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Properties

Label 98.2.e.a
Level $98$
Weight $2$
Character orbit 98.e
Analytic conductor $0.783$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,2,Mod(15,98)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("98.15"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 98.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.782533939809\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} + 15 x^{16} - 23 x^{15} + 72 x^{14} - 85 x^{13} + 432 x^{12} - 282 x^{11} + 1786 x^{10} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{13} q^{2} + ( - \beta_{14} + \beta_1) q^{3} + (\beta_{14} + \beta_{13} + \beta_{10} + \cdots - 1) q^{4} + (\beta_{17} - \beta_{14} - \beta_{11}) q^{5} + ( - \beta_{8} - \beta_{6} + \cdots - \beta_1) q^{6}+ \cdots + (3 \beta_{14} + 3 \beta_{11} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{2} + 3 q^{3} - 3 q^{4} - 6 q^{5} + 3 q^{6} - 7 q^{7} - 3 q^{8} + 10 q^{9} - 6 q^{10} - q^{11} - 4 q^{12} - 7 q^{14} - 9 q^{15} - 3 q^{16} - 11 q^{17} + 24 q^{18} + 36 q^{19} + q^{20} - 21 q^{21}+ \cdots + 106 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} + 15 x^{16} - 23 x^{15} + 72 x^{14} - 85 x^{13} + 432 x^{12} - 282 x^{11} + 1786 x^{10} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!35 \nu^{17} + \cdots + 71\!\cdots\!94 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!55 \nu^{17} + \cdots - 25\!\cdots\!02 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 62\!\cdots\!27 \nu^{17} + \cdots + 29\!\cdots\!45 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 79\!\cdots\!31 \nu^{17} + \cdots + 18\!\cdots\!12 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 98\!\cdots\!25 \nu^{17} + \cdots + 15\!\cdots\!71 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!90 \nu^{17} + \cdots + 29\!\cdots\!71 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 65\!\cdots\!97 \nu^{17} + \cdots - 53\!\cdots\!57 ) / 35\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 76\!\cdots\!84 \nu^{17} + \cdots - 13\!\cdots\!25 ) / 35\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 34\!\cdots\!38 \nu^{17} + \cdots + 15\!\cdots\!98 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37\!\cdots\!36 \nu^{17} + \cdots - 82\!\cdots\!95 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!95 \nu^{17} + \cdots - 60\!\cdots\!57 ) / 35\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\!\cdots\!06 \nu^{17} + \cdots - 31\!\cdots\!28 ) / 35\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 41\!\cdots\!05 \nu^{17} + \cdots + 39\!\cdots\!56 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 59\!\cdots\!48 \nu^{17} + \cdots - 32\!\cdots\!74 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 21\!\cdots\!01 \nu^{17} + \cdots - 90\!\cdots\!23 ) / 35\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 10\!\cdots\!83 \nu^{17} + \cdots - 28\!\cdots\!02 ) / 16\!\cdots\!91 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} + \beta_{16} - \beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - 5\beta_{10} + 2\beta_{9} - \beta_{8} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{11} - 6 \beta_{10} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{17} + 4 \beta_{16} - 4 \beta_{15} - 12 \beta_{14} + 6 \beta_{11} - 29 \beta_{10} + 45 \beta_{9} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32 \beta_{17} - 17 \beta_{15} - 55 \beta_{14} + 23 \beta_{13} + 20 \beta_{12} + 17 \beta_{11} + \cdots + 6 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 162 \beta_{17} - 78 \beta_{16} - 35 \beta_{15} - 242 \beta_{14} + 180 \beta_{13} + 162 \beta_{12} + \cdots - 180 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 581 \beta_{17} - 581 \beta_{16} - 695 \beta_{14} + 867 \beta_{13} + 738 \beta_{12} - 260 \beta_{11} + \cdots - 1248 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1788 \beta_{17} - 2877 \beta_{16} + 548 \beta_{15} - 1672 \beta_{14} + 3498 \beta_{13} + 2877 \beta_{12} + \cdots - 6220 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3907 \beta_{17} - 11007 \beta_{16} + 3907 \beta_{15} + 10554 \beta_{13} + 8778 \beta_{12} - 8778 \beta_{11} + \cdots - 23691 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 34258 \beta_{16} + 18780 \beta_{15} + 23343 \beta_{14} + 23343 \beta_{13} + 18780 \beta_{12} + \cdots - 74768 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 58350 \beta_{17} - 72822 \beta_{16} + 72822 \beta_{15} + 157800 \beta_{14} - 105493 \beta_{11} + \cdots - 157800 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 405403 \beta_{17} + 226992 \beta_{15} + 761719 \beta_{14} - 340865 \beta_{13} - 280742 \beta_{12} + \cdots - 132121 \beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1953735 \beta_{17} + 1084493 \beta_{16} + 483774 \beta_{15} + 2929197 \beta_{14} - 2350098 \beta_{13} + \cdots + 2350098 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 7543291 \beta_{17} + 7543291 \beta_{16} + 9062260 \beta_{14} - 11313804 \beta_{13} - 9402630 \beta_{12} + \cdots + 16356758 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 23320604 \beta_{17} + 36256752 \beta_{16} - 7185651 \beta_{15} + 19399238 \beta_{14} + \cdots + 78558012 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 49934244 \beta_{17} + 139836396 \beta_{16} - 49934244 \beta_{15} - 134834431 \beta_{13} + \cdots + 303034222 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 432441179 \beta_{16} - 239996980 \beta_{15} - 288598279 \beta_{14} - 288598279 \beta_{13} + \cdots + 937092022 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{10}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−1.30651 + 1.63831i
0.101116 − 0.126795i
1.35938 − 1.70461i
−0.392347 − 1.71899i
0.281648 + 1.23398i
0.432251 + 1.89382i
−1.88755 − 0.908997i
0.937338 + 0.451398i
3.47467 + 1.67331i
−1.88755 + 0.908997i
0.937338 − 0.451398i
3.47467 − 1.67331i
−0.392347 + 1.71899i
0.281648 − 1.23398i
0.432251 − 1.89382i
−1.30651 − 1.63831i
0.101116 + 0.126795i
1.35938 + 1.70461i
−0.222521 + 0.974928i −0.683017 + 0.856476i −0.900969 − 0.433884i −1.25326 + 1.57153i −0.683017 − 0.856476i −0.413525 + 2.61324i 0.623490 − 0.781831i 0.400524 + 1.75481i −1.25326 − 1.57153i
15.2 −0.222521 + 0.974928i 0.724605 − 0.908626i −0.900969 − 0.433884i 2.67921 − 3.35962i 0.724605 + 0.908626i −2.50613 + 0.848133i 0.623490 − 0.781831i 0.367014 + 1.60799i 2.67921 + 3.35962i
15.3 −0.222521 + 0.974928i 1.98287 − 2.48644i −0.900969 − 0.433884i −1.57994 + 1.98119i 1.98287 + 2.48644i 2.54314 − 0.729681i 0.623490 − 0.781831i −1.58305 − 6.93579i −1.57994 − 1.98119i
29.1 −0.900969 − 0.433884i −0.614868 − 2.69391i 0.623490 + 0.781831i 0.0678680 + 0.297349i −0.614868 + 2.69391i −1.53790 − 2.15287i −0.222521 − 0.974928i −4.17621 + 2.01115i 0.0678680 − 0.297349i
29.2 −0.900969 − 0.433884i 0.0591274 + 0.259054i 0.623490 + 0.781831i −0.871615 − 3.81880i 0.0591274 − 0.259054i 2.52187 + 0.800098i −0.222521 − 0.974928i 2.63929 − 1.27102i −0.871615 + 3.81880i
29.3 −0.900969 − 0.433884i 0.209730 + 0.918888i 0.623490 + 0.781831i 0.482195 + 2.11264i 0.209730 − 0.918888i −2.20649 + 1.45993i −0.222521 − 0.974928i 1.90254 − 0.916214i 0.482195 − 2.11264i
43.1 0.623490 − 0.781831i −2.78852 − 1.34288i −0.222521 − 0.974928i −0.309110 − 0.148859i −2.78852 + 1.34288i −2.03497 − 1.69083i −0.900969 − 0.433884i 4.10205 + 5.14381i −0.309110 + 0.148859i
43.2 0.623490 − 0.781831i 0.0363696 + 0.0175147i −0.222521 − 0.974928i 0.426714 + 0.205495i 0.0363696 − 0.0175147i 2.62504 − 0.330433i −0.900969 − 0.433884i −1.86945 − 2.34422i 0.426714 − 0.205495i
43.3 0.623490 − 0.781831i 2.57370 + 1.23943i −0.222521 − 0.974928i −2.64206 − 1.27235i 2.57370 − 1.23943i −2.49104 + 0.891482i −0.900969 − 0.433884i 3.21729 + 4.03436i −2.64206 + 1.27235i
57.1 0.623490 + 0.781831i −2.78852 + 1.34288i −0.222521 + 0.974928i −0.309110 + 0.148859i −2.78852 − 1.34288i −2.03497 + 1.69083i −0.900969 + 0.433884i 4.10205 − 5.14381i −0.309110 − 0.148859i
57.2 0.623490 + 0.781831i 0.0363696 − 0.0175147i −0.222521 + 0.974928i 0.426714 − 0.205495i 0.0363696 + 0.0175147i 2.62504 + 0.330433i −0.900969 + 0.433884i −1.86945 + 2.34422i 0.426714 + 0.205495i
57.3 0.623490 + 0.781831i 2.57370 − 1.23943i −0.222521 + 0.974928i −2.64206 + 1.27235i 2.57370 + 1.23943i −2.49104 − 0.891482i −0.900969 + 0.433884i 3.21729 − 4.03436i −2.64206 − 1.27235i
71.1 −0.900969 + 0.433884i −0.614868 + 2.69391i 0.623490 − 0.781831i 0.0678680 − 0.297349i −0.614868 − 2.69391i −1.53790 + 2.15287i −0.222521 + 0.974928i −4.17621 − 2.01115i 0.0678680 + 0.297349i
71.2 −0.900969 + 0.433884i 0.0591274 − 0.259054i 0.623490 − 0.781831i −0.871615 + 3.81880i 0.0591274 + 0.259054i 2.52187 − 0.800098i −0.222521 + 0.974928i 2.63929 + 1.27102i −0.871615 − 3.81880i
71.3 −0.900969 + 0.433884i 0.209730 − 0.918888i 0.623490 − 0.781831i 0.482195 − 2.11264i 0.209730 + 0.918888i −2.20649 − 1.45993i −0.222521 + 0.974928i 1.90254 + 0.916214i 0.482195 + 2.11264i
85.1 −0.222521 − 0.974928i −0.683017 − 0.856476i −0.900969 + 0.433884i −1.25326 − 1.57153i −0.683017 + 0.856476i −0.413525 − 2.61324i 0.623490 + 0.781831i 0.400524 − 1.75481i −1.25326 + 1.57153i
85.2 −0.222521 − 0.974928i 0.724605 + 0.908626i −0.900969 + 0.433884i 2.67921 + 3.35962i 0.724605 − 0.908626i −2.50613 − 0.848133i 0.623490 + 0.781831i 0.367014 − 1.60799i 2.67921 − 3.35962i
85.3 −0.222521 − 0.974928i 1.98287 + 2.48644i −0.900969 + 0.433884i −1.57994 − 1.98119i 1.98287 − 2.48644i 2.54314 + 0.729681i 0.623490 + 0.781831i −1.58305 + 6.93579i −1.57994 + 1.98119i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.2.e.a ✓ 18
3.b odd 2 1 882.2.u.j 18
4.b odd 2 1 784.2.u.c 18
7.b odd 2 1 686.2.e.a 18
7.c even 3 2 686.2.g.i 36
7.d odd 6 2 686.2.g.j 36
49.e even 7 1 inner 98.2.e.a ✓ 18
49.e even 7 1 4802.2.a.h 9
49.f odd 14 1 686.2.e.a 18
49.f odd 14 1 4802.2.a.e 9
49.g even 21 2 686.2.g.i 36
49.h odd 42 2 686.2.g.j 36
147.l odd 14 1 882.2.u.j 18
196.k odd 14 1 784.2.u.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.e.a ✓ 18 1.a even 1 1 trivial
98.2.e.a ✓ 18 49.e even 7 1 inner
686.2.e.a 18 7.b odd 2 1
686.2.e.a 18 49.f odd 14 1
686.2.g.i 36 7.c even 3 2
686.2.g.i 36 49.g even 21 2
686.2.g.j 36 7.d odd 6 2
686.2.g.j 36 49.h odd 42 2
784.2.u.c 18 4.b odd 2 1
784.2.u.c 18 196.k odd 14 1
882.2.u.j 18 3.b odd 2 1
882.2.u.j 18 147.l odd 14 1
4802.2.a.e 9 49.f odd 14 1
4802.2.a.h 9 49.e even 7 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 3 T_{3}^{17} + 4 T_{3}^{16} + 9 T_{3}^{15} + 12 T_{3}^{14} - 29 T_{3}^{13} + 277 T_{3}^{12} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} - 3 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{18} + 6 T^{17} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{18} + 7 T^{17} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 6126662529 \) Copy content Toggle raw display
$13$ \( T^{18} + 14 T^{16} + \cdots + 19882681 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 438986304 \) Copy content Toggle raw display
$19$ \( (T^{9} - 18 T^{8} + \cdots + 500387)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} - 5 T^{17} + \cdots + 1225449 \) Copy content Toggle raw display
$29$ \( T^{18} + 13 T^{17} + \cdots + 1347921 \) Copy content Toggle raw display
$31$ \( (T^{9} + 2 T^{8} - 42 T^{7} + \cdots - 13)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 1703523546481 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 661949588881161 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 63645702961 \) Copy content Toggle raw display
$47$ \( T^{18} - 36 T^{17} + \cdots + 9308601 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 97031627001 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 83822409441 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 4441105256449 \) Copy content Toggle raw display
$67$ \( (T^{9} - 17 T^{8} + \cdots - 149884139)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + 6 T^{17} + \cdots + 14085009 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 351446147128609 \) Copy content Toggle raw display
$79$ \( (T^{9} + T^{8} + \cdots + 5529329)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 10124977628529 \) Copy content Toggle raw display
$97$ \( (T^{9} - 28 T^{8} + \cdots + 8869)^{2} \) Copy content Toggle raw display
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