Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,6,Mod(1,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.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: | \(39.2940358542\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 246 x^{8} - 192 x^{7} + 20336 x^{6} + 25380 x^{5} - 639206 x^{4} - 722920 x^{3} + \cdots - 22888100 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3}\cdot 7^{5} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 246 x^{8} - 192 x^{7} + 20336 x^{6} + 25380 x^{5} - 639206 x^{4} - 722920 x^{3} + \cdots - 22888100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 433 \nu^{9} - 1947 \nu^{8} + 98973 \nu^{7} + 554383 \nu^{6} - 6721179 \nu^{5} + \cdots - 2613859980 ) / 69843200 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 83137 \nu^{9} + 200027 \nu^{8} - 19567597 \nu^{7} - 57068271 \nu^{6} + 1395513643 \nu^{5} + \cdots - 107486394420 ) / 9847891200 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 83137 \nu^{9} + 200027 \nu^{8} - 19567597 \nu^{7} - 57068271 \nu^{6} + 1395513643 \nu^{5} + \cdots - 590033063220 ) / 9847891200 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 76219 \nu^{9} + 438897 \nu^{8} - 21012719 \nu^{7} - 98631325 \nu^{6} + 1760844265 \nu^{5} + \cdots + 621408182180 ) / 4923945600 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 95695 \nu^{9} - 431909 \nu^{8} + 23439115 \nu^{7} + 118409649 \nu^{6} - 1626243037 \nu^{5} + \cdots - 737664945060 ) / 4923945600 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 271659 \nu^{9} - 2579273 \nu^{8} + 61777919 \nu^{7} + 579871301 \nu^{6} + \cdots - 1980608051620 ) / 9847891200 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 91631 \nu^{9} - 419483 \nu^{8} - 20161043 \nu^{7} + 77430063 \nu^{6} + 1507316885 \nu^{5} + \cdots - 489826503660 ) / 1969578240 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 25091 \nu^{9} + 50621 \nu^{8} - 5627811 \nu^{7} - 16286853 \nu^{6} + 381410769 \nu^{5} + \cdots + 64943739940 ) / 205164400 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta _1 + 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{4} - 2\beta_{3} + 4\beta_{2} + 80\beta _1 + 57 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} - 2\beta_{6} + 3\beta_{5} + 106\beta_{4} - 137\beta_{3} + 11\beta_{2} + 209\beta _1 + 3956 \)
|
| \(\nu^{5}\) | \(=\) |
\( 114 \beta_{9} - 112 \beta_{8} + 116 \beta_{7} + 264 \beta_{6} + 134 \beta_{5} + 336 \beta_{4} + \cdots + 10834 \)
|
| \(\nu^{6}\) | \(=\) |
\( 366 \beta_{9} - 76 \beta_{8} - 116 \beta_{6} + 480 \beta_{5} + 10577 \beta_{4} - 14513 \beta_{3} + \cdots + 369337 \)
|
| \(\nu^{7}\) | \(=\) |
\( 11979 \beta_{9} - 11273 \beta_{8} + 11413 \beta_{7} + 28830 \beta_{6} + 13957 \beta_{5} + 45098 \beta_{4} + \cdots + 1524905 \)
|
| \(\nu^{8}\) | \(=\) |
\( 50612 \beta_{9} - 17680 \beta_{8} - 1948 \beta_{7} + 11354 \beta_{6} + 64475 \beta_{5} + 1060034 \beta_{4} + \cdots + 36321428 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1265068 \beta_{9} - 1114544 \beta_{8} + 1075408 \beta_{7} + 3012396 \beta_{6} + 1388854 \beta_{5} + \cdots + 191858062 \)
|
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 |
|
−9.47445 | −20.3001 | 57.7653 | −25.0000 | 192.333 | 0 | −244.112 | 169.095 | 236.861 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.26930 | −1.27429 | 53.9199 | −25.0000 | 11.8118 | 0 | −203.182 | −241.376 | 231.732 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.72637 | 21.0476 | −18.1142 | −25.0000 | −78.4311 | 0 | 186.744 | 200.001 | 93.1592 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.48290 | −0.931256 | −19.8694 | −25.0000 | 3.24347 | 0 | 180.656 | −242.133 | 87.0725 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.633352 | −19.5171 | −31.5989 | −25.0000 | 12.3612 | 0 | 40.2804 | 137.918 | 15.8338 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.66104 | −29.0676 | −18.5968 | −25.0000 | −106.418 | 0 | −185.237 | 601.928 | −91.5259 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.37629 | 16.5213 | −3.09546 | −25.0000 | 88.8236 | 0 | −188.684 | 29.9545 | −134.407 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 7.46320 | 9.93685 | 23.6994 | −25.0000 | 74.1607 | 0 | −61.9494 | −144.259 | −186.580 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 9.75161 | −23.6367 | 63.0938 | −25.0000 | −230.496 | 0 | 303.214 | 315.692 | −243.790 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 10.3342 | −10.7786 | 74.7964 | −25.0000 | −111.389 | 0 | 442.268 | −126.821 | −258.356 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(7\) | \(1\) |
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 | 245.6.a.l | ✓ | 10 |
| 7.b | odd | 2 | 1 | 245.6.a.m | yes | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 245.6.a.l | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 245.6.a.m | yes | 10 | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):
|
\( T_{2}^{10} - 10 T_{2}^{9} - 201 T_{2}^{8} + 2040 T_{2}^{7} + 12314 T_{2}^{6} - 129840 T_{2}^{5} + \cdots - 10686592 \)
|
|
\( T_{3}^{10} + 58 T_{3}^{9} + 117 T_{3}^{8} - 44916 T_{3}^{7} - 571489 T_{3}^{6} + 9277842 T_{3}^{5} + \cdots - 12031198596 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 10 T^{9} + \cdots - 10686592 \)
|
| $3$ |
\( T^{10} + \cdots - 12031198596 \)
|
| $5$ |
\( (T + 25)^{10} \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( T^{10} + \cdots + 75\!\cdots\!00 \)
|
| $13$ |
\( T^{10} + \cdots - 10\!\cdots\!48 \)
|
| $17$ |
\( T^{10} + \cdots + 31\!\cdots\!28 \)
|
| $19$ |
\( T^{10} + \cdots - 53\!\cdots\!24 \)
|
| $23$ |
\( T^{10} + \cdots - 10\!\cdots\!96 \)
|
| $29$ |
\( T^{10} + \cdots + 20\!\cdots\!00 \)
|
| $31$ |
\( T^{10} + \cdots + 52\!\cdots\!48 \)
|
| $37$ |
\( T^{10} + \cdots - 14\!\cdots\!24 \)
|
| $41$ |
\( T^{10} + \cdots + 20\!\cdots\!16 \)
|
| $43$ |
\( T^{10} + \cdots - 31\!\cdots\!00 \)
|
| $47$ |
\( T^{10} + \cdots + 99\!\cdots\!00 \)
|
| $53$ |
\( T^{10} + \cdots - 52\!\cdots\!48 \)
|
| $59$ |
\( T^{10} + \cdots - 26\!\cdots\!64 \)
|
| $61$ |
\( T^{10} + \cdots - 11\!\cdots\!68 \)
|
| $67$ |
\( T^{10} + \cdots - 35\!\cdots\!32 \)
|
| $71$ |
\( T^{10} + \cdots + 13\!\cdots\!28 \)
|
| $73$ |
\( T^{10} + \cdots - 37\!\cdots\!32 \)
|
| $79$ |
\( T^{10} + \cdots + 56\!\cdots\!12 \)
|
| $83$ |
\( T^{10} + \cdots - 42\!\cdots\!00 \)
|
| $89$ |
\( T^{10} + \cdots - 31\!\cdots\!96 \)
|
| $97$ |
\( T^{10} + \cdots + 30\!\cdots\!88 \)
|
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