdots - 2860145863 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{6}\)

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
−5.21989 + 3.01370i
−5.17082 + 2.98538i
−2.73339 + 1.57812i
−0.209634 + 0.121032i
0.209634 − 0.121032i
2.73339 − 1.57812i
5.17082 − 2.98538i
5.21989 − 3.01370i
−5.21989 − 3.01370i
−5.17082 − 2.98538i
−2.73339 − 1.57812i
−0.209634 − 0.121032i
0.209634 + 0.121032i
2.73339 + 1.57812i
5.17082 + 2.98538i
5.21989 + 3.01370i
−5.21989 + 3.01370i 3.43915 − 8.31698i 10.1648 − 17.6060i −13.6433 + 7.87698i 7.11292 + 53.7783i 0 26.0965i −57.3444 − 57.2068i 47.4778 − 82.2339i
116.2 −5.17082 + 2.98538i 3.65772 + 8.22320i 9.82495 − 17.0173i 19.8909 − 11.4840i −43.4628 − 31.6011i 0 21.7927i −54.2421 + 60.1564i −68.5683 + 118.764i
116.3 −2.73339 + 1.57812i −8.34019 + 3.38250i −3.01906 + 5.22917i −14.1475 + 8.16806i 17.4590 − 22.4075i 0 − 69.5577i 58.1174 − 56.4213i 25.7804 − 44.6529i
116.4 −0.209634 + 0.121032i −4.53017 − 7.77673i −7.97070 + 13.8057i 26.4019 − 15.2431i 1.89091 + 1.08197i 0 − 7.73187i −39.9552 + 70.4598i −3.68982 + 6.39095i
116.5 0.209634 − 0.121032i 8.99993 + 0.0348727i −7.97070 + 13.8057i −26.4019 + 15.2431i 1.89091 − 1.08197i 0 7.73187i 80.9976 + 0.627705i −3.68982 + 6.39095i
116.6 2.73339 − 1.57812i 1.24077 + 8.91406i −3.01906 + 5.22917i 14.1475 − 8.16806i 17.4590 + 22.4075i 0 69.5577i −77.9210 + 22.1205i 25.7804 − 44.6529i
116.7 5.17082 − 2.98538i −8.95036 + 0.943921i 9.82495 − 17.0173i −19.8909 + 11.4840i −43.4628 + 31.6011i 0 − 21.7927i 79.2180 − 16.8969i −68.5683 + 118.764i
116.8 5.21989 − 3.01370i 5.48314 − 7.13689i 10.1648 − 17.6060i 13.6433 − 7.87698i 7.11292 − 53.7783i 0 − 26.0965i −20.8703 − 78.2651i 47.4778 − 82.2339i
128.1 −5.21989 − 3.01370i 3.43915 + 8.31698i 10.1648 + 17.6060i −13.6433 − 7.87698i 7.11292 − 53.7783i 0 − 26.0965i −57.3444 + 57.2068i 47.4778 + 82.2339i
128.2 −5.17082 − 2.98538i 3.65772 − 8.22320i 9.82495 + 17.0173i 19.8909 + 11.4840i −43.4628 + 31.6011i 0 − 21.7927i −54.2421 − 60.1564i −68.5683 − 118.764i
128.3 −2.73339 − 1.57812i −8.34019 − 3.38250i −3.01906 − 5.22917i −14.1475 − 8.16806i 17.4590 + 22.4075i 0 69.5577i 58.1174 + 56.4213i 25.7804 + 44.6529i
128.4 −0.209634 − 0.121032i −4.53017 + 7.77673i −7.97070 − 13.8057i 26.4019 + 15.2431i 1.89091 − 1.08197i 0 7.73187i −39.9552 − 70.4598i −3.68982 − 6.39095i
128.5 0.209634 + 0.121032i 8.99993 − 0.0348727i −7.97070 − 13.8057i −26.4019 − 15.2431i 1.89091 + 1.08197i 0 − 7.73187i 80.9976 − 0.627705i −3.68982 − 6.39095i
128.6 2.73339 + 1.57812i 1.24077 − 8.91406i −3.01906 − 5.22917i 14.1475 + 8.16806i 17.4590 − 22.4075i 0 − 69.5577i −77.9210 − 22.1205i 25.7804 + 44.6529i
128.7 5.17082 + 2.98538i −8.95036 − 0.943921i 9.82495 + 17.0173i −19.8909 − 11.4840i −43.4628 − 31.6011i 0 21.7927i 79.2180 + 16.8969i −68.5683 − 118.764i
128.8 5.21989 + 3.01370i 5.48314 + 7.13689i 10.1648 + 17.6060i 13.6433 + 7.87698i 7.11292 + 53.7783i 0 26.0965i −20.8703 + 78.2651i 47.4778 + 82.2339i
\(n\): 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.e 16
3.b odd 2 1 inner 147.5.h.e 16
7.b odd 2 1 147.5.h.c 16
7.c even 3 1 21.5.b.a ✓ 8
7.c even 3 1 inner 147.5.h.e 16
7.d odd 6 1 147.5.b.e 8
7.d odd 6 1 147.5.h.c 16
21.c even 2 1 147.5.h.c 16
21.g even 6 1 147.5.b.e 8
21.g even 6 1 147.5.h.c 16
21.h odd 6 1 21.5.b.a ✓ 8
21.h odd 6 1 inner 147.5.h.e 16
28.g odd 6 1 336.5.d.b 8
84.n even 6 1 336.5.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.b.a ✓ 8 7.c even 3 1
21.5.b.a ✓ 8 21.h odd 6 1
147.5.b.e 8 7.d odd 6 1
147.5.b.e 8 21.g even 6 1
147.5.h.c 16 7.b odd 2 1
147.5.h.c 16 7.d odd 6 1
147.5.h.c 16 21.c even 2 1
147.5.h.c 16 21.g even 6 1
147.5.h.e 16 1.a even 1 1 trivial
147.5.h.e 16 3.b odd 2 1 inner
147.5.h.e 16 7.c even 3 1 inner
147.5.h.e 16 21.h odd 6 1 inner
336.5.d.b 8 28.g odd 6 1
336.5.d.b 8 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 \(S_{5}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{16} - 82 T_{2}^{14} + 4707 T_{2}^{12} - 139354 T_{2}^{10} + 2999893 T_{2}^{8} - 26137356 T_{2}^{6} + \cdots + 571536 \) Copy content Toggle raw display
\( T_{13}^{4} - 210T_{13}^{3} - 50954T_{13}^{2} + 8164128T_{13} + 137696008 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 82 T^{14} + \cdots + 571536 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{4} - 210 T^{3} + \cdots + 137696008)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 33\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 69\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 12\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 55\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 25\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2360 T^{3} + \cdots + 117096670336)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 18\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 28\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 62\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 54\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 90\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 66\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 23\!\cdots\!48)^{4} \) Copy content Toggle raw display
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