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Properties

Label 9898.2.a.be
Level $9898$
Weight $2$
Character orbit 9898.a
Self dual yes
Analytic conductor $79.036$
Analytic rank $0$
Dimension $13$
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: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 24 x^{11} + 48 x^{10} + 200 x^{9} - 398 x^{8} - 682 x^{7} + 1358 x^{6} + 862 x^{5} + \cdots - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} - \beta_{2} q^{5} + \beta_1 q^{6} - q^{8} + (\beta_{11} - \beta_{10} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} - \beta_{2} q^{5} + \beta_1 q^{6} - q^{8} + (\beta_{11} - \beta_{10} + 1) q^{9} + \beta_{2} q^{10} + ( - \beta_{10} + 2) q^{11} - \beta_1 q^{12} + (\beta_{10} - \beta_{9} + \beta_{8} + \cdots - 1) q^{13}+ \cdots + ( - 2 \beta_{12} + \beta_{11} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{2} - 2 q^{3} + 13 q^{4} + q^{5} + 2 q^{6} - 13 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{2} - 2 q^{3} + 13 q^{4} + q^{5} + 2 q^{6} - 13 q^{8} + 13 q^{9} - q^{10} + 20 q^{11} - 2 q^{12} - 9 q^{13} + 6 q^{15} + 13 q^{16} - 13 q^{17} - 13 q^{18} - 2 q^{19} + q^{20} - 20 q^{22} + 12 q^{23} + 2 q^{24} + 12 q^{25} + 9 q^{26} + 4 q^{27} + 11 q^{29} - 6 q^{30} - 3 q^{31} - 13 q^{32} + 18 q^{33} + 13 q^{34} + 13 q^{36} - 11 q^{37} + 2 q^{38} - 9 q^{39} - q^{40} + 8 q^{41} + 19 q^{43} + 20 q^{44} - 33 q^{45} - 12 q^{46} - 2 q^{47} - 2 q^{48} - 12 q^{50} + 12 q^{51} - 9 q^{52} + 16 q^{53} - 4 q^{54} + 8 q^{55} + 62 q^{57} - 11 q^{58} - 17 q^{59} + 6 q^{60} - 9 q^{61} + 3 q^{62} + 13 q^{64} + 3 q^{65} - 18 q^{66} + 47 q^{67} - 13 q^{68} + 22 q^{69} + 14 q^{71} - 13 q^{72} - 3 q^{73} + 11 q^{74} - 45 q^{75} - 2 q^{76} + 9 q^{78} - 3 q^{79} + q^{80} + 17 q^{81} - 8 q^{82} + 29 q^{83} + 13 q^{85} - 19 q^{86} - 31 q^{87} - 20 q^{88} - 22 q^{89} + 33 q^{90} + 12 q^{92} + 9 q^{93} + 2 q^{94} + 14 q^{95} + 2 q^{96} + 22 q^{97} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 2 x^{12} - 24 x^{11} + 48 x^{10} + 200 x^{9} - 398 x^{8} - 682 x^{7} + 1358 x^{6} + 862 x^{5} + \cdots - 24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6611 \nu^{12} + 12976 \nu^{11} + 152504 \nu^{10} - 306816 \nu^{9} - 1179752 \nu^{8} + \cdots + 284828 ) / 88096 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18651 \nu^{12} - 40056 \nu^{11} - 434936 \nu^{10} + 952768 \nu^{9} + 3422264 \nu^{8} + \cdots - 713324 ) / 176192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 36611 \nu^{12} + 100280 \nu^{11} + 832824 \nu^{10} - 2402176 \nu^{9} - 6252440 \nu^{8} + \cdots + 1897036 ) / 176192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4788 \nu^{12} - 10705 \nu^{11} - 111266 \nu^{10} + 254164 \nu^{9} + 869612 \nu^{8} - 2069952 \nu^{7} + \cdots - 205922 ) / 22024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 52587 \nu^{12} + 138920 \nu^{11} + 1195352 \nu^{10} - 3323744 \nu^{9} - 8951384 \nu^{8} + \cdots + 3619084 ) / 176192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 54129 \nu^{12} + 146504 \nu^{11} + 1222568 \nu^{10} - 3499808 \nu^{9} - 9033576 \nu^{8} + \cdots + 4614212 ) / 176192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27359 \nu^{12} - 60282 \nu^{11} - 636492 \nu^{10} + 1433888 \nu^{9} + 4988448 \nu^{8} + \cdots - 1223040 ) / 44048 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 134907 \nu^{12} + 325096 \nu^{11} + 3107960 \nu^{10} - 7748448 \nu^{9} - 23898744 \nu^{8} + \cdots + 7812876 ) / 176192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 74941 \nu^{12} - 185448 \nu^{11} - 1720840 \nu^{10} + 4425088 \nu^{9} + 13137080 \nu^{8} + \cdots - 4545044 ) / 88096 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 74941 \nu^{12} - 185448 \nu^{11} - 1720840 \nu^{10} + 4425088 \nu^{9} + 13137080 \nu^{8} + \cdots - 4897428 ) / 88096 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 195649 \nu^{12} + 465904 \nu^{11} + 4509976 \nu^{10} - 11113792 \nu^{9} - 34702856 \nu^{8} + \cdots + 11558932 ) / 176192 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} - \beta_{6} + \beta_{3} + 6\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9 \beta_{11} - 8 \beta_{10} - \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{12} + 10 \beta_{11} + \beta_{10} + 12 \beta_{9} + 13 \beta_{8} - 15 \beta_{6} + \beta_{5} + \cdots - 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{12} + 80 \beta_{11} - 65 \beta_{10} - 15 \beta_{9} - 24 \beta_{8} + 27 \beta_{7} + 12 \beta_{6} + \cdots + 226 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 120 \beta_{12} + 79 \beta_{11} + 20 \beta_{10} + 119 \beta_{9} + 138 \beta_{8} - 3 \beta_{7} + \cdots - 266 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 33 \beta_{12} + 719 \beta_{11} - 549 \beta_{10} - 178 \beta_{9} - 241 \beta_{8} + 291 \beta_{7} + \cdots + 1938 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1122 \beta_{12} + 559 \beta_{11} + 271 \beta_{10} + 1138 \beta_{9} + 1374 \beta_{8} - 69 \beta_{7} + \cdots - 2720 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 352 \beta_{12} + 6522 \beta_{11} - 4786 \beta_{10} - 1953 \beta_{9} - 2373 \beta_{8} + 2921 \beta_{7} + \cdots + 17185 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 10166 \beta_{12} + 3502 \beta_{11} + 3182 \beta_{10} + 10848 \beta_{9} + 13346 \beta_{8} - 1120 \beta_{7} + \cdots - 27382 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2958 \beta_{12} + 59579 \beta_{11} - 42749 \beta_{10} - 20692 \beta_{9} - 23576 \beta_{8} + \cdots + 155846 \) 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
2.94654
2.90245
2.45377
1.49944
1.34356
0.900212
0.149322
−0.399293
−0.405302
−1.60000
−1.86606
−2.77531
−3.14934
−1.00000 −2.94654 1.00000 3.27166 2.94654 0 −1.00000 5.68208 −3.27166
1.2 −1.00000 −2.90245 1.00000 −2.25177 2.90245 0 −1.00000 5.42424 2.25177
1.3 −1.00000 −2.45377 1.00000 −3.65554 2.45377 0 −1.00000 3.02100 3.65554
1.4 −1.00000 −1.49944 1.00000 −2.72252 1.49944 0 −1.00000 −0.751689 2.72252
1.5 −1.00000 −1.34356 1.00000 0.907031 1.34356 0 −1.00000 −1.19484 −0.907031
1.6 −1.00000 −0.900212 1.00000 0.925547 0.900212 0 −1.00000 −2.18962 −0.925547
1.7 −1.00000 −0.149322 1.00000 −0.872935 0.149322 0 −1.00000 −2.97770 0.872935
1.8 −1.00000 0.399293 1.00000 3.34708 −0.399293 0 −1.00000 −2.84057 −3.34708
1.9 −1.00000 0.405302 1.00000 3.69644 −0.405302 0 −1.00000 −2.83573 −3.69644
1.10 −1.00000 1.60000 1.00000 −1.80333 −1.60000 0 −1.00000 −0.439988 1.80333
1.11 −1.00000 1.86606 1.00000 2.47812 −1.86606 0 −1.00000 0.482165 −2.47812
1.12 −1.00000 2.77531 1.00000 −1.88750 −2.77531 0 −1.00000 4.70232 1.88750
1.13 −1.00000 3.14934 1.00000 −0.432294 −3.14934 0 −1.00000 6.91832 0.432294
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.be 13
7.b odd 2 1 9898.2.a.bf 13
7.d odd 6 2 1414.2.e.e ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1414.2.e.e ✓ 26 7.d odd 6 2
9898.2.a.be 13 1.a even 1 1 trivial
9898.2.a.bf 13 7.b odd 2 1

Hecke kernels

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

\( T_{3}^{13} + 2 T_{3}^{12} - 24 T_{3}^{11} - 48 T_{3}^{10} + 200 T_{3}^{9} + 398 T_{3}^{8} - 682 T_{3}^{7} + \cdots + 24 \) Copy content Toggle raw display
\( T_{5}^{13} - T_{5}^{12} - 38 T_{5}^{11} + 23 T_{5}^{10} + 550 T_{5}^{9} - 100 T_{5}^{8} - 3765 T_{5}^{7} + \cdots + 2424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{13} \) Copy content Toggle raw display
$3$ \( T^{13} + 2 T^{12} + \cdots + 24 \) Copy content Toggle raw display
$5$ \( T^{13} - T^{12} + \cdots + 2424 \) Copy content Toggle raw display
$7$ \( T^{13} \) Copy content Toggle raw display
$11$ \( T^{13} - 20 T^{12} + \cdots + 768 \) Copy content Toggle raw display
$13$ \( T^{13} + 9 T^{12} + \cdots + 24648 \) Copy content Toggle raw display
$17$ \( T^{13} + 13 T^{12} + \cdots + 7176 \) Copy content Toggle raw display
$19$ \( T^{13} + 2 T^{12} + \cdots + 91130 \) Copy content Toggle raw display
$23$ \( T^{13} - 12 T^{12} + \cdots + 1802448 \) Copy content Toggle raw display
$29$ \( T^{13} - 11 T^{12} + \cdots - 722250 \) Copy content Toggle raw display
$31$ \( T^{13} + 3 T^{12} + \cdots + 244848 \) Copy content Toggle raw display
$37$ \( T^{13} + 11 T^{12} + \cdots + 6370408 \) Copy content Toggle raw display
$41$ \( T^{13} - 8 T^{12} + \cdots - 1133808 \) Copy content Toggle raw display
$43$ \( T^{13} - 19 T^{12} + \cdots + 3011662 \) Copy content Toggle raw display
$47$ \( T^{13} + 2 T^{12} + \cdots - 2446992 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 129411900 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 3331865664 \) Copy content Toggle raw display
$61$ \( T^{13} + 9 T^{12} + \cdots + 27580958 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 3820879240 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 151661340672 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 334998966976 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 27318551104 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 405610392 \) Copy content Toggle raw display
$89$ \( T^{13} + 22 T^{12} + \cdots - 79836192 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 225654720040 \) Copy content Toggle raw display
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