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Properties

Label 147.7.f.b
Level $147$
Weight $7$
Character orbit 147.f
Analytic conductor $33.818$
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] = [147,7,Mod(19,147)]
 
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, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.19");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 147.f (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: \(33.8179502921\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 212x^{6} - 1123x^{5} + 44168x^{4} - 138697x^{3} + 660109x^{2} + 680340x + 1040400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 - 1) q^{2} + ( - 9 \beta_{2} - 18) q^{3} + ( - \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{4}+ \cdots + (243 \beta_{2} + 243) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 - 1) q^{2} + ( - 9 \beta_{2} - 18) q^{3} + ( - \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{4}+ \cdots + (2187 \beta_{7} - 2187 \beta_{6} + \cdots + 207522) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{2} - 108 q^{3} - 173 q^{4} + 252 q^{5} - 1462 q^{8} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{2} - 108 q^{3} - 173 q^{4} + 252 q^{5} - 1462 q^{8} + 972 q^{9} + 4968 q^{10} + 3424 q^{11} + 4671 q^{12} - 4536 q^{15} - 14233 q^{16} + 5724 q^{17} + 1215 q^{18} - 10044 q^{19} - 7764 q^{22} + 12160 q^{23} + 19737 q^{24} + 38528 q^{25} - 70668 q^{26} + 60496 q^{29} - 44712 q^{30} - 119664 q^{31} - 2541 q^{32} - 92448 q^{33} - 84078 q^{36} + 19556 q^{37} - 108 q^{38} - 54432 q^{39} - 702324 q^{40} - 29272 q^{43} + 288942 q^{44} + 61236 q^{45} - 282486 q^{46} - 536904 q^{47} + 240154 q^{50} - 51516 q^{51} + 758484 q^{52} - 116144 q^{53} - 32805 q^{54} + 180792 q^{57} - 493842 q^{58} + 1007712 q^{59} + 187596 q^{60} - 794232 q^{61} - 690734 q^{64} + 418152 q^{65} + 104814 q^{66} + 1081924 q^{67} - 3256200 q^{68} - 506288 q^{71} - 177633 q^{72} - 450360 q^{73} - 256162 q^{74} - 1040256 q^{75} + 1272024 q^{78} + 46636 q^{79} - 49860 q^{80} - 236196 q^{81} - 1842372 q^{82} + 3740760 q^{85} - 1923226 q^{86} - 816696 q^{87} + 1038774 q^{88} + 237708 q^{89} - 9701580 q^{92} + 1076976 q^{93} + 2465856 q^{94} - 2077416 q^{95} + 68607 q^{96} + 1664064 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 212x^{6} - 1123x^{5} + 44168x^{4} - 138697x^{3} + 660109x^{2} + 680340x + 1040400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2099431151 \nu^{7} + 5187479451 \nu^{6} - 437949954552 \nu^{5} + 2987637671073 \nu^{4} + \cdots - 14\!\cdots\!40 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9082495 \nu^{7} - 20968969 \nu^{6} - 1852872025 \nu^{5} + 4069000805 \nu^{4} + \cdots - 6424259322060 ) / 6112279547031 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 113740445949 \nu^{7} - 641823046151 \nu^{6} - 24449220456858 \nu^{5} + \cdots - 32\!\cdots\!80 ) / 31\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11344105922 \nu^{7} - 60684979362 \nu^{6} + 2512890111134 \nu^{5} - 22453446873951 \nu^{4} + \cdots - 44\!\cdots\!90 ) / 222661612070415 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 188888287451 \nu^{7} - 56863000929 \nu^{6} - 37288475416142 \nu^{5} + \cdots - 29\!\cdots\!20 ) / 31\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 243335464867 \nu^{7} - 202814560837 \nu^{6} + 53668160863684 \nu^{5} + \cdots - 74\!\cdots\!40 ) / 31\!\cdots\!10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + 3\beta_{3} + 106\beta_{2} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{6} + 3\beta_{4} - 196\beta_{3} + 390 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -208\beta_{7} + 211\beta_{6} + 3\beta_{5} - 211\beta_{4} - 20956\beta_{2} + 1104\beta _1 - 20956 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 636\beta_{7} - 1092\beta_{6} + 1092\beta_{5} - 636\beta_{4} + 41233\beta_{3} + 132036\beta_{2} - 41233\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 43045\beta_{7} + 732\beta_{6} - 43045\beta_{5} + 43777\beta_{4} - 319383\beta_{3} + 4441690 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 322311 \beta_{7} + 193908 \beta_{6} - 128403 \beta_{5} - 193908 \beta_{4} - 36997758 \beta_{2} + \cdots - 36997758 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
6.44130 − 11.1567i
2.28617 − 3.95977i
−0.564972 + 0.978561i
−7.66250 + 13.2718i
6.44130 + 11.1567i
2.28617 + 3.95977i
−0.564972 − 0.978561i
−7.66250 − 13.2718i
−6.94130 + 12.0227i −13.5000 + 7.79423i −64.3633 − 111.480i −94.6080 − 54.6220i − 216.408i 0 898.572 121.500 − 210.444i 1313.41 − 758.295i
19.2 −2.78617 + 4.82579i −13.5000 + 7.79423i 16.4745 + 28.5346i 178.117 + 102.836i − 86.8643i 0 −540.233 121.500 − 210.444i −992.530 + 573.038i
19.3 0.0649724 − 0.112536i −13.5000 + 7.79423i 31.9916 + 55.4110i −109.506 − 63.2233i 2.02564i 0 16.6307 121.500 − 210.444i −14.2297 + 8.21554i
19.4 7.16250 − 12.4058i −13.5000 + 7.79423i −70.6028 − 122.288i 151.997 + 87.7554i 223.305i 0 −1105.97 121.500 − 210.444i 2177.35 − 1257.10i
31.1 −6.94130 − 12.0227i −13.5000 − 7.79423i −64.3633 + 111.480i −94.6080 + 54.6220i 216.408i 0 898.572 121.500 + 210.444i 1313.41 + 758.295i
31.2 −2.78617 − 4.82579i −13.5000 − 7.79423i 16.4745 − 28.5346i 178.117 − 102.836i 86.8643i 0 −540.233 121.500 + 210.444i −992.530 − 573.038i
31.3 0.0649724 + 0.112536i −13.5000 − 7.79423i 31.9916 − 55.4110i −109.506 + 63.2233i − 2.02564i 0 16.6307 121.500 + 210.444i −14.2297 − 8.21554i
31.4 7.16250 + 12.4058i −13.5000 − 7.79423i −70.6028 + 122.288i 151.997 − 87.7554i − 223.305i 0 −1105.97 121.500 + 210.444i 2177.35 + 1257.10i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.7.f.b 8
7.b odd 2 1 147.7.f.c 8
7.c even 3 1 21.7.d.a ✓ 8
7.c even 3 1 147.7.f.c 8
7.d odd 6 1 21.7.d.a ✓ 8
7.d odd 6 1 inner 147.7.f.b 8
21.g even 6 1 63.7.d.f 8
21.h odd 6 1 63.7.d.f 8
28.f even 6 1 336.7.f.a 8
28.g odd 6 1 336.7.f.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.d.a ✓ 8 7.c even 3 1
21.7.d.a ✓ 8 7.d odd 6 1
63.7.d.f 8 21.g even 6 1
63.7.d.f 8 21.h odd 6 1
147.7.f.b 8 1.a even 1 1 trivial
147.7.f.b 8 7.d odd 6 1 inner
147.7.f.c 8 7.b odd 2 1
147.7.f.c 8 7.c even 3 1
336.7.f.a 8 28.f even 6 1
336.7.f.a 8 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{8} + 5T_{2}^{7} + 227T_{2}^{6} + 1154T_{2}^{5} + 46070T_{2}^{4} + 217124T_{2}^{3} + 1199812T_{2}^{2} - 155808T_{2} + 20736 \) Copy content Toggle raw display
\( T_{5}^{8} - 252 T_{5}^{7} - 18762 T_{5}^{6} + 10062360 T_{5}^{5} + 624024900 T_{5}^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{7} + \cdots + 20736 \) Copy content Toggle raw display
$3$ \( (T^{2} + 27 T + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 32\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 94\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 55\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 76\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 36\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 39\!\cdots\!44 \) Copy content Toggle raw display
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