Properties

Label 98.6.c.c
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.

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{2} + (10 \zeta_{6} - 10) q^{3} + (16 \zeta_{6} - 16) q^{4} - 84 \zeta_{6} q^{5} - 40 q^{6} - 64 q^{8} + 143 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 \zeta_{6} q^{2} + (10 \zeta_{6} - 10) q^{3} + (16 \zeta_{6} - 16) q^{4} - 84 \zeta_{6} q^{5} - 40 q^{6} - 64 q^{8} + 143 \zeta_{6} q^{9} + ( - 336 \zeta_{6} + 336) q^{10} + ( - 336 \zeta_{6} + 336) q^{11} - 160 \zeta_{6} q^{12} + 584 q^{13} + 840 q^{15} - 256 \zeta_{6} q^{16} + ( - 1458 \zeta_{6} + 1458) q^{17} + (572 \zeta_{6} - 572) q^{18} - 470 \zeta_{6} q^{19} + 1344 q^{20} + 1344 q^{22} + 4200 \zeta_{6} q^{23} + ( - 640 \zeta_{6} + 640) q^{24} + (3931 \zeta_{6} - 3931) q^{25} + 2336 \zeta_{6} q^{26} - 3860 q^{27} + 4866 q^{29} + 3360 \zeta_{6} q^{30} + ( - 7372 \zeta_{6} + 7372) q^{31} + ( - 1024 \zeta_{6} + 1024) q^{32} + 3360 \zeta_{6} q^{33} + 5832 q^{34} - 2288 q^{36} - 14330 \zeta_{6} q^{37} + ( - 1880 \zeta_{6} + 1880) q^{38} + (5840 \zeta_{6} - 5840) q^{39} + 5376 \zeta_{6} q^{40} + 6222 q^{41} + 3704 q^{43} + 5376 \zeta_{6} q^{44} + ( - 12012 \zeta_{6} + 12012) q^{45} + (16800 \zeta_{6} - 16800) q^{46} + 1812 \zeta_{6} q^{47} + 2560 q^{48} - 15724 q^{50} + 14580 \zeta_{6} q^{51} + (9344 \zeta_{6} - 9344) q^{52} + ( - 37242 \zeta_{6} + 37242) q^{53} - 15440 \zeta_{6} q^{54} - 28224 q^{55} + 4700 q^{57} + 19464 \zeta_{6} q^{58} + (34302 \zeta_{6} - 34302) q^{59} + (13440 \zeta_{6} - 13440) q^{60} - 24476 \zeta_{6} q^{61} + 29488 q^{62} + 4096 q^{64} - 49056 \zeta_{6} q^{65} + (13440 \zeta_{6} - 13440) q^{66} + ( - 17452 \zeta_{6} + 17452) q^{67} + 23328 \zeta_{6} q^{68} - 42000 q^{69} + 28224 q^{71} - 9152 \zeta_{6} q^{72} + (3602 \zeta_{6} - 3602) q^{73} + ( - 57320 \zeta_{6} + 57320) q^{74} - 39310 \zeta_{6} q^{75} + 7520 q^{76} - 23360 q^{78} - 42872 \zeta_{6} q^{79} + (21504 \zeta_{6} - 21504) q^{80} + ( - 3851 \zeta_{6} + 3851) q^{81} + 24888 \zeta_{6} q^{82} - 35202 q^{83} - 122472 q^{85} + 14816 \zeta_{6} q^{86} + (48660 \zeta_{6} - 48660) q^{87} + (21504 \zeta_{6} - 21504) q^{88} - 26730 \zeta_{6} q^{89} + 48048 q^{90} - 67200 q^{92} + 73720 \zeta_{6} q^{93} + (7248 \zeta_{6} - 7248) q^{94} + (39480 \zeta_{6} - 39480) q^{95} + 10240 \zeta_{6} q^{96} - 16978 q^{97} + 48048 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 10 q^{3} - 16 q^{4} - 84 q^{5} - 80 q^{6} - 128 q^{8} + 143 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 10 q^{3} - 16 q^{4} - 84 q^{5} - 80 q^{6} - 128 q^{8} + 143 q^{9} + 336 q^{10} + 336 q^{11} - 160 q^{12} + 1168 q^{13} + 1680 q^{15} - 256 q^{16} + 1458 q^{17} - 572 q^{18} - 470 q^{19} + 2688 q^{20} + 2688 q^{22} + 4200 q^{23} + 640 q^{24} - 3931 q^{25} + 2336 q^{26} - 7720 q^{27} + 9732 q^{29} + 3360 q^{30} + 7372 q^{31} + 1024 q^{32} + 3360 q^{33} + 11664 q^{34} - 4576 q^{36} - 14330 q^{37} + 1880 q^{38} - 5840 q^{39} + 5376 q^{40} + 12444 q^{41} + 7408 q^{43} + 5376 q^{44} + 12012 q^{45} - 16800 q^{46} + 1812 q^{47} + 5120 q^{48} - 31448 q^{50} + 14580 q^{51} - 9344 q^{52} + 37242 q^{53} - 15440 q^{54} - 56448 q^{55} + 9400 q^{57} + 19464 q^{58} - 34302 q^{59} - 13440 q^{60} - 24476 q^{61} + 58976 q^{62} + 8192 q^{64} - 49056 q^{65} - 13440 q^{66} + 17452 q^{67} + 23328 q^{68} - 84000 q^{69} + 56448 q^{71} - 9152 q^{72} - 3602 q^{73} + 57320 q^{74} - 39310 q^{75} + 15040 q^{76} - 46720 q^{78} - 42872 q^{79} - 21504 q^{80} + 3851 q^{81} + 24888 q^{82} - 70404 q^{83} - 244944 q^{85} + 14816 q^{86} - 48660 q^{87} - 21504 q^{88} - 26730 q^{89} + 96096 q^{90} - 134400 q^{92} + 73720 q^{93} - 7248 q^{94} - 39480 q^{95} + 10240 q^{96} - 33956 q^{97} + 96096 q^{99}+O(q^{100}) \) 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 −5.00000 + 8.66025i −8.00000 + 13.8564i −42.0000 72.7461i −40.0000 0 −64.0000 71.5000 + 123.842i 168.000 290.985i
79.1 2.00000 3.46410i −5.00000 8.66025i −8.00000 13.8564i −42.0000 + 72.7461i −40.0000 0 −64.0000 71.5000 123.842i 168.000 + 290.985i
\(n\): 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.c 2
7.b odd 2 1 98.6.c.d 2
7.c even 3 1 14.6.a.a 1
7.c even 3 1 inner 98.6.c.c 2
7.d odd 6 1 98.6.a.a 1
7.d odd 6 1 98.6.c.d 2
21.g even 6 1 882.6.a.x 1
21.h odd 6 1 126.6.a.f 1
28.f even 6 1 784.6.a.i 1
28.g odd 6 1 112.6.a.c 1
35.j even 6 1 350.6.a.i 1
35.l odd 12 2 350.6.c.d 2
56.k odd 6 1 448.6.a.l 1
56.p even 6 1 448.6.a.e 1
84.n even 6 1 1008.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.a 1 7.c even 3 1
98.6.a.a 1 7.d odd 6 1
98.6.c.c 2 1.a even 1 1 trivial
98.6.c.c 2 7.c even 3 1 inner
98.6.c.d 2 7.b odd 2 1
98.6.c.d 2 7.d odd 6 1
112.6.a.c 1 28.g odd 6 1
126.6.a.f 1 21.h odd 6 1
350.6.a.i 1 35.j even 6 1
350.6.c.d 2 35.l odd 12 2
448.6.a.e 1 56.p even 6 1
448.6.a.l 1 56.k odd 6 1
784.6.a.i 1 28.f even 6 1
882.6.a.x 1 21.g even 6 1
1008.6.a.b 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 10T_{3} + 100 \) 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
$3$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$5$ \( T^{2} + 84T + 7056 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 336T + 112896 \) Copy content Toggle raw display
$13$ \( (T - 584)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1458 T + 2125764 \) Copy content Toggle raw display
$19$ \( T^{2} + 470T + 220900 \) Copy content Toggle raw display
$23$ \( T^{2} - 4200 T + 17640000 \) Copy content Toggle raw display
$29$ \( (T - 4866)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 7372 T + 54346384 \) Copy content Toggle raw display
$37$ \( T^{2} + 14330 T + 205348900 \) Copy content Toggle raw display
$41$ \( (T - 6222)^{2} \) Copy content Toggle raw display
$43$ \( (T - 3704)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 1812 T + 3283344 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 1386966564 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 1176627204 \) Copy content Toggle raw display
$61$ \( T^{2} + 24476 T + 599074576 \) Copy content Toggle raw display
$67$ \( T^{2} - 17452 T + 304572304 \) Copy content Toggle raw display
$71$ \( (T - 28224)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3602 T + 12974404 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 1838008384 \) Copy content Toggle raw display
$83$ \( (T + 35202)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 26730 T + 714492900 \) Copy content Toggle raw display
$97$ \( (T + 16978)^{2} \) Copy content Toggle raw display
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