Properties

Label 98.16.c.m
Level $98$
Weight $16$
Character orbit 98.c
Analytic conductor $139.840$
Analytic rank $0$
Dimension $10$
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] = [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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 11041039 x^{8} + 10251836788 x^{7} + 100086056086567 x^{6} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{4}\cdot 5^{2}\cdot 7^{8} \)
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 basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 128 \beta_{2} - 128) q^{2} + (\beta_{3} + 396 \beta_{2} + \beta_1) q^{3} + 16384 \beta_{2} q^{4} + (\beta_{4} - 60192 \beta_{2} - 7 \beta_1 - 60192) q^{5} + ( - 128 \beta_{3} + 50688) q^{6} + 2097152 q^{8} + (\beta_{9} - 5 \beta_{8} + 48 \beta_{4} - 3473847 \beta_{2} + \cdots - 3473847) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 128 \beta_{2} - 128) q^{2} + (\beta_{3} + 396 \beta_{2} + \beta_1) q^{3} + 16384 \beta_{2} q^{4} + (\beta_{4} - 60192 \beta_{2} - 7 \beta_1 - 60192) q^{5} + ( - 128 \beta_{3} + 50688) q^{6} + 2097152 q^{8} + (\beta_{9} - 5 \beta_{8} + 48 \beta_{4} - 3473847 \beta_{2} + \cdots - 3473847) q^{9}+ \cdots + ( - 9757149 \beta_{7} - 190388376 \beta_{6} + \cdots - 99023291730348) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 640 q^{2} - 1979 q^{3} - 81920 q^{4} - 300953 q^{5} + 506624 q^{6} + 20971520 q^{8} - 17367046 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 640 q^{2} - 1979 q^{3} - 81920 q^{4} - 300953 q^{5} + 506624 q^{6} + 20971520 q^{8} - 17367046 q^{9} - 38521984 q^{10} + 14189887 q^{11} - 32423936 q^{12} + 772602348 q^{13} + 1426775646 q^{15} - 1342177280 q^{16} + 145564295 q^{17} - 2222981888 q^{18} - 3550131629 q^{19} + 9861627904 q^{20} - 3632611072 q^{22} + 9869524537 q^{23} - 4150263808 q^{24} - 51839554204 q^{25} - 49446550272 q^{26} + 342946209778 q^{27} - 165255956188 q^{29} - 91313641344 q^{30} + 106825666677 q^{31} - 171798691840 q^{32} + 139826141649 q^{33} - 37264459520 q^{34} + 569083363328 q^{36} - 596799401515 q^{37} - 454416848512 q^{38} - 2827795559666 q^{39} - 631144185856 q^{40} - 1756981189740 q^{41} - 5305116572344 q^{43} + 232487108608 q^{44} - 10836341974062 q^{45} + 1263299140736 q^{46} - 4284391851525 q^{47} + 1062467534848 q^{48} + 13270925876224 q^{50} - 8298359375823 q^{51} - 6329158434816 q^{52} - 4037801378823 q^{53} - 21948557425792 q^{54} + 47232884322970 q^{55} - 75972269020058 q^{57} + 10576381196032 q^{58} - 6081930248483 q^{59} - 11688146092032 q^{60} - 29484338931189 q^{61} - 27347370669312 q^{62} + 43980465111040 q^{64} + 78038741223618 q^{65} + 17897746131072 q^{66} - 2851190345117 q^{67} + 2384925409280 q^{68} + 41705305747074 q^{69} + 190169727933856 q^{71} - 36421335252992 q^{72} - 284202883487269 q^{73} - 76390323393920 q^{74} + 151851953188876 q^{75} + 116330713219072 q^{76} + 723915663274496 q^{78} + 536529323488905 q^{79} - 80786455789568 q^{80} + 23023662902123 q^{81} + 112446796143360 q^{82} + 949641950127416 q^{83} + 24\!\cdots\!86 q^{85}+ \cdots - 989855214604404 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 11041039 x^{8} + 10251836788 x^{7} + 100086056086567 x^{6} + \cdots + 12\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31\!\cdots\!51 \nu^{9} + \cdots - 70\!\cdots\!25 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 31\!\cdots\!51 \nu^{9} + \cdots - 66\!\cdots\!75 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22\!\cdots\!69 \nu^{9} + \cdots + 52\!\cdots\!75 ) / 50\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 34\!\cdots\!78 \nu^{9} + \cdots - 29\!\cdots\!00 ) / 44\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 83\!\cdots\!97 \nu^{9} + \cdots + 12\!\cdots\!25 ) / 88\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!11 \nu^{9} + \cdots - 11\!\cdots\!25 ) / 60\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!87 \nu^{9} + \cdots - 40\!\cdots\!75 ) / 60\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 84\!\cdots\!11 \nu^{9} + \cdots - 25\!\cdots\!25 ) / 88\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 37\!\cdots\!79 \nu^{9} + \cdots + 11\!\cdots\!25 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + 5\beta_{8} - \beta_{7} - 5\beta_{6} + 48\beta_{5} + 1408\beta_{3} + 17665939\beta_{2} + 1408\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6772\beta_{7} - 21805\beta_{6} + 65454\beta_{5} + 65454\beta_{4} + 28389097\beta_{3} - 24663073015 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3162531 \beta_{9} - 99020985 \beta_{8} + 1029376683 \beta_{4} - 250596026708216 \beta_{2} - 41678035415 \beta _1 - 250596026708216 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 64533809240 \beta_{9} - 292096583735 \beta_{8} - 64533809240 \beta_{7} + 292096583735 \beta_{6} - 1215766884720 \beta_{5} + \cdots - 240699192189409 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 43465737693923 \beta_{7} + 987326574800540 \beta_{6} + \cdots + 21\!\cdots\!23 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 57\!\cdots\!87 \beta_{9} + \cdots + 43\!\cdots\!05 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 80\!\cdots\!79 \beta_{9} + \cdots + 55\!\cdots\!60 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 51\!\cdots\!35 \beta_{7} + \cdots - 48\!\cdots\!54 ) / 8 \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
1594.44 + 2761.64i
778.235 + 1347.94i
−253.251 438.643i
−879.415 1523.19i
−1239.01 2146.02i
1594.44 2761.64i
778.235 1347.94i
−253.251 + 438.643i
−879.415 + 1523.19i
−1239.01 + 2146.02i
−64.0000 110.851i −3386.37 + 5865.37i −8192.00 + 14189.0i −127571. 220960.i 866911. 0 2.09715e6 −1.57606e7 2.72981e7i −1.63291e7 + 2.82829e7i
67.2 −64.0000 110.851i −1753.97 + 3037.97i −8192.00 + 14189.0i 36165.1 + 62639.7i 449016. 0 2.09715e6 1.02163e6 + 1.76951e6i 4.62913e6 8.01788e6i
67.3 −64.0000 110.851i 309.001 535.206i −8192.00 + 14189.0i −8985.53 15563.4i −79104.3 0 2.09715e6 6.98349e6 + 1.20958e7i −1.15015e6 + 1.99212e6i
67.4 −64.0000 110.851i 1561.33 2704.30i −8192.00 + 14189.0i 101823. + 176362.i −399701. 0 2.09715e6 2.29895e6 + 3.98189e6i 1.30333e7 2.25744e7i
67.5 −64.0000 110.851i 2280.51 3949.96i −8192.00 + 14189.0i −151908. 263112.i −583811. 0 2.09715e6 −3.22701e6 5.58934e6i −1.94442e7 + 3.36783e7i
79.1 −64.0000 + 110.851i −3386.37 5865.37i −8192.00 14189.0i −127571. + 220960.i 866911. 0 2.09715e6 −1.57606e7 + 2.72981e7i −1.63291e7 2.82829e7i
79.2 −64.0000 + 110.851i −1753.97 3037.97i −8192.00 14189.0i 36165.1 62639.7i 449016. 0 2.09715e6 1.02163e6 1.76951e6i 4.62913e6 + 8.01788e6i
79.3 −64.0000 + 110.851i 309.001 + 535.206i −8192.00 14189.0i −8985.53 + 15563.4i −79104.3 0 2.09715e6 6.98349e6 1.20958e7i −1.15015e6 1.99212e6i
79.4 −64.0000 + 110.851i 1561.33 + 2704.30i −8192.00 14189.0i 101823. 176362.i −399701. 0 2.09715e6 2.29895e6 3.98189e6i 1.30333e7 + 2.25744e7i
79.5 −64.0000 + 110.851i 2280.51 + 3949.96i −8192.00 14189.0i −151908. + 263112.i −583811. 0 2.09715e6 −3.22701e6 + 5.58934e6i −1.94442e7 3.36783e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.5
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.m 10
7.b odd 2 1 14.16.c.a 10
7.c even 3 1 98.16.a.k 5
7.c even 3 1 inner 98.16.c.m 10
7.d odd 6 1 14.16.c.a 10
7.d odd 6 1 98.16.a.j 5
21.c even 2 1 126.16.g.e 10
21.g even 6 1 126.16.g.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.16.c.a 10 7.b odd 2 1
14.16.c.a 10 7.d odd 6 1
98.16.a.j 5 7.d odd 6 1
98.16.a.k 5 7.c even 3 1
98.16.c.m 10 1.a even 1 1 trivial
98.16.c.m 10 7.c even 3 1 inner
126.16.g.e 10 21.c even 2 1
126.16.g.e 10 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 1979 T_{3}^{9} + 46514011 T_{3}^{8} - 62709703674 T_{3}^{7} + \cdots + 43\!\cdots\!25 \) acting on \(S_{16}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 128 T + 16384)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + 1979 T^{9} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{10} + 300953 T^{9} + \cdots + 42\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 14189887 T^{9} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{5} - 386301174 T^{4} + \cdots - 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} - 145564295 T^{9} + \cdots + 42\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{10} + 3550131629 T^{9} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{10} - 9869524537 T^{9} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{5} + 82627978094 T^{4} + \cdots - 36\!\cdots\!72)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 106825666677 T^{9} + \cdots + 69\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{10} + 596799401515 T^{9} + \cdots + 39\!\cdots\!09 \) Copy content Toggle raw display
$41$ \( (T^{5} + 878490594870 T^{4} + \cdots - 58\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 2652558286172 T^{4} + \cdots - 29\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 4284391851525 T^{9} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{10} + 4037801378823 T^{9} + \cdots + 24\!\cdots\!29 \) Copy content Toggle raw display
$59$ \( T^{10} + 6081930248483 T^{9} + \cdots + 76\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{10} + 29484338931189 T^{9} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{10} + 2851190345117 T^{9} + \cdots + 61\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{5} - 95084863966928 T^{4} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 284202883487269 T^{9} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{10} - 536529323488905 T^{9} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{5} - 474820975063708 T^{4} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 160148042721333 T^{9} + \cdots + 85\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
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