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Properties

Label 147.8.e.l
Level $147$
Weight $8$
Character orbit 147.e
Analytic conductor $45.921$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,8,Mod(67,147)]
 
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([0, 4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.67");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 147.e (of order \(3\), 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: \(45.9205987462\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 181x^{6} - 1656x^{5} - 6927x^{4} - 5022x^{3} - 974106x^{2} + 14431284x + 117021996 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{5}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 4 \beta_1 + 4) q^{2} - 27 \beta_1 q^{3} + ( - \beta_{5} + 3 \beta_{3} + \cdots - 110 \beta_1) q^{4}+ \cdots + (729 \beta_1 - 729) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 4 \beta_1 + 4) q^{2} - 27 \beta_1 q^{3} + ( - \beta_{5} + 3 \beta_{3} + \cdots - 110 \beta_1) q^{4}+ \cdots + (10935 \beta_{6} - 5103 \beta_{4} + \cdots - 372519) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 15 q^{2} - 108 q^{3} - 437 q^{4} - 504 q^{5} - 810 q^{6} - 4290 q^{8} - 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 15 q^{2} - 108 q^{3} - 437 q^{4} - 504 q^{5} - 810 q^{6} - 4290 q^{8} - 2916 q^{9} + 5724 q^{10} + 1920 q^{11} - 11799 q^{12} + 36288 q^{13} + 27216 q^{15} - 52529 q^{16} + 19584 q^{17} + 10935 q^{18} - 31320 q^{19} + 339624 q^{20} + 223880 q^{22} - 101160 q^{23} + 57915 q^{24} - 74788 q^{25} + 107820 q^{26} + 157464 q^{27} + 389664 q^{29} + 154548 q^{30} - 78840 q^{31} + 732585 q^{32} + 51840 q^{33} - 155520 q^{34} + 637146 q^{36} - 128640 q^{37} - 716292 q^{38} - 489888 q^{39} + 2649780 q^{40} + 730080 q^{41} + 899040 q^{43} - 113220 q^{44} - 367416 q^{45} - 1033664 q^{46} + 1575792 q^{47} + 2836566 q^{48} - 10783866 q^{50} + 528768 q^{51} - 5747652 q^{52} + 1448160 q^{53} + 295245 q^{54} + 6166800 q^{55} + 1691280 q^{57} + 2570950 q^{58} - 3280320 q^{59} - 4584924 q^{60} - 606960 q^{61} + 24129072 q^{62} + 5218274 q^{64} - 1318464 q^{65} - 3022380 q^{66} - 3492880 q^{67} + 1476432 q^{68} + 5462640 q^{69} + 1968 q^{71} + 1563705 q^{72} - 10981440 q^{73} + 14177022 q^{74} - 2019276 q^{75} + 37928520 q^{76} - 5822280 q^{78} - 4654544 q^{79} - 37026324 q^{80} - 2125764 q^{81} + 10402560 q^{82} + 16252992 q^{83} + 33584112 q^{85} + 6392244 q^{86} - 5260464 q^{87} + 10447140 q^{88} + 11272320 q^{89} - 8345592 q^{90} + 6578880 q^{92} - 2128680 q^{93} - 37697400 q^{94} - 11132208 q^{95} + 19779795 q^{96} + 13144896 q^{97} - 2799360 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 181x^{6} - 1656x^{5} - 6927x^{4} - 5022x^{3} - 974106x^{2} + 14431284x + 117021996 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 25059185 \nu^{7} + 108747808 \nu^{6} - 5617275632 \nu^{5} + 65009823678 \nu^{4} + \cdots - 259361458146126 ) / 221945762174598 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 437745646 \nu^{7} - 6282593783 \nu^{6} - 63426488729 \nu^{5} - 615039630675 \nu^{4} + \cdots - 15\!\cdots\!84 ) / 22\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11221432713 \nu^{7} - 68116573765 \nu^{6} + 2696798507969 \nu^{5} - 33879956437576 \nu^{4} + \cdots + 23\!\cdots\!56 ) / 12\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 715663987 \nu^{7} - 21507406346 \nu^{6} + 63421202737 \nu^{5} - 3799713860178 \nu^{4} + \cdots + 28\!\cdots\!22 ) / 379338671952009 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 185020841309 \nu^{7} - 1527110359795 \nu^{6} + 58930909012805 \nu^{5} + \cdots + 17\!\cdots\!46 ) / 19\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31544025160 \nu^{7} + 90319016705 \nu^{6} - 5057516969179 \nu^{5} + \cdots - 33\!\cdots\!26 ) / 22\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 331354851584 \nu^{7} + 4957556385286 \nu^{6} - 90523623551666 \nu^{5} + \cdots - 74\!\cdots\!42 ) / 16\!\cdots\!22 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + 4\beta_{6} + \beta_{5} + 2\beta_{4} + 82\beta_{3} + 44\beta_{2} - 20\beta _1 + 40 ) / 126 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 11\beta_{7} + 7\beta_{6} - 5\beta_{5} + 77\beta_{4} - 305\beta_{3} - 466\beta_{2} - 5087\beta _1 - 3017 ) / 126 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 112 \beta_{7} + 25 \beta_{6} + 28 \beta_{5} - 103 \beta_{4} - 13412 \beta_{3} - 73 \beta_{2} + \cdots + 102625 ) / 126 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3089 \beta_{7} + 1732 \beta_{6} + 9031 \beta_{5} - 1990 \beta_{4} + 97669 \beta_{3} + 120275 \beta_{2} + \cdots + 1148800 ) / 126 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 47369 \beta_{7} + 51469 \beta_{6} + 70987 \beta_{5} + 116717 \beta_{4} + 1693405 \beta_{3} + \cdots - 18431867 ) / 126 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 111136 \beta_{7} - 76763 \beta_{6} - 1499696 \beta_{5} + 109175 \beta_{4} - 32225168 \beta_{3} + \cdots + 22212817 ) / 126 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 952765 \beta_{7} - 6997316 \beta_{6} + 250249 \beta_{5} - 22581442 \beta_{4} - 162265157 \beta_{3} + \cdots + 4599936112 ) / 126 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−2.82369 − 14.4305i
9.87242 + 2.98493i
−6.68281 + 0.882730i
0.634077 + 10.5628i
−2.82369 + 14.4305i
9.87242 − 2.98493i
−6.68281 − 0.882730i
0.634077 − 10.5628i
−9.08530 − 15.7362i −13.5000 + 23.3827i −101.085 + 175.085i −51.7027 − 89.5517i 490.606 0 1347.73 −364.500 − 631.333i −939.469 + 1627.21i
67.2 −0.351190 − 0.608279i −13.5000 + 23.3827i 63.7533 − 110.424i −84.1641 − 145.777i 18.9643 0 −179.463 −364.500 − 631.333i −59.1152 + 102.391i
67.3 6.10587 + 10.5757i −13.5000 + 23.3827i −10.5634 + 18.2963i 142.335 + 246.532i −329.717 0 1305.11 −364.500 − 631.333i −1738.16 + 3010.58i
67.4 10.8306 + 18.7592i −13.5000 + 23.3827i −170.605 + 295.496i −258.468 − 447.680i −584.853 0 −4618.37 −364.500 − 631.333i 5598.75 − 9697.31i
79.1 −9.08530 + 15.7362i −13.5000 − 23.3827i −101.085 − 175.085i −51.7027 + 89.5517i 490.606 0 1347.73 −364.500 + 631.333i −939.469 − 1627.21i
79.2 −0.351190 + 0.608279i −13.5000 − 23.3827i 63.7533 + 110.424i −84.1641 + 145.777i 18.9643 0 −179.463 −364.500 + 631.333i −59.1152 − 102.391i
79.3 6.10587 − 10.5757i −13.5000 − 23.3827i −10.5634 − 18.2963i 142.335 − 246.532i −329.717 0 1305.11 −364.500 + 631.333i −1738.16 − 3010.58i
79.4 10.8306 − 18.7592i −13.5000 − 23.3827i −170.605 − 295.496i −258.468 + 447.680i −584.853 0 −4618.37 −364.500 + 631.333i 5598.75 + 9697.31i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.8.e.l 8
7.b odd 2 1 147.8.e.m 8
7.c even 3 1 147.8.a.g yes 4
7.c even 3 1 inner 147.8.e.l 8
7.d odd 6 1 147.8.a.f ✓ 4
7.d odd 6 1 147.8.e.m 8
21.g even 6 1 441.8.a.v 4
21.h odd 6 1 441.8.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.8.a.f ✓ 4 7.d odd 6 1
147.8.a.g yes 4 7.c even 3 1
147.8.e.l 8 1.a even 1 1 trivial
147.8.e.l 8 7.c even 3 1 inner
147.8.e.m 8 7.b odd 2 1
147.8.e.m 8 7.d odd 6 1
441.8.a.u 4 21.h odd 6 1
441.8.a.v 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{8} - 15 T_{2}^{7} + 587 T_{2}^{6} - 3690 T_{2}^{5} + 196068 T_{2}^{4} - 1549440 T_{2}^{3} + \cdots + 11397376 \) Copy content Toggle raw display
\( T_{5}^{8} + 504 T_{5}^{7} + 320652 T_{5}^{6} + 38304576 T_{5}^{5} + 25117829136 T_{5}^{4} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 15 T^{7} + \cdots + 11397376 \) Copy content Toggle raw display
$3$ \( (T^{2} + 27 T + 729)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 48\!\cdots\!64)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 96\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 22\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 47\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 79\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 61\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 84\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 40\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 73\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 39\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 13\!\cdots\!36)^{2} \) Copy content Toggle raw display
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