Properties

Label 98.6.c.b
Level $98$
Weight $6$
Character orbit 98.c
Analytic conductor $15.718$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,6,Mod(67,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.67");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 98.c (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: \(15.7176143417\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q - 4 \zeta_{6} q^{2} + ( - 8 \zeta_{6} + 8) q^{3} + (16 \zeta_{6} - 16) q^{4} + 10 \zeta_{6} q^{5} - 32 q^{6} + 64 q^{8} + 179 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 4 \zeta_{6} q^{2} + ( - 8 \zeta_{6} + 8) q^{3} + (16 \zeta_{6} - 16) q^{4} + 10 \zeta_{6} q^{5} - 32 q^{6} + 64 q^{8} + 179 \zeta_{6} q^{9} + ( - 40 \zeta_{6} + 40) q^{10} + ( - 340 \zeta_{6} + 340) q^{11} + 128 \zeta_{6} q^{12} + 294 q^{13} + 80 q^{15} - 256 \zeta_{6} q^{16} + ( - 1226 \zeta_{6} + 1226) q^{17} + ( - 716 \zeta_{6} + 716) q^{18} + 2432 \zeta_{6} q^{19} - 160 q^{20} - 1360 q^{22} - 2000 \zeta_{6} q^{23} + ( - 512 \zeta_{6} + 512) q^{24} + ( - 3025 \zeta_{6} + 3025) q^{25} - 1176 \zeta_{6} q^{26} + 3376 q^{27} - 6746 q^{29} - 320 \zeta_{6} q^{30} + ( - 8856 \zeta_{6} + 8856) q^{31} + (1024 \zeta_{6} - 1024) q^{32} - 2720 \zeta_{6} q^{33} - 4904 q^{34} - 2864 q^{36} - 9182 \zeta_{6} q^{37} + ( - 9728 \zeta_{6} + 9728) q^{38} + ( - 2352 \zeta_{6} + 2352) q^{39} + 640 \zeta_{6} q^{40} + 14574 q^{41} + 8108 q^{43} + 5440 \zeta_{6} q^{44} + (1790 \zeta_{6} - 1790) q^{45} + (8000 \zeta_{6} - 8000) q^{46} - 312 \zeta_{6} q^{47} - 2048 q^{48} - 12100 q^{50} - 9808 \zeta_{6} q^{51} + (4704 \zeta_{6} - 4704) q^{52} + ( - 14634 \zeta_{6} + 14634) q^{53} - 13504 \zeta_{6} q^{54} + 3400 q^{55} + 19456 q^{57} + 26984 \zeta_{6} q^{58} + (27656 \zeta_{6} - 27656) q^{59} + (1280 \zeta_{6} - 1280) q^{60} + 34338 \zeta_{6} q^{61} - 35424 q^{62} + 4096 q^{64} + 2940 \zeta_{6} q^{65} + (10880 \zeta_{6} - 10880) q^{66} + (12316 \zeta_{6} - 12316) q^{67} + 19616 \zeta_{6} q^{68} - 16000 q^{69} + 36920 q^{71} + 11456 \zeta_{6} q^{72} + (61718 \zeta_{6} - 61718) q^{73} + (36728 \zeta_{6} - 36728) q^{74} - 24200 \zeta_{6} q^{75} - 38912 q^{76} - 9408 q^{78} + 64752 \zeta_{6} q^{79} + ( - 2560 \zeta_{6} + 2560) q^{80} + (16489 \zeta_{6} - 16489) q^{81} - 58296 \zeta_{6} q^{82} + 77056 q^{83} + 12260 q^{85} - 32432 \zeta_{6} q^{86} + (53968 \zeta_{6} - 53968) q^{87} + ( - 21760 \zeta_{6} + 21760) q^{88} - 8166 \zeta_{6} q^{89} + 7160 q^{90} + 32000 q^{92} - 70848 \zeta_{6} q^{93} + (1248 \zeta_{6} - 1248) q^{94} + (24320 \zeta_{6} - 24320) q^{95} + 8192 \zeta_{6} q^{96} - 20650 q^{97} + 60860 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{3} - 16 q^{4} + 10 q^{5} - 64 q^{6} + 128 q^{8} + 179 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 8 q^{3} - 16 q^{4} + 10 q^{5} - 64 q^{6} + 128 q^{8} + 179 q^{9} + 40 q^{10} + 340 q^{11} + 128 q^{12} + 588 q^{13} + 160 q^{15} - 256 q^{16} + 1226 q^{17} + 716 q^{18} + 2432 q^{19} - 320 q^{20} - 2720 q^{22} - 2000 q^{23} + 512 q^{24} + 3025 q^{25} - 1176 q^{26} + 6752 q^{27} - 13492 q^{29} - 320 q^{30} + 8856 q^{31} - 1024 q^{32} - 2720 q^{33} - 9808 q^{34} - 5728 q^{36} - 9182 q^{37} + 9728 q^{38} + 2352 q^{39} + 640 q^{40} + 29148 q^{41} + 16216 q^{43} + 5440 q^{44} - 1790 q^{45} - 8000 q^{46} - 312 q^{47} - 4096 q^{48} - 24200 q^{50} - 9808 q^{51} - 4704 q^{52} + 14634 q^{53} - 13504 q^{54} + 6800 q^{55} + 38912 q^{57} + 26984 q^{58} - 27656 q^{59} - 1280 q^{60} + 34338 q^{61} - 70848 q^{62} + 8192 q^{64} + 2940 q^{65} - 10880 q^{66} - 12316 q^{67} + 19616 q^{68} - 32000 q^{69} + 73840 q^{71} + 11456 q^{72} - 61718 q^{73} - 36728 q^{74} - 24200 q^{75} - 77824 q^{76} - 18816 q^{78} + 64752 q^{79} + 2560 q^{80} - 16489 q^{81} - 58296 q^{82} + 154112 q^{83} + 24520 q^{85} - 32432 q^{86} - 53968 q^{87} + 21760 q^{88} - 8166 q^{89} + 14320 q^{90} + 64000 q^{92} - 70848 q^{93} - 1248 q^{94} - 24320 q^{95} + 8192 q^{96} - 41300 q^{97} + 121720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\zeta_{6}\)

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
−2.00000 3.46410i 4.00000 6.92820i −8.00000 + 13.8564i 5.00000 + 8.66025i −32.0000 0 64.0000 89.5000 + 155.019i 20.0000 34.6410i
79.1 −2.00000 + 3.46410i 4.00000 + 6.92820i −8.00000 13.8564i 5.00000 8.66025i −32.0000 0 64.0000 89.5000 155.019i 20.0000 + 34.6410i
\(n\): 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 98.6.c.b 2
7.b odd 2 1 98.6.c.a 2
7.c even 3 1 98.6.a.b 1
7.c even 3 1 inner 98.6.c.b 2
7.d odd 6 1 14.6.a.b 1
7.d odd 6 1 98.6.c.a 2
21.g even 6 1 126.6.a.c 1
21.h odd 6 1 882.6.a.g 1
28.f even 6 1 112.6.a.d 1
28.g odd 6 1 784.6.a.h 1
35.i odd 6 1 350.6.a.b 1
35.k even 12 2 350.6.c.f 2
56.j odd 6 1 448.6.a.f 1
56.m even 6 1 448.6.a.k 1
84.j odd 6 1 1008.6.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 7.d odd 6 1
98.6.a.b 1 7.c even 3 1
98.6.c.a 2 7.b odd 2 1
98.6.c.a 2 7.d odd 6 1
98.6.c.b 2 1.a even 1 1 trivial
98.6.c.b 2 7.c even 3 1 inner
112.6.a.d 1 28.f even 6 1
126.6.a.c 1 21.g even 6 1
350.6.a.b 1 35.i odd 6 1
350.6.c.f 2 35.k even 12 2
448.6.a.f 1 56.j odd 6 1
448.6.a.k 1 56.m even 6 1
784.6.a.h 1 28.g odd 6 1
882.6.a.g 1 21.h odd 6 1
1008.6.a.n 1 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8T_{3} + 64 \) acting on \(S_{6}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$5$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 340T + 115600 \) Copy content Toggle raw display
$13$ \( (T - 294)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1226 T + 1503076 \) Copy content Toggle raw display
$19$ \( T^{2} - 2432 T + 5914624 \) Copy content Toggle raw display
$23$ \( T^{2} + 2000 T + 4000000 \) Copy content Toggle raw display
$29$ \( (T + 6746)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8856 T + 78428736 \) Copy content Toggle raw display
$37$ \( T^{2} + 9182 T + 84309124 \) Copy content Toggle raw display
$41$ \( (T - 14574)^{2} \) Copy content Toggle raw display
$43$ \( (T - 8108)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 312T + 97344 \) Copy content Toggle raw display
$53$ \( T^{2} - 14634 T + 214153956 \) Copy content Toggle raw display
$59$ \( T^{2} + 27656 T + 764854336 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 1179098244 \) Copy content Toggle raw display
$67$ \( T^{2} + 12316 T + 151683856 \) Copy content Toggle raw display
$71$ \( (T - 36920)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 3809111524 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 4192821504 \) Copy content Toggle raw display
$83$ \( (T - 77056)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 8166 T + 66683556 \) Copy content Toggle raw display
$97$ \( (T + 20650)^{2} \) Copy content Toggle raw display
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