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Properties

Label 98.13.d.b
Level $98$
Weight $13$
Character orbit 98.d
Analytic conductor $89.571$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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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: \( x^{16} + 51570 x^{14} + 1743306357 x^{12} + 34303771893750 x^{10} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{54}\cdot 3^{8}\cdot 7^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{10} + \beta_{9}) q^{3} + 2048 \beta_1 q^{4} + ( - \beta_{12} + 11 \beta_{9}) q^{5} + (\beta_{15} + 2 \beta_{11} - 5 \beta_{10}) q^{6} + (2048 \beta_{3} - 2048 \beta_{2}) q^{8} + (\beta_{6} + 5 \beta_{5} + \cdots + 184863) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{10} + \beta_{9}) q^{3} + 2048 \beta_1 q^{4} + ( - \beta_{12} + 11 \beta_{9}) q^{5} + (\beta_{15} + 2 \beta_{11} - 5 \beta_{10}) q^{6} + (2048 \beta_{3} - 2048 \beta_{2}) q^{8} + (\beta_{6} + 5 \beta_{5} + \cdots + 184863) q^{9}+ \cdots + (4168263 \beta_{7} + \cdots - 225119282058) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16384 q^{4} + 1478904 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16384 q^{4} + 1478904 q^{9} + 213840 q^{11} + 131764608 q^{15} - 33554432 q^{16} - 32547840 q^{18} - 442675200 q^{22} - 156731760 q^{23} - 191237000 q^{25} + 617707296 q^{29} + 2203567104 q^{30} - 6057590784 q^{36} + 3243600880 q^{37} - 13521315264 q^{39} + 42012604000 q^{43} + 437944320 q^{44} - 9664610304 q^{46} + 52518979584 q^{50} + 80965832832 q^{51} - 180445637520 q^{53} - 126291924480 q^{57} + 94193264640 q^{58} - 134926958592 q^{60} + 137438953472 q^{64} + 424890168192 q^{65} - 369211259440 q^{67} + 1148116288608 q^{71} - 66657976320 q^{72} - 450517137408 q^{74} - 502001694720 q^{78} + 607826610128 q^{79} - 919051941384 q^{81} - 494404521216 q^{85} + 413092638720 q^{86} + 453299404800 q^{88} + 641973288960 q^{92} - 2292312458880 q^{93} + 1053641981376 q^{95} - 3601908512928 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 51570 x^{14} + 1743306357 x^{12} + 34303771893750 x^{10} + \cdots + 24\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 86\!\cdots\!38 \nu^{14} + \cdots + 28\!\cdots\!54 ) / 38\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58\!\cdots\!28 \nu^{14} + \cdots + 67\!\cdots\!64 ) / 21\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\!\cdots\!97 \nu^{14} + \cdots + 57\!\cdots\!16 ) / 21\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 57\!\cdots\!67 \nu^{14} + \cdots + 10\!\cdots\!12 ) / 66\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16\!\cdots\!57 \nu^{14} + \cdots - 66\!\cdots\!28 ) / 41\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 60\!\cdots\!72 \nu^{14} + \cdots - 86\!\cdots\!28 ) / 66\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!30 \nu^{14} + \cdots - 90\!\cdots\!44 ) / 66\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27\!\cdots\!77 \nu^{15} + \cdots + 15\!\cdots\!56 \nu ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 76\!\cdots\!67 \nu^{15} + \cdots + 32\!\cdots\!28 \nu ) / 37\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 49\!\cdots\!61 \nu^{15} + \cdots - 17\!\cdots\!04 \nu ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 21\!\cdots\!09 \nu^{15} + \cdots - 97\!\cdots\!44 \nu ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 26\!\cdots\!57 \nu^{15} + \cdots - 18\!\cdots\!72 \nu ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28\!\cdots\!83 \nu^{15} + \cdots + 17\!\cdots\!80 \nu ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14\!\cdots\!49 \nu^{15} + \cdots + 10\!\cdots\!04 \nu ) / 37\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14\!\cdots\!09 \nu^{15} + \cdots + 53\!\cdots\!08 \nu ) / 37\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6 \beta_{15} - 3 \beta_{14} - 4 \beta_{13} - 10 \beta_{12} + 20 \beta_{11} - 62 \beta_{10} + \cdots + 8 \beta_{8} ) / 2688 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 8\beta_{4} - 982\beta_{3} + 2165940\beta_1 ) / 168 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 61147 \beta_{15} + 122294 \beta_{14} + 57464 \beta_{13} + 391588 \beta_{12} + \cdots - 28732 \beta_{8} ) / 2688 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7595\beta_{6} - 68510\beta_{5} - 68510\beta_{4} - 1255148\beta_{2} - 11580069396\beta _1 - 11580069396 ) / 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1221225651 \beta_{15} - 1221225651 \beta_{14} - 246101484 \beta_{13} - 3719995090 \beta_{12} + \cdots - 246101484 \beta_{8} ) / 2688 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 467649979 \beta_{7} + 467649979 \beta_{6} + 4296957850 \beta_{5} - 515735519472 \beta_{3} + \cdots + 622587124261620 ) / 168 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 47901037661686 \beta_{15} - 23950518830843 \beta_{14} - 2692784311228 \beta_{13} + \cdots + 5385568622456 \beta_{8} ) / 2688 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3237766366385 \beta_{7} + 28355382552250 \beta_{4} + \cdots + 38\!\cdots\!88 \beta_1 ) / 56 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 46\!\cdots\!35 \beta_{15} + \cdots - 36\!\cdots\!52 \beta_{8} ) / 2688 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 20\!\cdots\!39 \beta_{6} + \cdots - 22\!\cdots\!00 ) / 168 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 89\!\cdots\!23 \beta_{15} + \cdots - 58\!\cdots\!08 \beta_{8} ) / 2688 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 14\!\cdots\!65 \beta_{7} + \cdots + 14\!\cdots\!72 ) / 56 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 34\!\cdots\!62 \beta_{15} + \cdots + 19\!\cdots\!76 \beta_{8} ) / 2688 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 90\!\cdots\!67 \beta_{7} + \cdots + 82\!\cdots\!60 \beta_1 ) / 168 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 33\!\cdots\!79 \beta_{15} + \cdots - 16\!\cdots\!84 \beta_{8} ) / 2688 \) 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
68.6895 − 118.974i
−43.1345 + 74.7111i
43.1345 − 74.7111i
−68.6895 + 118.974i
−69.9637 + 121.181i
−37.6663 + 65.2399i
37.6663 − 65.2399i
69.9637 − 121.181i
68.6895 + 118.974i
−43.1345 − 74.7111i
43.1345 + 74.7111i
−68.6895 − 118.974i
−69.9637 − 121.181i
−37.6663 − 65.2399i
37.6663 + 65.2399i
69.9637 + 121.181i
−22.6274 + 39.1918i −928.383 + 536.002i −1024.00 − 1773.62i 987.816 + 570.316i − 48513.4i 0 92681.9 308876. − 534989.i −44703.4 + 25809.5i
19.2 −22.6274 + 39.1918i −278.763 + 160.944i −1024.00 − 1773.62i −14851.9 − 8574.77i − 14567.0i 0 92681.9 −213914. + 370511.i 672122. − 388050.i
19.3 −22.6274 + 39.1918i 278.763 − 160.944i −1024.00 − 1773.62i 14851.9 + 8574.77i 14567.0i 0 92681.9 −213914. + 370511.i −672122. + 388050.i
19.4 −22.6274 + 39.1918i 928.383 − 536.002i −1024.00 − 1773.62i −987.816 − 570.316i 48513.4i 0 92681.9 308876. − 534989.i 44703.4 − 25809.5i
19.5 22.6274 − 39.1918i −1096.13 + 632.850i −1024.00 − 1773.62i −20081.7 − 11594.2i 57279.1i 0 −92681.9 535279. − 927130.i −908792. + 524691.i
19.6 22.6274 − 39.1918i −88.3706 + 51.0208i −1024.00 − 1773.62i 5989.93 + 3458.29i 4617.87i 0 −92681.9 −260514. + 451224.i 271073. − 156504.i
19.7 22.6274 − 39.1918i 88.3706 − 51.0208i −1024.00 − 1773.62i −5989.93 − 3458.29i − 4617.87i 0 −92681.9 −260514. + 451224.i −271073. + 156504.i
19.8 22.6274 − 39.1918i 1096.13 − 632.850i −1024.00 − 1773.62i 20081.7 + 11594.2i − 57279.1i 0 −92681.9 535279. − 927130.i 908792. − 524691.i
31.1 −22.6274 − 39.1918i −928.383 − 536.002i −1024.00 + 1773.62i 987.816 − 570.316i 48513.4i 0 92681.9 308876. + 534989.i −44703.4 − 25809.5i
31.2 −22.6274 − 39.1918i −278.763 − 160.944i −1024.00 + 1773.62i −14851.9 + 8574.77i 14567.0i 0 92681.9 −213914. − 370511.i 672122. + 388050.i
31.3 −22.6274 − 39.1918i 278.763 + 160.944i −1024.00 + 1773.62i 14851.9 − 8574.77i − 14567.0i 0 92681.9 −213914. − 370511.i −672122. − 388050.i
31.4 −22.6274 − 39.1918i 928.383 + 536.002i −1024.00 + 1773.62i −987.816 + 570.316i − 48513.4i 0 92681.9 308876. + 534989.i 44703.4 + 25809.5i
31.5 22.6274 + 39.1918i −1096.13 − 632.850i −1024.00 + 1773.62i −20081.7 + 11594.2i − 57279.1i 0 −92681.9 535279. + 927130.i −908792. − 524691.i
31.6 22.6274 + 39.1918i −88.3706 − 51.0208i −1024.00 + 1773.62i 5989.93 − 3458.29i − 4617.87i 0 −92681.9 −260514. − 451224.i 271073. + 156504.i
31.7 22.6274 + 39.1918i 88.3706 + 51.0208i −1024.00 + 1773.62i −5989.93 + 3458.29i 4617.87i 0 −92681.9 −260514. − 451224.i −271073. − 156504.i
31.8 22.6274 + 39.1918i 1096.13 + 632.850i −1024.00 + 1773.62i 20081.7 − 11594.2i 57279.1i 0 −92681.9 535279. + 927130.i 908792. + 524691.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.b odd 2 1 inner
7.c even 3 1 inner
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.b 16
7.b odd 2 1 inner 98.13.d.b 16
7.c even 3 1 14.13.b.a ✓ 8
7.c even 3 1 inner 98.13.d.b 16
7.d odd 6 1 14.13.b.a ✓ 8
7.d odd 6 1 inner 98.13.d.b 16
21.g even 6 1 126.13.c.a 8
21.h odd 6 1 126.13.c.a 8
28.f even 6 1 112.13.c.c 8
28.g odd 6 1 112.13.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.13.b.a ✓ 8 7.c even 3 1
14.13.b.a ✓ 8 7.d odd 6 1
98.13.d.b 16 1.a even 1 1 trivial
98.13.d.b 16 7.b odd 2 1 inner
98.13.d.b 16 7.c even 3 1 inner
98.13.d.b 16 7.d odd 6 1 inner
112.13.c.c 8 28.f even 6 1
112.13.c.c 8 28.g odd 6 1
126.13.c.a 8 21.g even 6 1
126.13.c.a 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 2865216 T_{3}^{14} + 6053675160096 T_{3}^{12} + \cdots + 39\!\cdots\!16 \) 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
$3$ \( T^{16} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 25\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 25\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 19\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 26\!\cdots\!16)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 28\!\cdots\!36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 24\!\cdots\!64)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 43\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 40\!\cdots\!16)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 16\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 49\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
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