Properties

Label 98.16.c.d
Level $98$
Weight $16$
Character orbit 98.c
Analytic conductor $139.840$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,16,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, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.67");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 16 \)
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: \(139.839634998\)
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 2)
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.

\(f(q)\) \(=\) \( q + 128 \zeta_{6} q^{2} + ( - 6252 \zeta_{6} + 6252) q^{3} + (16384 \zeta_{6} - 16384) q^{4} + 90510 \zeta_{6} q^{5} + 800256 q^{6} - 2097152 q^{8} - 24738597 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 128 \zeta_{6} q^{2} + ( - 6252 \zeta_{6} + 6252) q^{3} + (16384 \zeta_{6} - 16384) q^{4} + 90510 \zeta_{6} q^{5} + 800256 q^{6} - 2097152 q^{8} - 24738597 \zeta_{6} q^{9} + (11585280 \zeta_{6} - 11585280) q^{10} + ( - 95889948 \zeta_{6} + 95889948) q^{11} + 102432768 \zeta_{6} q^{12} + 59782138 q^{13} + 565868520 q^{15} - 268435456 \zeta_{6} q^{16} + (1355814414 \zeta_{6} - 1355814414) q^{17} + ( - 3166540416 \zeta_{6} + 3166540416) q^{18} + 3783593180 \zeta_{6} q^{19} - 1482915840 q^{20} + 12273913344 q^{22} + 11608845528 \zeta_{6} q^{23} + (13111394304 \zeta_{6} - 13111394304) q^{24} + ( - 22325518025 \zeta_{6} + 22325518025) q^{25} + 7652113664 \zeta_{6} q^{26} - 64956341880 q^{27} - 28959105930 q^{29} + 72431170560 \zeta_{6} q^{30} + ( - 253685353952 \zeta_{6} + 253685353952) q^{31} + ( - 34359738368 \zeta_{6} + 34359738368) q^{32} - 599503954896 \zeta_{6} q^{33} - 173544244992 q^{34} + 405317173248 q^{36} - 817641294446 \zeta_{6} q^{37} + (484299927040 \zeta_{6} - 484299927040) q^{38} + ( - 373757926776 \zeta_{6} + 373757926776) q^{39} - 189813227520 \zeta_{6} q^{40} + 682333284198 q^{41} + 366945604292 q^{43} + 1571060908032 \zeta_{6} q^{44} + ( - 2239090414470 \zeta_{6} + 2239090414470) q^{45} + (1485932227584 \zeta_{6} - 1485932227584) q^{46} + 695741581776 \zeta_{6} q^{47} - 1678258470912 q^{48} + 2857666307200 q^{50} + 8476551716328 \zeta_{6} q^{51} + (979470548992 \zeta_{6} - 979470548992) q^{52} + (12993372468702 \zeta_{6} - 12993372468702) q^{53} - 8314411760640 \zeta_{6} q^{54} + 8678999193480 q^{55} + 23655024561360 q^{57} - 3706765559040 \zeta_{6} q^{58} + ( - 9209035340340 \zeta_{6} + 9209035340340) q^{59} + (9271189831680 \zeta_{6} - 9271189831680) q^{60} - 42338641200298 \zeta_{6} q^{61} + 32471725305856 q^{62} + 4398046511104 q^{64} + 5410881310380 \zeta_{6} q^{65} + ( - 76736506226688 \zeta_{6} + 76736506226688) q^{66} + (30029787950636 \zeta_{6} - 30029787950636) q^{67} - 22213663358976 \zeta_{6} q^{68} + 72578502241056 q^{69} + 115328696975352 q^{71} + 51880598175744 \zeta_{6} q^{72} + ( - 43787346432122 \zeta_{6} + 43787346432122) q^{73} + ( - 104658085689088 \zeta_{6} + 104658085689088) q^{74} - 139579138692300 \zeta_{6} q^{75} - 61990390661120 q^{76} + 47841014627328 q^{78} - 79603813043120 \zeta_{6} q^{79} + ( - 24296093122560 \zeta_{6} + 24296093122560) q^{80} + (51135221770281 \zeta_{6} - 51135221770281) q^{81} + 87338660377344 \zeta_{6} q^{82} + 3417068864868 q^{83} - 122714762611140 q^{85} + 46969037349376 \zeta_{6} q^{86} + (181052330274360 \zeta_{6} - 181052330274360) q^{87} + (201095796228096 \zeta_{6} - 201095796228096) q^{88} - 377306179184790 \zeta_{6} q^{89} + 286603573052160 q^{90} - 190199325130752 q^{92} - 15\!\cdots\!04 \zeta_{6} q^{93} + \cdots - 23\!\cdots\!56 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 128 q^{2} + 6252 q^{3} - 16384 q^{4} + 90510 q^{5} + 1600512 q^{6} - 4194304 q^{8} - 24738597 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 128 q^{2} + 6252 q^{3} - 16384 q^{4} + 90510 q^{5} + 1600512 q^{6} - 4194304 q^{8} - 24738597 q^{9} - 11585280 q^{10} + 95889948 q^{11} + 102432768 q^{12} + 119564276 q^{13} + 1131737040 q^{15} - 268435456 q^{16} - 1355814414 q^{17} + 3166540416 q^{18} + 3783593180 q^{19} - 2965831680 q^{20} + 24547826688 q^{22} + 11608845528 q^{23} - 13111394304 q^{24} + 22325518025 q^{25} + 7652113664 q^{26} - 129912683760 q^{27} - 57918211860 q^{29} + 72431170560 q^{30} + 253685353952 q^{31} + 34359738368 q^{32} - 599503954896 q^{33} - 347088489984 q^{34} + 810634346496 q^{36} - 817641294446 q^{37} - 484299927040 q^{38} + 373757926776 q^{39} - 189813227520 q^{40} + 1364666568396 q^{41} + 733891208584 q^{43} + 1571060908032 q^{44} + 2239090414470 q^{45} - 1485932227584 q^{46} + 695741581776 q^{47} - 3356516941824 q^{48} + 5715332614400 q^{50} + 8476551716328 q^{51} - 979470548992 q^{52} - 12993372468702 q^{53} - 8314411760640 q^{54} + 17357998386960 q^{55} + 47310049122720 q^{57} - 3706765559040 q^{58} + 9209035340340 q^{59} - 9271189831680 q^{60} - 42338641200298 q^{61} + 64943450611712 q^{62} + 8796093022208 q^{64} + 5410881310380 q^{65} + 76736506226688 q^{66} - 30029787950636 q^{67} - 22213663358976 q^{68} + 145157004482112 q^{69} + 230657393950704 q^{71} + 51880598175744 q^{72} + 43787346432122 q^{73} + 104658085689088 q^{74} - 139579138692300 q^{75} - 123980781322240 q^{76} + 95682029254656 q^{78} - 79603813043120 q^{79} + 24296093122560 q^{80} - 51135221770281 q^{81} + 87338660377344 q^{82} + 6834137729736 q^{83} - 245429525222280 q^{85} + 46969037349376 q^{86} - 181052330274360 q^{87} - 201095796228096 q^{88} - 377306179184790 q^{89} + 573207146104320 q^{90} - 380398650261504 q^{92} - 15\!\cdots\!04 q^{93}+ \cdots - 47\!\cdots\!12 q^{99}+O(q^{100}) \) 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
64.0000 + 110.851i 3126.00 5414.39i −8192.00 + 14189.0i 45255.0 + 78384.0i 800256. 0 −2.09715e6 −1.23693e7 2.14243e7i −5.79264e6 + 1.00331e7i
79.1 64.0000 110.851i 3126.00 + 5414.39i −8192.00 14189.0i 45255.0 78384.0i 800256. 0 −2.09715e6 −1.23693e7 + 2.14243e7i −5.79264e6 1.00331e7i
\(n\): 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 98.16.c.d 2
7.b odd 2 1 98.16.c.a 2
7.c even 3 1 98.16.a.a 1
7.c even 3 1 inner 98.16.c.d 2
7.d odd 6 1 2.16.a.a 1
7.d odd 6 1 98.16.c.a 2
21.g even 6 1 18.16.a.e 1
28.f even 6 1 16.16.a.a 1
35.i odd 6 1 50.16.a.b 1
35.k even 12 2 50.16.b.a 2
56.j odd 6 1 64.16.a.a 1
56.m even 6 1 64.16.a.k 1
84.j odd 6 1 144.16.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.16.a.a 1 7.d odd 6 1
16.16.a.a 1 28.f even 6 1
18.16.a.e 1 21.g even 6 1
50.16.a.b 1 35.i odd 6 1
50.16.b.a 2 35.k even 12 2
64.16.a.a 1 56.j odd 6 1
64.16.a.k 1 56.m even 6 1
98.16.a.a 1 7.c even 3 1
98.16.c.a 2 7.b odd 2 1
98.16.c.a 2 7.d odd 6 1
98.16.c.d 2 1.a even 1 1 trivial
98.16.c.d 2 7.c even 3 1 inner
144.16.a.d 1 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 6252T_{3} + 39087504 \) acting on \(S_{16}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 128T + 16384 \) Copy content Toggle raw display
$3$ \( T^{2} - 6252 T + 39087504 \) Copy content Toggle raw display
$5$ \( T^{2} - 90510 T + 8192060100 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 95889948 T + 91\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T - 59782138)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 1355814414 T + 18\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{2} - 3783593180 T + 14\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} - 11608845528 T + 13\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T + 28959105930)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 253685353952 T + 64\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{2} + 817641294446 T + 66\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T - 682333284198)^{2} \) Copy content Toggle raw display
$43$ \( (T - 366945604292)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 695741581776 T + 48\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{2} + 12993372468702 T + 16\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} - 9209035340340 T + 84\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + 42338641200298 T + 17\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{2} + 30029787950636 T + 90\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T - 115328696975352)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 43787346432122 T + 19\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{2} + 79603813043120 T + 63\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T - 3417068864868)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 377306179184790 T + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T - 166982186657374)^{2} \) Copy content Toggle raw display
show more
show less