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Properties

Label 9898.2.a.bh
Level $9898$
Weight $2$
Character orbit 9898.a
Self dual yes
Analytic conductor $79.036$
Analytic rank $1$
Dimension $19$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9898,2,Mod(1,9898)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9898, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9898.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9898 = 2 \cdot 7^{2} \cdot 101 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9898.a (trivial)

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: yes
Analytic conductor: \(79.0359279207\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 38 x^{17} + 587 x^{15} - 18 x^{14} - 4771 x^{13} + 525 x^{12} + 22017 x^{11} - 5563 x^{10} + \cdots - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1414)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_{8} q^{5} + \beta_1 q^{6} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_{8} q^{5} + \beta_1 q^{6} - q^{8} + (\beta_{2} + 1) q^{9} - \beta_{8} q^{10} + (\beta_{14} - 1) q^{11} - \beta_1 q^{12} + \beta_{15} q^{13} + (\beta_{15} + \beta_{12} - 1) q^{15} + q^{16} + (\beta_{18} - \beta_{17} + \beta_{14} + \cdots - 1) q^{17}+ \cdots + ( - \beta_{18} - 2 \beta_{17} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 19 q^{2} + 19 q^{4} + q^{5} - 19 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 19 q^{2} + 19 q^{4} + q^{5} - 19 q^{8} + 19 q^{9} - q^{10} - 14 q^{11} + 3 q^{13} - 8 q^{15} + 19 q^{16} + 5 q^{17} - 19 q^{18} + 2 q^{19} + q^{20} + 14 q^{22} - 16 q^{23} + 18 q^{25} - 3 q^{26} - 15 q^{29} + 8 q^{30} - 9 q^{31} - 19 q^{32} - 10 q^{33} - 5 q^{34} + 19 q^{36} - 3 q^{37} - 2 q^{38} - 15 q^{39} - q^{40} + 6 q^{41} - 39 q^{43} - 14 q^{44} - 11 q^{45} + 16 q^{46} - 6 q^{47} - 18 q^{50} - 9 q^{51} + 3 q^{52} - 26 q^{53} + 25 q^{55} - 40 q^{57} + 15 q^{58} - 23 q^{59} - 8 q^{60} - 15 q^{61} + 9 q^{62} + 19 q^{64} - 27 q^{65} + 10 q^{66} - 13 q^{67} + 5 q^{68} + 40 q^{69} - 22 q^{71} - 19 q^{72} - 31 q^{73} + 3 q^{74} - 45 q^{75} + 2 q^{76} + 15 q^{78} + 13 q^{79} + q^{80} + 27 q^{81} - 6 q^{82} + 21 q^{83} - 19 q^{85} + 39 q^{86} - 29 q^{87} + 14 q^{88} - 10 q^{89} + 11 q^{90} - 16 q^{92} + 21 q^{93} + 6 q^{94} - 38 q^{95} + 54 q^{97} - 98 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 38 x^{17} + 587 x^{15} - 18 x^{14} - 4771 x^{13} + 525 x^{12} + 22017 x^{11} - 5563 x^{10} + \cdots - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 91870836739 \nu^{18} + 2162036152591 \nu^{17} - 7059131935611 \nu^{16} + \cdots - 5433704953143 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 564495428677 \nu^{18} + 769586213989 \nu^{17} - 19259660170764 \nu^{16} + \cdots - 131841369986685 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1076072020608 \nu^{18} + 1085328065911 \nu^{17} - 39470337114352 \nu^{16} + \cdots + 25309874963925 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1320719346236 \nu^{18} - 3120075865661 \nu^{17} + 52505854473948 \nu^{16} + \cdots - 224238745409046 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1349165487341 \nu^{18} + 1845566911481 \nu^{17} + 48511175122721 \nu^{16} + \cdots - 29653793907801 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1551251494676 \nu^{18} - 756281650114 \nu^{17} + 54666586944449 \nu^{16} + \cdots + 64113438576993 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 531339120391 \nu^{18} - 294347558355 \nu^{17} + 20304765015505 \nu^{16} + \cdots - 14232690716134 ) / 8176071187036 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 879858024511 \nu^{18} - 1182102433994 \nu^{17} + 34153178101297 \nu^{16} + \cdots - 67091561334030 ) / 12264106780554 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1814367521226 \nu^{18} - 108698724056 \nu^{17} - 65816292679852 \nu^{16} + \cdots - 52067876094567 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2550449381386 \nu^{18} + 716732117028 \nu^{17} + 92736687990962 \nu^{16} + \cdots - 77676999550602 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2867734328714 \nu^{18} - 4320108733884 \nu^{17} + 109277031692668 \nu^{16} + \cdots - 83581031499240 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3141553295699 \nu^{18} - 5251029723965 \nu^{17} + 120652480786992 \nu^{16} + \cdots + 62295562964163 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3306731031500 \nu^{18} + 3564237736211 \nu^{17} - 125077533421314 \nu^{16} + \cdots + 116166476563794 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 3529170928951 \nu^{18} - 734731265799 \nu^{17} + 132685674936692 \nu^{16} + \cdots - 215771645982474 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 3657046310895 \nu^{18} - 5056190594044 \nu^{17} + 139878920408518 \nu^{16} + \cdots - 47109915601269 ) / 24528213561108 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 6574922261874 \nu^{18} - 132852861883 \nu^{17} - 239528456796971 \nu^{16} + \cdots + 96441051570084 ) / 24528213561108 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{13} + \beta_{10} - \beta_{6} + \beta_{2} + 7\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{18} + \beta_{17} - 2 \beta_{16} - 3 \beta_{15} - \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{18} + 2 \beta_{17} + 11 \beta_{15} - 5 \beta_{14} + 12 \beta_{13} - 6 \beta_{12} - 2 \beta_{11} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 46 \beta_{18} + 17 \beta_{17} - 37 \beta_{16} - 52 \beta_{15} - 21 \beta_{14} + 36 \beta_{13} + \cdots + 235 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 62 \beta_{18} + 38 \beta_{17} + 3 \beta_{16} + 109 \beta_{15} - 92 \beta_{14} + 115 \beta_{13} + \cdots + 35 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 563 \beta_{18} + 217 \beta_{17} - 513 \beta_{16} - 683 \beta_{15} - 290 \beta_{14} + 478 \beta_{13} + \cdots + 2167 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 909 \beta_{18} + 523 \beta_{17} + 81 \beta_{16} + 1083 \beta_{15} - 1241 \beta_{14} + 1039 \beta_{13} + \cdots + 634 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6426 \beta_{18} + 2507 \beta_{17} - 6371 \beta_{16} - 8191 \beta_{15} - 3473 \beta_{14} + 5674 \beta_{13} + \cdots + 21135 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 11738 \beta_{18} + 6344 \beta_{17} + 1429 \beta_{16} + 11007 \beta_{15} - 14920 \beta_{14} + \cdots + 7473 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 71412 \beta_{18} + 27686 \beta_{17} - 74977 \beta_{16} - 94496 \beta_{15} - 39004 \beta_{14} + \cdots + 213402 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 142777 \beta_{18} + 72223 \beta_{17} + 21183 \beta_{16} + 114715 \beta_{15} - 169788 \beta_{14} + \cdots + 75435 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 785272 \beta_{18} + 298812 \beta_{17} - 856751 \beta_{16} - 1069395 \beta_{15} - 423721 \beta_{14} + \cdots + 2202496 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1681969 \beta_{18} + 793538 \beta_{17} + 287927 \beta_{16} + 1222761 \beta_{15} - 1876111 \beta_{14} + \cdots + 691665 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 8602724 \beta_{18} + 3185564 \beta_{17} - 9622765 \beta_{16} - 11973139 \beta_{15} - 4518302 \beta_{14} + \cdots + 23058052 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 19447424 \beta_{18} + 8537951 \beta_{17} + 3731284 \beta_{16} + 13276383 \beta_{15} - 20386637 \beta_{14} + \cdots + 5789241 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 94166895 \beta_{18} + 33735166 \beta_{17} - 106949729 \beta_{16} - 133174515 \beta_{15} + \cdots + 243719365 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.25489
3.00017
2.69259
2.22289
1.55569
1.08966
0.889409
0.837839
0.401290
0.236263
0.0429343
−0.101496
−1.67563
−1.89953
−2.16118
−2.18566
−2.24228
−2.62946
−3.32839
−1.00000 −3.25489 1.00000 3.57805 3.25489 0 −1.00000 7.59428 −3.57805
1.2 −1.00000 −3.00017 1.00000 −2.17869 3.00017 0 −1.00000 6.00105 2.17869
1.3 −1.00000 −2.69259 1.00000 −1.88099 2.69259 0 −1.00000 4.25005 1.88099
1.4 −1.00000 −2.22289 1.00000 −0.114847 2.22289 0 −1.00000 1.94126 0.114847
1.5 −1.00000 −1.55569 1.00000 −0.973664 1.55569 0 −1.00000 −0.579838 0.973664
1.6 −1.00000 −1.08966 1.00000 3.26400 1.08966 0 −1.00000 −1.81265 −3.26400
1.7 −1.00000 −0.889409 1.00000 −3.74730 0.889409 0 −1.00000 −2.20895 3.74730
1.8 −1.00000 −0.837839 1.00000 3.09324 0.837839 0 −1.00000 −2.29803 −3.09324
1.9 −1.00000 −0.401290 1.00000 4.01277 0.401290 0 −1.00000 −2.83897 −4.01277
1.10 −1.00000 −0.236263 1.00000 −2.49454 0.236263 0 −1.00000 −2.94418 2.49454
1.11 −1.00000 −0.0429343 1.00000 2.25888 0.0429343 0 −1.00000 −2.99816 −2.25888
1.12 −1.00000 0.101496 1.00000 −1.55036 −0.101496 0 −1.00000 −2.98970 1.55036
1.13 −1.00000 1.67563 1.00000 0.146495 −1.67563 0 −1.00000 −0.192262 −0.146495
1.14 −1.00000 1.89953 1.00000 0.625715 −1.89953 0 −1.00000 0.608208 −0.625715
1.15 −1.00000 2.16118 1.00000 −0.753052 −2.16118 0 −1.00000 1.67070 0.753052
1.16 −1.00000 2.18566 1.00000 2.82665 −2.18566 0 −1.00000 1.77709 −2.82665
1.17 −1.00000 2.24228 1.00000 −3.53306 −2.24228 0 −1.00000 2.02784 3.53306
1.18 −1.00000 2.62946 1.00000 −2.23129 −2.62946 0 −1.00000 3.91403 2.23129
1.19 −1.00000 3.32839 1.00000 0.652007 −3.32839 0 −1.00000 8.07821 −0.652007
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( +1 \)
\(101\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9898.2.a.bh 19
7.b odd 2 1 9898.2.a.bg 19
7.c even 3 2 1414.2.e.f ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1414.2.e.f ✓ 38 7.c even 3 2
9898.2.a.bg 19 7.b odd 2 1
9898.2.a.bh 19 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9898))\):

\( T_{3}^{19} - 38 T_{3}^{17} + 587 T_{3}^{15} + 18 T_{3}^{14} - 4771 T_{3}^{13} - 525 T_{3}^{12} + \cdots + 9 \) Copy content Toggle raw display
\( T_{5}^{19} - T_{5}^{18} - 56 T_{5}^{17} + 35 T_{5}^{16} + 1295 T_{5}^{15} - 297 T_{5}^{14} - 15936 T_{5}^{13} + \cdots - 2181 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{19} \) Copy content Toggle raw display
$3$ \( T^{19} - 38 T^{17} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{19} - T^{18} + \cdots - 2181 \) Copy content Toggle raw display
$7$ \( T^{19} \) Copy content Toggle raw display
$11$ \( T^{19} + \cdots - 147544128 \) Copy content Toggle raw display
$13$ \( T^{19} + \cdots + 167726223 \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots + 49386705696 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots - 703269376 \) Copy content Toggle raw display
$23$ \( T^{19} + 16 T^{18} + \cdots + 16229376 \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots - 117707410944 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots + 313670862336 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots + 17974431849941 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots + 11181513999360 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots + 273511777238016 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots - 236185996800 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 106952641847808 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots + 1195894834688 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 8232859886771 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 55436276889600 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots - 319403119104 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots + 25\!\cdots\!43 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots - 200846001916416 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots + 43692289663712 \) Copy content Toggle raw display
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