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Properties

Label 98.7.d.d
Level $98$
Weight $7$
Character orbit 98.d
Analytic conductor $22.545$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,7,Mod(19,98)] 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("98.19"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 98.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-256,0,0,0,0,7392] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.5453001947\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 3076 x^{14} - 19104 x^{13} + 3961602 x^{12} - 18288096 x^{11} + \cdots + 12\!\cdots\!11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4}\cdot 7^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

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

\(f(q)\) \(=\) \( q + 4 \beta_{4} q^{2} + ( - \beta_{10} + 2 \beta_{8} - 2 \beta_{3}) q^{3} + (32 \beta_{2} - 32) q^{4} + (\beta_{12} + 3 \beta_{11} + \cdots + \beta_{3}) q^{5} + (8 \beta_{12} + 4 \beta_{8} + \cdots + 8 \beta_{5}) q^{6}+ \cdots + (3705 \beta_{15} + 3705 \beta_{14} + \cdots + 1368900) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 256 q^{4} + 7392 q^{9} + 648 q^{11} - 57264 q^{15} - 8192 q^{16} + 11232 q^{18} - 41792 q^{22} + 5128 q^{23} + 78096 q^{25} - 70592 q^{29} - 86208 q^{30} - 473088 q^{36} - 238208 q^{37} + 544920 q^{39}+ \cdots + 21902400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 3076 x^{14} - 19104 x^{13} + 3961602 x^{12} - 18288096 x^{11} + \cdots + 12\!\cdots\!11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 29\!\cdots\!46 \nu^{15} + \cdots + 85\!\cdots\!81 ) / 27\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 43\!\cdots\!92 \nu^{15} + \cdots - 33\!\cdots\!29 ) / 24\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!70 \nu^{15} + \cdots + 83\!\cdots\!11 ) / 50\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 75\!\cdots\!20 \nu^{15} + \cdots + 10\!\cdots\!81 ) / 26\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\!\cdots\!00 \nu^{15} + \cdots - 16\!\cdots\!50 ) / 72\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 33\!\cdots\!77 \nu^{15} + \cdots + 34\!\cdots\!68 ) / 35\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28\!\cdots\!38 \nu^{15} + \cdots + 21\!\cdots\!47 ) / 21\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31\!\cdots\!10 \nu^{15} + \cdots - 10\!\cdots\!28 ) / 21\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33\!\cdots\!24 \nu^{15} + \cdots - 22\!\cdots\!20 ) / 11\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\!\cdots\!28 \nu^{15} + \cdots + 21\!\cdots\!09 ) / 35\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 71\!\cdots\!78 \nu^{15} + \cdots + 13\!\cdots\!06 ) / 21\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!46 \nu^{15} + \cdots + 62\!\cdots\!96 ) / 50\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 20\!\cdots\!70 \nu^{15} + \cdots - 13\!\cdots\!44 ) / 37\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 65\!\cdots\!66 \nu^{15} + \cdots + 44\!\cdots\!14 ) / 11\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 67\!\cdots\!24 \nu^{15} + \cdots + 46\!\cdots\!03 ) / 11\!\cdots\!46 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} - \beta_{13} + 2\beta_{9} + \beta_{4} + 6\beta_{3} + 20\beta _1 + 21 ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{14} - 3 \beta_{13} - 28 \beta_{11} + 28 \beta_{10} + 6 \beta_{9} + 28 \beta_{7} + \cdots - 15939 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 193 \beta_{15} + 184 \beta_{14} + 320 \beta_{13} + 3 \beta_{12} - 63 \beta_{11} + 63 \beta_{10} + \cdots - 20937 ) / 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1359 \beta_{15} + 1287 \beta_{14} + 3313 \beta_{13} + 288 \beta_{12} + 37408 \beta_{11} + \cdots + 7493871 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 193383 \beta_{15} - 162153 \beta_{14} - 349077 \beta_{13} - 38556 \beta_{12} + 241710 \beta_{11} + \cdots + 40984461 ) / 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 640482 \beta_{15} - 533517 \beta_{14} - 1314409 \beta_{13} - 512700 \beta_{12} - 16981678 \beta_{11} + \cdots - 1901727156 ) / 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 109778227 \beta_{15} + 73631557 \beta_{14} + 170762969 \beta_{13} + 77797440 \beta_{12} + \cdots - 31854027093 ) / 42 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1060954649 \beta_{15} + 712674941 \beta_{14} + 1732188651 \beta_{13} + 1974343056 \beta_{12} + \cdots + 1924320188259 ) / 42 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 32590481836 \beta_{15} - 15357301132 \beta_{14} - 36521204974 \beta_{13} - 51581771028 \beta_{12} + \cdots + 10891018244085 ) / 21 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 825625810659 \beta_{15} - 396799456719 \beta_{14} - 960270845217 \beta_{13} - 2820492890256 \beta_{12} + \cdots - 932716090950693 ) / 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 39095316607171 \beta_{15} + 10046297832307 \beta_{14} + 24087677745739 \beta_{13} + \cdots - 13\!\cdots\!61 ) / 42 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 309047831771564 \beta_{15} + 85211671604285 \beta_{14} + 205927923084795 \beta_{13} + \cdots + 20\!\cdots\!45 ) / 21 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 23\!\cdots\!25 \beta_{15} - 917920201388849 \beta_{14} + \cdots + 76\!\cdots\!69 ) / 42 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 44\!\cdots\!37 \beta_{15} + \cdots - 16\!\cdots\!07 ) / 42 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 68\!\cdots\!95 \beta_{15} + \cdots - 18\!\cdots\!03 ) / 21 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.455720 + 13.7734i
−0.869934 − 13.7734i
0.455720 − 13.0080i
−0.869934 + 13.0080i
2.80731 − 23.2668i
−0.393100 + 23.2668i
2.80731 + 25.1146i
−0.393100 − 25.1146i
0.455720 − 13.7734i
−0.869934 + 13.7734i
0.455720 + 13.0080i
−0.869934 − 13.0080i
2.80731 + 23.2668i
−0.393100 − 23.2668i
2.80731 − 25.1146i
−0.393100 + 25.1146i
−2.82843 + 4.89898i −44.0738 + 25.4460i −16.0000 − 27.7128i −74.6013 − 43.0711i − 287.889i 0 181.019 930.501 − 1611.67i 422.009 − 243.647i
19.2 −2.82843 + 4.89898i −30.1544 + 17.4097i −16.0000 − 27.7128i 192.311 + 111.031i − 196.968i 0 181.019 241.694 − 418.626i −1087.88 + 628.087i
19.3 −2.82843 + 4.89898i 30.1544 − 17.4097i −16.0000 − 27.7128i −192.311 − 111.031i 196.968i 0 181.019 241.694 − 418.626i 1087.88 − 628.087i
19.4 −2.82843 + 4.89898i 44.0738 − 25.4460i −16.0000 − 27.7128i 74.6013 + 43.0711i 287.889i 0 181.019 930.501 − 1611.67i −422.009 + 243.647i
19.5 2.82843 − 4.89898i −44.5743 + 25.7350i −16.0000 − 27.7128i 164.748 + 95.1175i 291.158i 0 −181.019 960.079 − 1662.91i 931.957 − 538.066i
19.6 2.82843 − 4.89898i −10.9700 + 6.33351i −16.0000 − 27.7128i 80.4360 + 46.4397i 71.6555i 0 −181.019 −284.273 + 492.376i 455.015 − 262.703i
19.7 2.82843 − 4.89898i 10.9700 − 6.33351i −16.0000 − 27.7128i −80.4360 − 46.4397i − 71.6555i 0 −181.019 −284.273 + 492.376i −455.015 + 262.703i
19.8 2.82843 − 4.89898i 44.5743 − 25.7350i −16.0000 − 27.7128i −164.748 − 95.1175i − 291.158i 0 −181.019 960.079 − 1662.91i −931.957 + 538.066i
31.1 −2.82843 − 4.89898i −44.0738 − 25.4460i −16.0000 + 27.7128i −74.6013 + 43.0711i 287.889i 0 181.019 930.501 + 1611.67i 422.009 + 243.647i
31.2 −2.82843 − 4.89898i −30.1544 − 17.4097i −16.0000 + 27.7128i 192.311 − 111.031i 196.968i 0 181.019 241.694 + 418.626i −1087.88 − 628.087i
31.3 −2.82843 − 4.89898i 30.1544 + 17.4097i −16.0000 + 27.7128i −192.311 + 111.031i − 196.968i 0 181.019 241.694 + 418.626i 1087.88 + 628.087i
31.4 −2.82843 − 4.89898i 44.0738 + 25.4460i −16.0000 + 27.7128i 74.6013 − 43.0711i − 287.889i 0 181.019 930.501 + 1611.67i −422.009 − 243.647i
31.5 2.82843 + 4.89898i −44.5743 − 25.7350i −16.0000 + 27.7128i 164.748 − 95.1175i − 291.158i 0 −181.019 960.079 + 1662.91i 931.957 + 538.066i
31.6 2.82843 + 4.89898i −10.9700 − 6.33351i −16.0000 + 27.7128i 80.4360 − 46.4397i − 71.6555i 0 −181.019 −284.273 − 492.376i 455.015 + 262.703i
31.7 2.82843 + 4.89898i 10.9700 + 6.33351i −16.0000 + 27.7128i −80.4360 + 46.4397i 71.6555i 0 −181.019 −284.273 − 492.376i −455.015 − 262.703i
31.8 2.82843 + 4.89898i 44.5743 + 25.7350i −16.0000 + 27.7128i −164.748 + 95.1175i 291.158i 0 −181.019 960.079 + 1662.91i −931.957 − 538.066i
\(n\): e.g. 2-40 or 80-90
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 98.7.d.d 16
7.b odd 2 1 inner 98.7.d.d 16
7.c even 3 1 98.7.b.b ✓ 8
7.c even 3 1 inner 98.7.d.d 16
7.d odd 6 1 98.7.b.b ✓ 8
7.d odd 6 1 inner 98.7.d.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.7.b.b ✓ 8 7.c even 3 1
98.7.b.b ✓ 8 7.d odd 6 1
98.7.d.d 16 1.a even 1 1 trivial
98.7.d.d 16 7.b odd 2 1 inner
98.7.d.d 16 7.c even 3 1 inner
98.7.d.d 16 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 6612 T_{3}^{14} + 29470158 T_{3}^{12} - 73332953640 T_{3}^{10} + 132661157386500 T_{3}^{8} + \cdots + 17\!\cdots\!36 \) acting on \(S_{7}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 32 T^{2} + 1024)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 38\!\cdots\!16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 28\!\cdots\!44)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 54\!\cdots\!08)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 39\!\cdots\!24)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 35\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 95\!\cdots\!88)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 63\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 71\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 62\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 18\!\cdots\!56)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 73\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 64\!\cdots\!16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 14\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 67\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 28\!\cdots\!36)^{2} \) Copy content Toggle raw display
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