Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,9,Mod(19,98)]
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([5]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.19");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.d (of order \(6\), 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: | \(39.9231037860\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + \cdots + 36\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{4}\cdot 7^{4} \) |
| Twist minimal: | no (minimal twist has level 14) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + \cdots + 36\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 24\!\cdots\!14 \nu^{11} + \cdots + 26\!\cdots\!50 ) / 12\!\cdots\!50 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 20\!\cdots\!84 \nu^{11} + \cdots - 15\!\cdots\!00 ) / 14\!\cdots\!75 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 41\!\cdots\!32 \nu^{11} + \cdots - 11\!\cdots\!00 ) / 22\!\cdots\!75 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 41\!\cdots\!63 \nu^{11} + \cdots + 17\!\cdots\!00 ) / 44\!\cdots\!50 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\!\cdots\!87 \nu^{11} + \cdots + 24\!\cdots\!00 ) / 14\!\cdots\!50 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 15\!\cdots\!52 \nu^{11} + \cdots + 57\!\cdots\!00 ) / 89\!\cdots\!75 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 21\!\cdots\!74 \nu^{11} + \cdots - 62\!\cdots\!50 ) / 44\!\cdots\!50 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 21\!\cdots\!04 \nu^{11} + \cdots - 53\!\cdots\!50 ) / 44\!\cdots\!75 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 17\!\cdots\!06 \nu^{11} + \cdots + 72\!\cdots\!00 ) / 29\!\cdots\!50 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 62\!\cdots\!73 \nu^{11} + \cdots - 29\!\cdots\!00 ) / 44\!\cdots\!50 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 54\!\cdots\!63 \nu^{11} + \cdots + 15\!\cdots\!00 ) / 14\!\cdots\!50 \)
|
| \(\nu\) | \(=\) |
\( ( - 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - 4 \beta_{7} + 34 \beta_{5} + 17 \beta_{4} - 18 \beta_{3} + \cdots + 110 ) / 336 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 6 \beta_{11} + 12 \beta_{10} + 55 \beta_{9} + 96 \beta_{8} - 55 \beta_{7} - 96 \beta_{6} + \cdots - 55 ) / 336 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 890 \beta_{11} + 890 \beta_{10} + 5218 \beta_{9} - 2192 \beta_{8} + 2609 \beta_{7} - 4384 \beta_{6} + \cdots - 2484496 ) / 336 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 18992 \beta_{11} - 9496 \beta_{10} + 47835 \beta_{9} - 156872 \beta_{8} + 95670 \beta_{7} + \cdots - 104507805 ) / 168 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 580054 \beta_{11} - 1160108 \beta_{10} - 2005661 \beta_{9} - 2464220 \beta_{8} + 2005661 \beta_{7} + \cdots + 2005661 ) / 168 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2709928 \beta_{11} - 2709928 \beta_{10} - 23180026 \beta_{9} + 17758860 \beta_{8} + \cdots + 21331647928 ) / 24 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 1790529994 \beta_{11} + 895264997 \beta_{10} - 3336899693 \beta_{9} + 9145290168 \beta_{8} + \cdots + 5421811581227 ) / 168 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 8497690459 \beta_{11} + 16995380918 \beta_{10} + 34394833955 \beta_{9} + 50867224385 \beta_{8} + \cdots - 34394833955 ) / 42 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 1470767816921 \beta_{11} + 1470767816921 \beta_{10} + 11305285154420 \beta_{9} - 8017247786748 \beta_{8} + \cdots - 94\!\cdots\!32 ) / 168 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 58853308093290 \beta_{11} - 29426654046645 \beta_{10} + 116731168654249 \beta_{9} + \cdots - 19\!\cdots\!63 ) / 84 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 24\!\cdots\!83 \beta_{11} + \cdots + 96\!\cdots\!32 ) / 168 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−5.65685 | + | 9.79796i | −123.742 | + | 71.4423i | −64.0000 | − | 110.851i | 758.365 | + | 437.842i | − | 1616.56i | 0 | 1448.15 | 6927.51 | − | 11998.8i | −8579.92 | + | 4953.62i | |||||||||||||||||||||||||||||||||||||||||
| 19.2 | −5.65685 | + | 9.79796i | −24.5958 | + | 14.2004i | −64.0000 | − | 110.851i | −753.641 | − | 435.115i | − | 321.318i | 0 | 1448.15 | −2877.20 | + | 4983.45i | 8526.48 | − | 4922.77i | ||||||||||||||||||||||||||||||||||||||||||
| 19.3 | −5.65685 | + | 9.79796i | 90.8670 | − | 52.4621i | −64.0000 | − | 110.851i | −32.9005 | − | 18.9951i | 1187.08i | 0 | 1448.15 | 2224.04 | − | 3852.15i | 372.227 | − | 214.905i | |||||||||||||||||||||||||||||||||||||||||||
| 19.4 | 5.65685 | − | 9.79796i | −126.451 | + | 73.0064i | −64.0000 | − | 110.851i | −372.872 | − | 215.278i | 1651.95i | 0 | −1448.15 | 7379.38 | − | 12781.5i | −4218.56 | + | 2435.59i | |||||||||||||||||||||||||||||||||||||||||||
| 19.5 | 5.65685 | − | 9.79796i | 21.7328 | − | 12.5474i | −64.0000 | − | 110.851i | 492.158 | + | 284.147i | − | 283.915i | 0 | −1448.15 | −2965.62 | + | 5136.61i | 5568.13 | − | 3214.76i | ||||||||||||||||||||||||||||||||||||||||||
| 19.6 | 5.65685 | − | 9.79796i | 81.1886 | − | 46.8743i | −64.0000 | − | 110.851i | −928.109 | − | 535.844i | − | 1060.64i | 0 | −1448.15 | 1113.90 | − | 1929.33i | −10500.4 | + | 6062.38i | ||||||||||||||||||||||||||||||||||||||||||
| 31.1 | −5.65685 | − | 9.79796i | −123.742 | − | 71.4423i | −64.0000 | + | 110.851i | 758.365 | − | 437.842i | 1616.56i | 0 | 1448.15 | 6927.51 | + | 11998.8i | −8579.92 | − | 4953.62i | |||||||||||||||||||||||||||||||||||||||||||
| 31.2 | −5.65685 | − | 9.79796i | −24.5958 | − | 14.2004i | −64.0000 | + | 110.851i | −753.641 | + | 435.115i | 321.318i | 0 | 1448.15 | −2877.20 | − | 4983.45i | 8526.48 | + | 4922.77i | |||||||||||||||||||||||||||||||||||||||||||
| 31.3 | −5.65685 | − | 9.79796i | 90.8670 | + | 52.4621i | −64.0000 | + | 110.851i | −32.9005 | + | 18.9951i | − | 1187.08i | 0 | 1448.15 | 2224.04 | + | 3852.15i | 372.227 | + | 214.905i | ||||||||||||||||||||||||||||||||||||||||||
| 31.4 | 5.65685 | + | 9.79796i | −126.451 | − | 73.0064i | −64.0000 | + | 110.851i | −372.872 | + | 215.278i | − | 1651.95i | 0 | −1448.15 | 7379.38 | + | 12781.5i | −4218.56 | − | 2435.59i | ||||||||||||||||||||||||||||||||||||||||||
| 31.5 | 5.65685 | + | 9.79796i | 21.7328 | + | 12.5474i | −64.0000 | + | 110.851i | 492.158 | − | 284.147i | 283.915i | 0 | −1448.15 | −2965.62 | − | 5136.61i | 5568.13 | + | 3214.76i | |||||||||||||||||||||||||||||||||||||||||||
| 31.6 | 5.65685 | + | 9.79796i | 81.1886 | + | 46.8743i | −64.0000 | + | 110.851i | −928.109 | + | 535.844i | 1060.64i | 0 | −1448.15 | 1113.90 | + | 1929.33i | −10500.4 | − | 6062.38i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 98.9.d.b | 12 | |
| 7.b | odd | 2 | 1 | 14.9.d.a | ✓ | 12 | |
| 7.c | even | 3 | 1 | 14.9.d.a | ✓ | 12 | |
| 7.c | even | 3 | 1 | 98.9.b.c | 12 | ||
| 7.d | odd | 6 | 1 | 98.9.b.c | 12 | ||
| 7.d | odd | 6 | 1 | inner | 98.9.d.b | 12 | |
| 21.c | even | 2 | 1 | 126.9.n.b | 12 | ||
| 21.h | odd | 6 | 1 | 126.9.n.b | 12 | ||
| 28.d | even | 2 | 1 | 112.9.s.c | 12 | ||
| 28.g | odd | 6 | 1 | 112.9.s.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.9.d.a | ✓ | 12 | 7.b | odd | 2 | 1 | |
| 14.9.d.a | ✓ | 12 | 7.c | even | 3 | 1 | |
| 98.9.b.c | 12 | 7.c | even | 3 | 1 | ||
| 98.9.b.c | 12 | 7.d | odd | 6 | 1 | ||
| 98.9.d.b | 12 | 1.a | even | 1 | 1 | trivial | |
| 98.9.d.b | 12 | 7.d | odd | 6 | 1 | inner | |
| 112.9.s.c | 12 | 28.d | even | 2 | 1 | ||
| 112.9.s.c | 12 | 28.g | odd | 6 | 1 | ||
| 126.9.n.b | 12 | 21.c | even | 2 | 1 | ||
| 126.9.n.b | 12 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 162 T_{3}^{11} - 18363 T_{3}^{10} - 4391982 T_{3}^{9} + 378913158 T_{3}^{8} + \cdots + 21\!\cdots\!81 \)
acting on \(S_{9}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 128 T^{2} + 16384)^{3} \)
|
| $3$ |
\( T^{12} + \cdots + 21\!\cdots\!81 \)
|
| $5$ |
\( T^{12} + \cdots + 57\!\cdots\!25 \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} + \cdots + 48\!\cdots\!81 \)
|
| $13$ |
\( T^{12} + \cdots + 35\!\cdots\!24 \)
|
| $17$ |
\( T^{12} + \cdots + 30\!\cdots\!01 \)
|
| $19$ |
\( T^{12} + \cdots + 94\!\cdots\!25 \)
|
| $23$ |
\( T^{12} + \cdots + 25\!\cdots\!41 \)
|
| $29$ |
\( (T^{6} + \cdots - 79\!\cdots\!12)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 95\!\cdots\!29 \)
|
| $37$ |
\( T^{12} + \cdots + 20\!\cdots\!21 \)
|
| $41$ |
\( T^{12} + \cdots + 50\!\cdots\!96 \)
|
| $43$ |
\( (T^{6} + \cdots - 32\!\cdots\!64)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 85\!\cdots\!49 \)
|
| $53$ |
\( T^{12} + \cdots + 10\!\cdots\!41 \)
|
| $59$ |
\( T^{12} + \cdots + 28\!\cdots\!01 \)
|
| $61$ |
\( T^{12} + \cdots + 13\!\cdots\!69 \)
|
| $67$ |
\( T^{12} + \cdots + 23\!\cdots\!89 \)
|
| $71$ |
\( (T^{6} + \cdots - 21\!\cdots\!64)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 26\!\cdots\!49 \)
|
| $79$ |
\( T^{12} + \cdots + 92\!\cdots\!25 \)
|
| $83$ |
\( T^{12} + \cdots + 17\!\cdots\!84 \)
|
| $89$ |
\( T^{12} + \cdots + 13\!\cdots\!21 \)
|
| $97$ |
\( T^{12} + \cdots + 52\!\cdots\!16 \)
|
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