Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [61,6,Mod(1,61)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 61 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 61.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: | \(9.78341300859\) |
| Analytic rank: | \(1\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 2 x^{10} - 229 x^{9} + 328 x^{8} + 19183 x^{7} - 17298 x^{6} - 718243 x^{5} + 321732 x^{4} + \cdots - 10818720 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 2 x^{10} - 229 x^{9} + 328 x^{8} + 19183 x^{7} - 17298 x^{6} - 718243 x^{5} + 321732 x^{4} + \cdots - 10818720 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 48648949 \nu^{10} - 21285584747 \nu^{9} + 4238770550 \nu^{8} + 4468245266890 \nu^{7} + \cdots - 99\!\cdots\!00 ) / 257784015998976 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7951781 \nu^{10} - 847359455 \nu^{9} - 704696522 \nu^{8} + 172580067202 \nu^{7} + \cdots - 881317438459936 ) / 10741000666624 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 272686279 \nu^{10} + 672603231 \nu^{9} - 63068397798 \nu^{8} - 206423950450 \nu^{7} + \cdots + 765031459938208 ) / 42964002666496 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1909737467 \nu^{10} + 4705620587 \nu^{9} - 398979924422 \nu^{8} - 1337423753194 \nu^{7} + \cdots + 16\!\cdots\!28 ) / 257784015998976 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 2394542797 \nu^{10} + 18953294771 \nu^{9} + 470910183706 \nu^{8} + \cdots + 12\!\cdots\!84 ) / 257784015998976 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2504964101 \nu^{10} - 14925477547 \nu^{9} - 507189158330 \nu^{8} + 2768356666730 \nu^{7} + \cdots - 43\!\cdots\!88 ) / 257784015998976 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 2504964101 \nu^{10} - 14925477547 \nu^{9} - 507189158330 \nu^{8} + 2768356666730 \nu^{7} + \cdots - 15\!\cdots\!80 ) / 257784015998976 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 861168741 \nu^{10} - 5081812955 \nu^{9} - 174069682794 \nu^{8} + 1023827004394 \nu^{7} + \cdots - 39\!\cdots\!00 ) / 85928005332992 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 745062463 \nu^{10} - 1846792171 \nu^{9} + 164822625466 \nu^{8} + 489455520722 \nu^{7} + \cdots + 31\!\cdots\!08 ) / 64446003999744 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} - \beta_{7} + \beta _1 + 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{10} - \beta_{9} - 2\beta_{8} - \beta_{7} + 3\beta_{5} - 3\beta_{4} + \beta_{3} + 3\beta_{2} + 63\beta _1 + 22 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2 \beta_{10} + 85 \beta_{8} - 78 \beta_{7} + 7 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} + \cdots + 2653 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 317 \beta_{10} - 79 \beta_{9} - 192 \beta_{8} - 95 \beta_{7} + 58 \beta_{6} + 323 \beta_{5} + \cdots + 3072 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 40 \beta_{10} - 142 \beta_{9} + 6659 \beta_{8} - 5792 \beta_{7} + 873 \beta_{6} + 581 \beta_{5} + \cdots + 193023 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 27651 \beta_{10} - 4721 \beta_{9} - 14558 \beta_{8} - 9097 \beta_{7} + 8468 \beta_{6} + 28407 \beta_{5} + \cdots + 346314 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 24202 \beta_{10} - 24780 \beta_{9} + 515321 \beta_{8} - 435366 \beta_{7} + 83023 \beta_{6} + \cdots + 14866021 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2298489 \beta_{10} - 229475 \beta_{9} - 999948 \beta_{8} - 916939 \beta_{7} + 891654 \beta_{6} + \cdots + 35813676 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 3646060 \beta_{10} - 2930362 \beta_{9} + 39937327 \beta_{8} - 33355260 \beta_{7} + 7200261 \beta_{6} + \cdots + 1174650491 \)
|
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 |
|
−9.70013 | 18.7856 | 62.0925 | −66.3101 | −182.223 | 20.2041 | −291.901 | 109.900 | 643.217 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.36970 | −28.5720 | 55.7913 | −51.2793 | 267.711 | 132.317 | −222.918 | 573.361 | 480.471 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −6.87405 | −10.8865 | 15.2526 | 10.3300 | 74.8343 | 113.931 | 115.123 | −124.484 | −71.0092 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −5.27117 | −26.3971 | −4.21475 | 100.474 | 139.144 | −57.9925 | 190.894 | 453.810 | −529.614 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −4.99171 | 17.8279 | −7.08282 | 20.7662 | −88.9918 | −212.179 | 195.090 | 74.8348 | −103.659 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.15158 | 0.962237 | −30.6739 | 27.6033 | −1.10810 | 175.817 | 72.1742 | −242.074 | −31.7875 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.99703 | 16.2235 | −23.0178 | −69.0097 | 48.6225 | −91.0191 | −164.890 | 20.2035 | −206.824 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.42744 | 7.03345 | −20.2526 | −26.3029 | 24.1067 | −63.8432 | −179.093 | −193.531 | −90.1515 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 5.53582 | −16.1415 | −1.35467 | 69.2247 | −89.3567 | −94.3410 | −184.646 | 17.5494 | 383.216 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 8.12124 | −11.7012 | 33.9545 | −45.6792 | −95.0279 | −182.149 | 15.8726 | −106.083 | −370.971 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 8.27682 | −13.1344 | 36.5057 | −98.8167 | −108.711 | 172.253 | 37.2930 | −70.4872 | −817.888 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(61\) | \(1\) |
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 | 61.6.a.a | ✓ | 11 |
| 3.b | odd | 2 | 1 | 549.6.a.b | 11 | ||
| 4.b | odd | 2 | 1 | 976.6.a.g | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 61.6.a.a | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 549.6.a.b | 11 | 3.b | odd | 2 | 1 | ||
| 976.6.a.g | 11 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 9 T_{2}^{10} - 194 T_{2}^{9} - 1658 T_{2}^{8} + 13653 T_{2}^{7} + 106973 T_{2}^{6} + \cdots - 72359744 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(61))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + 9 T^{10} + \cdots - 72359744 \)
|
| $3$ |
\( T^{11} + \cdots - 749020807296 \)
|
| $5$ |
\( T^{11} + \cdots - 11\!\cdots\!68 \)
|
| $7$ |
\( T^{11} + \cdots - 11\!\cdots\!31 \)
|
| $11$ |
\( T^{11} + \cdots - 17\!\cdots\!20 \)
|
| $13$ |
\( T^{11} + \cdots + 89\!\cdots\!32 \)
|
| $17$ |
\( T^{11} + \cdots - 10\!\cdots\!48 \)
|
| $19$ |
\( T^{11} + \cdots - 24\!\cdots\!36 \)
|
| $23$ |
\( T^{11} + \cdots - 22\!\cdots\!81 \)
|
| $29$ |
\( T^{11} + \cdots + 99\!\cdots\!52 \)
|
| $31$ |
\( T^{11} + \cdots + 10\!\cdots\!60 \)
|
| $37$ |
\( T^{11} + \cdots - 22\!\cdots\!56 \)
|
| $41$ |
\( T^{11} + \cdots + 31\!\cdots\!75 \)
|
| $43$ |
\( T^{11} + \cdots - 51\!\cdots\!56 \)
|
| $47$ |
\( T^{11} + \cdots + 12\!\cdots\!12 \)
|
| $53$ |
\( T^{11} + \cdots + 12\!\cdots\!28 \)
|
| $59$ |
\( T^{11} + \cdots + 34\!\cdots\!04 \)
|
| $61$ |
\( (T + 3721)^{11} \)
|
| $67$ |
\( T^{11} + \cdots - 19\!\cdots\!68 \)
|
| $71$ |
\( T^{11} + \cdots - 34\!\cdots\!68 \)
|
| $73$ |
\( T^{11} + \cdots - 10\!\cdots\!69 \)
|
| $79$ |
\( T^{11} + \cdots + 67\!\cdots\!36 \)
|
| $83$ |
\( T^{11} + \cdots - 17\!\cdots\!32 \)
|
| $89$ |
\( T^{11} + \cdots + 15\!\cdots\!44 \)
|
| $97$ |
\( T^{11} + \cdots + 17\!\cdots\!56 \)
|
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