Properties

Label 147.8.e.a
Level $147$
Weight $8$
Character orbit 147.e
Analytic conductor $45.921$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 6 \zeta_{6} q^{2} + (27 \zeta_{6} - 27) q^{3} + ( - 92 \zeta_{6} + 92) q^{4} + 390 \zeta_{6} q^{5} + 162 q^{6} - 1320 q^{8} - 729 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 6 \zeta_{6} q^{2} + (27 \zeta_{6} - 27) q^{3} + ( - 92 \zeta_{6} + 92) q^{4} + 390 \zeta_{6} q^{5} + 162 q^{6} - 1320 q^{8} - 729 \zeta_{6} q^{9} + ( - 2340 \zeta_{6} + 2340) q^{10} + ( - 948 \zeta_{6} + 948) q^{11} + 2484 \zeta_{6} q^{12} + 5098 q^{13} - 10530 q^{15} - 3856 \zeta_{6} q^{16} + ( - 28386 \zeta_{6} + 28386) q^{17} + (4374 \zeta_{6} - 4374) q^{18} - 8620 \zeta_{6} q^{19} + 35880 q^{20} - 5688 q^{22} + 15288 \zeta_{6} q^{23} + ( - 35640 \zeta_{6} + 35640) q^{24} + (73975 \zeta_{6} - 73975) q^{25} - 30588 \zeta_{6} q^{26} + 19683 q^{27} + 36510 q^{29} + 63180 \zeta_{6} q^{30} + (276808 \zeta_{6} - 276808) q^{31} + (192096 \zeta_{6} - 192096) q^{32} + 25596 \zeta_{6} q^{33} - 170316 q^{34} - 67068 q^{36} - 268526 \zeta_{6} q^{37} + (51720 \zeta_{6} - 51720) q^{38} + (137646 \zeta_{6} - 137646) q^{39} - 514800 \zeta_{6} q^{40} + 629718 q^{41} + 685772 q^{43} - 87216 \zeta_{6} q^{44} + ( - 284310 \zeta_{6} + 284310) q^{45} + ( - 91728 \zeta_{6} + 91728) q^{46} + 583296 \zeta_{6} q^{47} + 104112 q^{48} + 443850 q^{50} + 766422 \zeta_{6} q^{51} + ( - 469016 \zeta_{6} + 469016) q^{52} + ( - 428058 \zeta_{6} + 428058) q^{53} - 118098 \zeta_{6} q^{54} + 369720 q^{55} + 232740 q^{57} - 219060 \zeta_{6} q^{58} + ( - 1306380 \zeta_{6} + 1306380) q^{59} + (968760 \zeta_{6} - 968760) q^{60} + 300662 \zeta_{6} q^{61} + 1660848 q^{62} + 659008 q^{64} + 1988220 \zeta_{6} q^{65} + ( - 153576 \zeta_{6} + 153576) q^{66} + ( - 507244 \zeta_{6} + 507244) q^{67} - 2611512 \zeta_{6} q^{68} - 412776 q^{69} + 5560632 q^{71} + 962280 \zeta_{6} q^{72} + ( - 1369082 \zeta_{6} + 1369082) q^{73} + (1611156 \zeta_{6} - 1611156) q^{74} - 1997325 \zeta_{6} q^{75} - 793040 q^{76} + 825876 q^{78} + 6913720 \zeta_{6} q^{79} + ( - 1503840 \zeta_{6} + 1503840) q^{80} + (531441 \zeta_{6} - 531441) q^{81} - 3778308 \zeta_{6} q^{82} + 4376748 q^{83} + 11070540 q^{85} - 4114632 \zeta_{6} q^{86} + (985770 \zeta_{6} - 985770) q^{87} + (1251360 \zeta_{6} - 1251360) q^{88} - 8528310 \zeta_{6} q^{89} - 1705860 q^{90} + 1406496 q^{92} - 7473816 \zeta_{6} q^{93} + ( - 3499776 \zeta_{6} + 3499776) q^{94} + ( - 3361800 \zeta_{6} + 3361800) q^{95} - 5186592 \zeta_{6} q^{96} + 8826814 q^{97} - 691092 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} - 27 q^{3} + 92 q^{4} + 390 q^{5} + 324 q^{6} - 2640 q^{8} - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{2} - 27 q^{3} + 92 q^{4} + 390 q^{5} + 324 q^{6} - 2640 q^{8} - 729 q^{9} + 2340 q^{10} + 948 q^{11} + 2484 q^{12} + 10196 q^{13} - 21060 q^{15} - 3856 q^{16} + 28386 q^{17} - 4374 q^{18} - 8620 q^{19} + 71760 q^{20} - 11376 q^{22} + 15288 q^{23} + 35640 q^{24} - 73975 q^{25} - 30588 q^{26} + 39366 q^{27} + 73020 q^{29} + 63180 q^{30} - 276808 q^{31} - 192096 q^{32} + 25596 q^{33} - 340632 q^{34} - 134136 q^{36} - 268526 q^{37} - 51720 q^{38} - 137646 q^{39} - 514800 q^{40} + 1259436 q^{41} + 1371544 q^{43} - 87216 q^{44} + 284310 q^{45} + 91728 q^{46} + 583296 q^{47} + 208224 q^{48} + 887700 q^{50} + 766422 q^{51} + 469016 q^{52} + 428058 q^{53} - 118098 q^{54} + 739440 q^{55} + 465480 q^{57} - 219060 q^{58} + 1306380 q^{59} - 968760 q^{60} + 300662 q^{61} + 3321696 q^{62} + 1318016 q^{64} + 1988220 q^{65} + 153576 q^{66} + 507244 q^{67} - 2611512 q^{68} - 825552 q^{69} + 11121264 q^{71} + 962280 q^{72} + 1369082 q^{73} - 1611156 q^{74} - 1997325 q^{75} - 1586080 q^{76} + 1651752 q^{78} + 6913720 q^{79} + 1503840 q^{80} - 531441 q^{81} - 3778308 q^{82} + 8753496 q^{83} + 22141080 q^{85} - 4114632 q^{86} - 985770 q^{87} - 1251360 q^{88} - 8528310 q^{89} - 3411720 q^{90} + 2812992 q^{92} - 7473816 q^{93} + 3499776 q^{94} + 3361800 q^{95} - 5186592 q^{96} + 17653628 q^{97} - 1382184 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 0.866025i
−3.00000 5.19615i −13.5000 + 23.3827i 46.0000 79.6743i 195.000 + 337.750i 162.000 0 −1320.00 −364.500 631.333i 1170.00 2026.50i
79.1 −3.00000 + 5.19615i −13.5000 23.3827i 46.0000 + 79.6743i 195.000 337.750i 162.000 0 −1320.00 −364.500 + 631.333i 1170.00 + 2026.50i
\(n\): e.g. 2-40 or 990-1000
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.a 2
7.b odd 2 1 147.8.e.b 2
7.c even 3 1 147.8.a.b 1
7.c even 3 1 inner 147.8.e.a 2
7.d odd 6 1 3.8.a.a 1
7.d odd 6 1 147.8.e.b 2
21.g even 6 1 9.8.a.a 1
21.h odd 6 1 441.8.a.a 1
28.f even 6 1 48.8.a.g 1
35.i odd 6 1 75.8.a.a 1
35.k even 12 2 75.8.b.c 2
56.j odd 6 1 192.8.a.i 1
56.m even 6 1 192.8.a.a 1
63.i even 6 1 81.8.c.c 2
63.k odd 6 1 81.8.c.a 2
63.s even 6 1 81.8.c.c 2
63.t odd 6 1 81.8.c.a 2
77.i even 6 1 363.8.a.b 1
84.j odd 6 1 144.8.a.b 1
91.s odd 6 1 507.8.a.a 1
105.p even 6 1 225.8.a.i 1
105.w odd 12 2 225.8.b.f 2
168.ba even 6 1 576.8.a.w 1
168.be odd 6 1 576.8.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.8.a.a 1 7.d odd 6 1
9.8.a.a 1 21.g even 6 1
48.8.a.g 1 28.f even 6 1
75.8.a.a 1 35.i odd 6 1
75.8.b.c 2 35.k even 12 2
81.8.c.a 2 63.k odd 6 1
81.8.c.a 2 63.t odd 6 1
81.8.c.c 2 63.i even 6 1
81.8.c.c 2 63.s even 6 1
144.8.a.b 1 84.j odd 6 1
147.8.a.b 1 7.c even 3 1
147.8.e.a 2 1.a even 1 1 trivial
147.8.e.a 2 7.c even 3 1 inner
147.8.e.b 2 7.b odd 2 1
147.8.e.b 2 7.d odd 6 1
192.8.a.a 1 56.m even 6 1
192.8.a.i 1 56.j odd 6 1
225.8.a.i 1 105.p even 6 1
225.8.b.f 2 105.w odd 12 2
363.8.a.b 1 77.i even 6 1
441.8.a.a 1 21.h odd 6 1
507.8.a.a 1 91.s odd 6 1
576.8.a.w 1 168.ba even 6 1
576.8.a.x 1 168.be odd 6 1

Hecke kernels

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

\( T_{2}^{2} + 6T_{2} + 36 \) Copy content Toggle raw display
\( T_{5}^{2} - 390T_{5} + 152100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$3$ \( T^{2} + 27T + 729 \) Copy content Toggle raw display
$5$ \( T^{2} - 390T + 152100 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 948T + 898704 \) Copy content Toggle raw display
$13$ \( (T - 5098)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 28386 T + 805764996 \) Copy content Toggle raw display
$19$ \( T^{2} + 8620 T + 74304400 \) Copy content Toggle raw display
$23$ \( T^{2} - 15288 T + 233722944 \) Copy content Toggle raw display
$29$ \( (T - 36510)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 276808 T + 76622668864 \) Copy content Toggle raw display
$37$ \( T^{2} + 268526 T + 72106212676 \) Copy content Toggle raw display
$41$ \( (T - 629718)^{2} \) Copy content Toggle raw display
$43$ \( (T - 685772)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 583296 T + 340234223616 \) Copy content Toggle raw display
$53$ \( T^{2} - 428058 T + 183233651364 \) Copy content Toggle raw display
$59$ \( T^{2} - 1306380 T + 1706628704400 \) Copy content Toggle raw display
$61$ \( T^{2} - 300662 T + 90397638244 \) Copy content Toggle raw display
$67$ \( T^{2} - 507244 T + 257296475536 \) Copy content Toggle raw display
$71$ \( (T - 5560632)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 1369082 T + 1874385522724 \) Copy content Toggle raw display
$79$ \( T^{2} - 6913720 T + 47799524238400 \) Copy content Toggle raw display
$83$ \( (T - 4376748)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 8528310 T + 72732071456100 \) Copy content Toggle raw display
$97$ \( (T - 8826814)^{2} \) Copy content Toggle raw display
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