↑

Properties

Label 98.13.d.a
Level $98$
Weight $13$
Character orbit 98.d
Analytic conductor $89.571$
Analytic rank $0$
Dimension $16$
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] = [98,13,Mod(19,98)]
 
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([5]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.19");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 98.d (of order \(6\), 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: \(89.5713940931\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} - 2204556 x^{14} - 87623088 x^{13} + 1948666431190 x^{12} + 195028079162640 x^{11} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{8}\cdot 7^{8} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{5} + 5 \beta_{3} - 2 \beta_{2}) q^{3} + 2048 \beta_1 q^{4} + ( - \beta_{7} + 3 \beta_{4} + \cdots - 756) q^{5}+ \cdots + ( - 2 \beta_{15} - 2 \beta_{14} + \cdots + 295281) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{5} + 5 \beta_{3} - 2 \beta_{2}) q^{3} + 2048 \beta_1 q^{4} + ( - \beta_{7} + 3 \beta_{4} + \cdots - 756) q^{5}+ \cdots + (61299 \beta_{14} + \cdots + 245073772161) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16384 q^{4} - 18144 q^{5} + 2362248 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16384 q^{4} - 18144 q^{5} + 2362248 q^{9} + 2290176 q^{10} + 2072088 q^{11} - 27163728 q^{15} - 33554432 q^{16} + 101561040 q^{17} - 4592640 q^{18} - 174931848 q^{19} + 62776320 q^{22} + 25630560 q^{23} - 242221056 q^{24} + 521205808 q^{25} + 1434682368 q^{26} + 532360944 q^{29} - 2151917568 q^{30} - 4583818344 q^{31} - 6054957720 q^{33} - 9675767808 q^{36} + 5764524040 q^{37} + 149506560 q^{38} + 10526083272 q^{39} - 4690280448 q^{40} - 66929432000 q^{43} + 4243636224 q^{44} - 57253352184 q^{45} + 7203213312 q^{46} - 18116171640 q^{47} - 99248080896 q^{50} - 23299256376 q^{51} - 8269578240 q^{52} + 39134161800 q^{53} - 105152205312 q^{54} - 328243960080 q^{57} + 29637396480 q^{58} - 201845459088 q^{59} + 27815657472 q^{60} - 336780254328 q^{61} + 137438953472 q^{64} + 158322703896 q^{65} - 268884080640 q^{66} + 107767119920 q^{67} - 207997009920 q^{68} - 1150259029344 q^{71} - 9405726720 q^{72} + 738414283320 q^{73} + 4902778368 q^{74} - 1537028640000 q^{75} - 786088888320 q^{78} + 227632064768 q^{79} + 76101451776 q^{80} - 391984178400 q^{81} + 302578053120 q^{82} + 710209696080 q^{85} - 38105192448 q^{86} - 1957017683880 q^{87} - 64282951680 q^{88} + 2485007442792 q^{89} - 104982773760 q^{92} - 458668768680 q^{93} + 2021298693120 q^{94} - 186503862960 q^{95} + 496068722688 q^{96} + 3921180354576 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 2204556 x^{14} - 87623088 x^{13} + 1948666431190 x^{12} + 195028079162640 x^{11} + \cdots + 24\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 33\!\cdots\!20 \nu^{15} + \cdots - 28\!\cdots\!00 ) / 33\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 60\!\cdots\!71 \nu^{15} + \cdots - 39\!\cdots\!45 ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 91\!\cdots\!71 \nu^{15} + \cdots + 91\!\cdots\!90 ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!39 \nu^{15} + \cdots - 91\!\cdots\!75 ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!53 \nu^{15} + \cdots + 99\!\cdots\!20 ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 28\!\cdots\!52 \nu^{15} + \cdots - 59\!\cdots\!10 ) / 48\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34\!\cdots\!24 \nu^{15} + \cdots - 10\!\cdots\!90 ) / 48\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28\!\cdots\!11 \nu^{15} + \cdots - 34\!\cdots\!25 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 38\!\cdots\!63 \nu^{15} + \cdots + 36\!\cdots\!05 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 80\!\cdots\!27 \nu^{15} + \cdots + 72\!\cdots\!15 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 89\!\cdots\!77 \nu^{15} + \cdots - 86\!\cdots\!50 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 19\!\cdots\!59 \nu^{15} + \cdots + 16\!\cdots\!30 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!81 \nu^{15} + \cdots - 11\!\cdots\!60 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 15\!\cdots\!93 \nu^{15} + \cdots - 13\!\cdots\!90 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19\!\cdots\!96 \nu^{15} + \cdots + 15\!\cdots\!45 ) / 11\!\cdots\!80 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - 7\beta_{3} + 7\beta_{2} - 3\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 3 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \cdots + 826719 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{15} - 1650 \beta_{14} + 1671 \beta_{13} + 406 \beta_{12} + 1433 \beta_{11} - 1829 \beta_{10} + \cdots + 57968502 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1612 \beta_{15} + 1661672 \beta_{14} - 1666426 \beta_{13} + 1024264 \beta_{12} - 1526536 \beta_{11} + \cdots + 361538501343 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5117320 \beta_{15} - 1361854928 \beta_{14} + 1381505158 \beta_{13} + 261277260 \beta_{12} + \cdots + 19195727425833 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1552323690 \beta_{15} + 925356336552 \beta_{14} - 930028528653 \beta_{13} + 541222171732 \beta_{12} + \cdots + 18\!\cdots\!80 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3783175801132 \beta_{15} - 914572651260432 \beta_{14} + 929352138757983 \beta_{13} + \cdots + 82\!\cdots\!62 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10\!\cdots\!76 \beta_{15} + \cdots + 96\!\cdots\!48 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 25\!\cdots\!92 \beta_{15} + \cdots - 49\!\cdots\!21 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 58\!\cdots\!70 \beta_{15} + \cdots + 53\!\cdots\!07 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 17\!\cdots\!90 \beta_{15} + \cdots - 56\!\cdots\!58 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 29\!\cdots\!12 \beta_{15} + \cdots + 29\!\cdots\!09 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11\!\cdots\!12 \beta_{15} + \cdots - 46\!\cdots\!34 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 13\!\cdots\!64 \beta_{15} + \cdots + 16\!\cdots\!15 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 70\!\cdots\!78 \beta_{15} + \cdots - 34\!\cdots\!12 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
738.105 + 0.866025i
468.905 + 0.866025i
−272.613 + 0.866025i
−496.819 + 0.866025i
657.612 + 0.866025i
−22.0174 + 0.866025i
−295.555 + 0.866025i
−773.618 + 0.866025i
738.105 − 0.866025i
468.905 − 0.866025i
−272.613 − 0.866025i
−496.819 − 0.866025i
657.612 − 0.866025i
−22.0174 − 0.866025i
−295.555 − 0.866025i
−773.618 − 0.866025i
−22.6274 + 39.1918i −1106.41 + 638.785i −1024.00 − 1773.62i −24805.4 − 14321.4i − 57816.2i 0 92681.9 550372. − 953273.i 1.12257e6 − 648114.i
19.2 −22.6274 + 39.1918i −702.607 + 405.650i −1024.00 − 1773.62i 18305.2 + 10568.5i − 36715.3i 0 92681.9 63384.0 − 109784.i −828399. + 478276.i
19.3 −22.6274 + 39.1918i 409.670 − 236.523i −1024.00 − 1773.62i −18219.6 − 10519.1i 21407.6i 0 92681.9 −153834. + 266449.i 824525. − 476040.i
19.4 −22.6274 + 39.1918i 745.979 − 430.691i −1024.00 − 1773.62i 7532.28 + 4348.76i 38981.7i 0 92681.9 105269. − 182331.i −340872. + 196803.i
19.5 22.6274 − 39.1918i −985.668 + 569.076i −1024.00 − 1773.62i 12133.2 + 7005.11i 51506.9i 0 −92681.9 381974. − 661598.i 549087. − 317015.i
19.6 22.6274 − 39.1918i 33.7761 − 19.5006i −1024.00 − 1773.62i −2481.87 − 1432.91i − 1764.99i 0 −92681.9 −264960. + 458924.i −112317. + 64846.2i
19.7 22.6274 − 39.1918i 444.082 − 256.391i −1024.00 − 1773.62i 12687.9 + 7325.35i − 23205.9i 0 −92681.9 −134248. + 232524.i 574187. − 331507.i
19.8 22.6274 − 39.1918i 1161.18 − 670.406i −1024.00 − 1773.62i −14223.7 − 8212.03i − 60678.2i 0 −92681.9 633167. − 1.09668e6i −643689. + 371634.i
31.1 −22.6274 − 39.1918i −1106.41 − 638.785i −1024.00 + 1773.62i −24805.4 + 14321.4i 57816.2i 0 92681.9 550372. + 953273.i 1.12257e6 + 648114.i
31.2 −22.6274 − 39.1918i −702.607 − 405.650i −1024.00 + 1773.62i 18305.2 − 10568.5i 36715.3i 0 92681.9 63384.0 + 109784.i −828399. − 478276.i
31.3 −22.6274 − 39.1918i 409.670 + 236.523i −1024.00 + 1773.62i −18219.6 + 10519.1i − 21407.6i 0 92681.9 −153834. − 266449.i 824525. + 476040.i
31.4 −22.6274 − 39.1918i 745.979 + 430.691i −1024.00 + 1773.62i 7532.28 − 4348.76i − 38981.7i 0 92681.9 105269. + 182331.i −340872. − 196803.i
31.5 22.6274 + 39.1918i −985.668 − 569.076i −1024.00 + 1773.62i 12133.2 − 7005.11i − 51506.9i 0 −92681.9 381974. + 661598.i 549087. + 317015.i
31.6 22.6274 + 39.1918i 33.7761 + 19.5006i −1024.00 + 1773.62i −2481.87 + 1432.91i 1764.99i 0 −92681.9 −264960. − 458924.i −112317. − 64846.2i
31.7 22.6274 + 39.1918i 444.082 + 256.391i −1024.00 + 1773.62i 12687.9 − 7325.35i 23205.9i 0 −92681.9 −134248. − 232524.i 574187. + 331507.i
31.8 22.6274 + 39.1918i 1161.18 + 670.406i −1024.00 + 1773.62i −14223.7 + 8212.03i 60678.2i 0 −92681.9 633167. + 1.09668e6i −643689. − 371634.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.13.d.a 16
7.b odd 2 1 14.13.d.a ✓ 16
7.c even 3 1 14.13.d.a ✓ 16
7.c even 3 1 98.13.b.c 16
7.d odd 6 1 98.13.b.c 16
7.d odd 6 1 inner 98.13.d.a 16
21.c even 2 1 126.13.n.a 16
21.h odd 6 1 126.13.n.a 16
28.d even 2 1 112.13.s.c 16
28.g odd 6 1 112.13.s.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.13.d.a ✓ 16 7.b odd 2 1
14.13.d.a ✓ 16 7.c even 3 1
98.13.b.c 16 7.c even 3 1
98.13.b.c 16 7.d odd 6 1
98.13.d.a 16 1.a even 1 1 trivial
98.13.d.a 16 7.d odd 6 1 inner
112.13.s.c 16 28.d even 2 1
112.13.s.c 16 28.g odd 6 1
126.13.n.a 16 21.c even 2 1
126.13.n.a 16 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 3306888 T_{3}^{14} + 7637014892322 T_{3}^{12} - 276426941444640 T_{3}^{11} + \cdots + 16\!\cdots\!41 \) acting on \(S_{13}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2048 T^{2} + 4194304)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 16\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 29\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 73\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 72\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 45\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 24\!\cdots\!84)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 41\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 90\!\cdots\!01 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 53\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 19\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 56\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 97\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 25\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 80\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 68\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 78\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
show more
show less