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Properties

Label 98.10.c.n
Level $98$
Weight $10$
Character orbit 98.c
Analytic conductor $50.474$
Analytic rank $0$
Dimension $12$
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] = [98,10,Mod(67,98)]
 
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([4]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.67");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), 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: \(50.4735119441\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 14308 x^{10} + 60936 x^{9} + 76758531 x^{8} - 346934000 x^{7} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4}\cdot 7^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 16 \beta_1 q^{2} + (\beta_{10} + \beta_{8}) q^{3} + (256 \beta_1 - 256) q^{4} + ( - \beta_{6} + 3 \beta_{4} + 2 \beta_{2}) q^{5} + ( - 16 \beta_{10} + \cdots + 16 \beta_{2}) q^{6}+ \cdots + ( - \beta_{9} + 6 \beta_{7} + \cdots - 12094 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 \beta_1 q^{2} + (\beta_{10} + \beta_{8}) q^{3} + (256 \beta_1 - 256) q^{4} + ( - \beta_{6} + 3 \beta_{4} + 2 \beta_{2}) q^{5} + ( - 16 \beta_{10} + \cdots + 16 \beta_{2}) q^{6}+ \cdots + (70915 \beta_{5} - 323205 \beta_{3} + 887229172) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 96 q^{2} - 1536 q^{4} + 49152 q^{8} - 72550 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 96 q^{2} - 1536 q^{4} + 49152 q^{8} - 72550 q^{9} - 110664 q^{11} - 1045456 q^{15} - 393216 q^{16} - 1160800 q^{18} + 3541248 q^{22} - 243384 q^{23} - 17125446 q^{25} + 20927280 q^{29} + 8363648 q^{30} - 6291456 q^{32} + 37145600 q^{36} - 24332568 q^{37} - 42635880 q^{39} + 149517168 q^{43} - 28329984 q^{44} - 3894144 q^{46} + 548014272 q^{50} - 260246248 q^{51} - 137815308 q^{53} + 899327920 q^{57} - 167418240 q^{58} + 133818368 q^{60} + 201326592 q^{64} - 565976460 q^{65} + 202943376 q^{67} + 1228585536 q^{71} - 297164800 q^{72} - 389321088 q^{74} + 1364348160 q^{78} + 182370096 q^{79} + 432881842 q^{81} + 4140991464 q^{85} - 1196137344 q^{86} - 453279744 q^{88} + 124612608 q^{92} - 190722192 q^{93} + 1311555480 q^{95} + 10645173584 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 14308 x^{10} + 60936 x^{9} + 76758531 x^{8} - 346934000 x^{7} + \cdots + 15\!\cdots\!52 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 25508 \nu^{11} + 5406369 \nu^{10} - 398562328 \nu^{9} - 96945055801 \nu^{8} + \cdots + 85\!\cdots\!96 ) / 48\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16\!\cdots\!03 \nu^{11} + \cdots + 18\!\cdots\!60 ) / 29\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 20\!\cdots\!77 \nu^{11} + \cdots - 11\!\cdots\!76 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 94\!\cdots\!99 \nu^{11} + \cdots + 22\!\cdots\!56 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 84\!\cdots\!69 \nu^{11} + \cdots + 39\!\cdots\!32 ) / 46\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21\!\cdots\!47 \nu^{11} + \cdots - 64\!\cdots\!88 ) / 36\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!16 \nu^{11} + \cdots - 44\!\cdots\!72 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49\!\cdots\!59 \nu^{11} + \cdots + 32\!\cdots\!72 ) / 29\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!76 \nu^{11} + \cdots + 46\!\cdots\!28 ) / 69\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 52\!\cdots\!59 \nu^{11} + \cdots + 40\!\cdots\!92 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!27 \nu^{11} + \cdots - 11\!\cdots\!76 ) / 36\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{5} + 5\beta_{3} - 12\beta_{2} + 195 ) / 588 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -28\beta_{6} - 58\beta_{5} - 336\beta_{4} + 275\beta_{3} + 16\beta_{2} + 1176\beta _1 + 1402269 ) / 588 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12 \beta_{9} - 24 \beta_{8} + 30 \beta_{7} - 1302 \beta_{6} + 7102 \beta_{5} + 19656 \beta_{4} + \cdots - 545625 ) / 588 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 112 \beta_{11} - 1344 \beta_{10} - 696 \beta_{9} + 64 \beta_{8} + 3300 \beta_{7} + \cdots + 5006069805 ) / 588 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8680 \beta_{11} + 131040 \beta_{10} + 142000 \beta_{9} - 582992 \beta_{8} + 363400 \beta_{7} + \cdots - 3230851155 ) / 588 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2021824 \beta_{11} - 23791488 \beta_{10} - 6282660 \beta_{9} + 2139328 \beta_{8} + \cdots + 17830542876699 ) / 588 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 107852724 \beta_{11} + 1652658672 \beta_{10} + 1069023844 \beta_{9} - 7301818272 \beta_{8} + \cdots - 16091146957425 ) / 588 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 20212694432 \beta_{11} - 239050111104 \beta_{10} - 42312520768 \beta_{9} + 24440347904 \beta_{8} + \cdots + 63\!\cdots\!97 ) / 588 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 918324236256 \beta_{11} + 14283429245568 \beta_{10} + 6567426935424 \beta_{9} + \cdots - 73\!\cdots\!35 ) / 588 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 154613522963648 \beta_{11} + \cdots + 22\!\cdots\!95 ) / 588 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 63\!\cdots\!68 \beta_{11} + \cdots - 31\!\cdots\!85 ) / 588 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.549990 − 1.22474i
−59.2520 + 1.22474i
58.9949 − 1.22474i
60.4091 + 1.22474i
−60.6662 − 1.22474i
1.96420 + 1.22474i
0.549990 + 1.22474i
−59.2520 − 1.22474i
58.9949 + 1.22474i
60.4091 − 1.22474i
−60.6662 + 1.22474i
1.96420 − 1.22474i
−8.00000 − 13.8564i −122.591 + 212.333i −128.000 + 221.703i 1331.30 + 2305.88i 3922.90 0 4096.00 −20215.5 − 35014.2i 21300.8 − 36894.1i
67.2 −8.00000 − 13.8564i −71.5530 + 123.933i −128.000 + 221.703i −237.173 − 410.796i 2289.70 0 4096.00 −398.157 − 689.628i −3794.77 + 6572.74i
67.3 −8.00000 − 13.8564i −60.6851 + 105.110i −128.000 + 221.703i −1333.01 − 2308.84i 1941.92 0 4096.00 2476.14 + 4288.79i −21328.1 + 36941.4i
67.4 −8.00000 − 13.8564i 60.6851 − 105.110i −128.000 + 221.703i 1333.01 + 2308.84i −1941.92 0 4096.00 2476.14 + 4288.79i 21328.1 − 36941.4i
67.5 −8.00000 − 13.8564i 71.5530 − 123.933i −128.000 + 221.703i 237.173 + 410.796i −2289.70 0 4096.00 −398.157 − 689.628i 3794.77 − 6572.74i
67.6 −8.00000 − 13.8564i 122.591 − 212.333i −128.000 + 221.703i −1331.30 − 2305.88i −3922.90 0 4096.00 −20215.5 − 35014.2i −21300.8 + 36894.1i
79.1 −8.00000 + 13.8564i −122.591 − 212.333i −128.000 − 221.703i 1331.30 − 2305.88i 3922.90 0 4096.00 −20215.5 + 35014.2i 21300.8 + 36894.1i
79.2 −8.00000 + 13.8564i −71.5530 − 123.933i −128.000 − 221.703i −237.173 + 410.796i 2289.70 0 4096.00 −398.157 + 689.628i −3794.77 − 6572.74i
79.3 −8.00000 + 13.8564i −60.6851 − 105.110i −128.000 − 221.703i −1333.01 + 2308.84i 1941.92 0 4096.00 2476.14 − 4288.79i −21328.1 − 36941.4i
79.4 −8.00000 + 13.8564i 60.6851 + 105.110i −128.000 − 221.703i 1333.01 − 2308.84i −1941.92 0 4096.00 2476.14 − 4288.79i 21328.1 + 36941.4i
79.5 −8.00000 + 13.8564i 71.5530 + 123.933i −128.000 − 221.703i 237.173 − 410.796i −2289.70 0 4096.00 −398.157 + 689.628i 3794.77 + 6572.74i
79.6 −8.00000 + 13.8564i 122.591 + 212.333i −128.000 − 221.703i −1331.30 + 2305.88i −3922.90 0 4096.00 −20215.5 + 35014.2i −21300.8 − 36894.1i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.6
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 98.10.c.n 12
7.b odd 2 1 inner 98.10.c.n 12
7.c even 3 1 98.10.a.l ✓ 6
7.c even 3 1 inner 98.10.c.n 12
7.d odd 6 1 98.10.a.l ✓ 6
7.d odd 6 1 inner 98.10.c.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.10.a.l ✓ 6 7.c even 3 1
98.10.a.l ✓ 6 7.d odd 6 1
98.10.c.n 12 1.a even 1 1 trivial
98.10.c.n 12 7.b odd 2 1 inner
98.10.c.n 12 7.c even 3 1 inner
98.10.c.n 12 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 95324 T_{3}^{10} + 6668374732 T_{3}^{8} + 194251315665456 T_{3}^{6} + \cdots + 32\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16 T + 256)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 17\!\cdots\!36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 69\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots + 15\!\cdots\!76)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 53\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots + 20\!\cdots\!48)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 53\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 52\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots + 12\!\cdots\!48)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 61\!\cdots\!52)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 13\!\cdots\!72)^{2} \) Copy content Toggle raw display
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