Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [605,4,Mod(1,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 605.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: | \(35.6961555535\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{11} - 67 x^{10} + 65 x^{9} + 1558 x^{8} - 1475 x^{7} - 14915 x^{6} + 12951 x^{5} + \cdots + 27856 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5\cdot 11^{2} \) |
| Twist minimal: | no (minimal twist has level 55) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - x^{11} - 67 x^{10} + 65 x^{9} + 1558 x^{8} - 1475 x^{7} - 14915 x^{6} + 12951 x^{5} + \cdots + 27856 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1107391 \nu^{11} + 7673220 \nu^{10} + 70067177 \nu^{9} - 487545538 \nu^{8} + \cdots - 25808536144 ) / 3033701760 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1547143 \nu^{11} - 5364860 \nu^{10} - 98589101 \nu^{9} + 346707534 \nu^{8} + 2080085148 \nu^{7} + \cdots + 54810001872 ) / 1516850880 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3213147 \nu^{11} - 5066460 \nu^{10} - 213226829 \nu^{9} + 325869666 \nu^{8} + \cdots + 58802064848 ) / 3033701760 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4650909 \nu^{11} - 4351580 \nu^{10} - 298006803 \nu^{9} + 269668622 \nu^{8} + \cdots - 115681745104 ) / 3033701760 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 732784 \nu^{11} - 1688045 \nu^{10} - 46784388 \nu^{9} + 107260687 \nu^{8} + 990422074 \nu^{7} + \cdots + 11388669576 ) / 379212720 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 3817717 \nu^{11} + 5069480 \nu^{10} + 246143239 \nu^{9} - 330363646 \nu^{8} + \cdots - 44614066288 ) / 1516850880 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 4071101 \nu^{11} - 5076600 \nu^{10} - 264479807 \nu^{9} + 326978918 \nu^{8} + \cdots + 57175693584 ) / 1516850880 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 551436 \nu^{11} - 1451975 \nu^{10} - 35084837 \nu^{9} + 93506228 \nu^{8} + 742205606 \nu^{7} + \cdots + 12332638144 ) / 189606360 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 10483571 \nu^{11} - 18333020 \nu^{10} - 668387117 \nu^{9} + 1173224538 \nu^{8} + \cdots + 156036728784 ) / 3033701760 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{10} - \beta_{7} + \beta_{4} - \beta_{3} + 21\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{10} + 3\beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - 3\beta_{5} + 3\beta_{4} + 27\beta_{2} - \beta _1 + 225 \)
|
| \(\nu^{5}\) | \(=\) |
\( 40 \beta_{11} - 35 \beta_{10} - 5 \beta_{9} + 6 \beta_{8} - 40 \beta_{7} + 4 \beta_{6} + 9 \beta_{5} + \cdots - 20 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 22 \beta_{11} - 82 \beta_{10} + 154 \beta_{9} + 50 \beta_{8} + 42 \beta_{7} - 26 \beta_{6} + \cdots + 5342 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1268 \beta_{11} - 1042 \beta_{10} - 168 \beta_{9} + 308 \beta_{8} - 1320 \beta_{7} + 204 \beta_{6} + \cdots + 203 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1177 \beta_{11} - 2701 \beta_{10} + 5728 \beta_{9} + 1752 \beta_{8} + 1249 \beta_{7} - 568 \beta_{6} + \cdots + 135553 \)
|
| \(\nu^{9}\) | \(=\) |
\( 37662 \beta_{11} - 29810 \beta_{10} - 4155 \beta_{9} + 11547 \beta_{8} - 40611 \beta_{7} + 7485 \beta_{6} + \cdots + 28119 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 43872 \beta_{11} - 83631 \beta_{10} + 188539 \beta_{9} + 54844 \beta_{8} + 32146 \beta_{7} + \cdots + 3583236 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1089982 \beta_{11} - 845122 \beta_{10} - 85396 \beta_{9} + 384822 \beta_{8} - 1207126 \beta_{7} + \cdots + 1372748 \)
|
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 |
|
−5.39954 | 6.27988 | 21.1550 | −5.00000 | −33.9084 | 1.92176 | −71.0308 | 12.4368 | 26.9977 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.36383 | −6.47441 | 11.0430 | −5.00000 | 28.2532 | 24.2107 | −13.2791 | 14.9180 | 21.8191 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.48072 | −4.49803 | 4.11544 | −5.00000 | 15.6564 | 0.859797 | 13.5211 | −6.76775 | 17.4036 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.17093 | 8.94509 | −3.28705 | −5.00000 | −19.4192 | −13.6976 | 24.5034 | 53.0147 | 10.8547 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.85253 | −5.02253 | −4.56812 | −5.00000 | 9.30441 | −33.4856 | 23.2829 | −1.77419 | 9.26267 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.696762 | 2.36331 | −7.51452 | −5.00000 | −1.64666 | 26.2619 | 10.8099 | −21.4148 | 3.48381 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.747441 | 1.20483 | −7.44133 | −5.00000 | 0.900538 | 18.4546 | −11.5415 | −25.5484 | −3.73720 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.25858 | −9.09998 | −6.41597 | −5.00000 | −11.4531 | −16.3005 | −18.1437 | 55.8097 | −6.29291 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.43938 | 9.32704 | −5.92820 | −5.00000 | 13.4251 | −14.6375 | −20.0479 | 59.9937 | −7.19688 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 3.94598 | 5.15950 | 7.57072 | −5.00000 | 20.3593 | −18.9445 | −1.69392 | −0.379584 | −19.7299 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 4.31191 | −6.66658 | 10.5926 | −5.00000 | −28.7457 | 17.4967 | 11.1791 | 17.4433 | −21.5596 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 5.26103 | −0.518116 | 19.6784 | −5.00000 | −2.72582 | −33.1399 | 61.4406 | −26.7316 | −26.3051 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 605.4.a.q | 12 | |
| 11.b | odd | 2 | 1 | 605.4.a.s | 12 | ||
| 11.d | odd | 10 | 2 | 55.4.g.a | ✓ | 24 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.4.g.a | ✓ | 24 | 11.d | odd | 10 | 2 | |
| 605.4.a.q | 12 | 1.a | even | 1 | 1 | trivial | |
| 605.4.a.s | 12 | 11.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(605))\):
|
\( T_{2}^{12} + T_{2}^{11} - 67 T_{2}^{10} - 65 T_{2}^{9} + 1558 T_{2}^{8} + 1475 T_{2}^{7} - 14915 T_{2}^{6} + \cdots + 27856 \)
|
|
\( T_{3}^{12} - T_{3}^{11} - 227 T_{3}^{10} + 98 T_{3}^{9} + 18918 T_{3}^{8} + 620 T_{3}^{7} + \cdots + 35387420 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + T^{11} + \cdots + 27856 \)
|
| $3$ |
\( T^{12} - T^{11} + \cdots + 35387420 \)
|
| $5$ |
\( (T + 5)^{12} \)
|
| $7$ |
\( T^{12} + \cdots + 23307327589120 \)
|
| $11$ |
\( T^{12} \)
|
| $13$ |
\( T^{12} + \cdots + 60\!\cdots\!00 \)
|
| $17$ |
\( T^{12} + \cdots + 20\!\cdots\!16 \)
|
| $19$ |
\( T^{12} + \cdots - 41\!\cdots\!25 \)
|
| $23$ |
\( T^{12} + \cdots - 43\!\cdots\!96 \)
|
| $29$ |
\( T^{12} + \cdots + 38\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots + 53\!\cdots\!44 \)
|
| $37$ |
\( T^{12} + \cdots + 54\!\cdots\!84 \)
|
| $41$ |
\( T^{12} + \cdots + 44\!\cdots\!01 \)
|
| $43$ |
\( T^{12} + \cdots - 37\!\cdots\!76 \)
|
| $47$ |
\( T^{12} + \cdots - 14\!\cdots\!04 \)
|
| $53$ |
\( T^{12} + \cdots + 65\!\cdots\!76 \)
|
| $59$ |
\( T^{12} + \cdots - 97\!\cdots\!25 \)
|
| $61$ |
\( T^{12} + \cdots - 21\!\cdots\!80 \)
|
| $67$ |
\( T^{12} + \cdots - 51\!\cdots\!36 \)
|
| $71$ |
\( T^{12} + \cdots + 12\!\cdots\!84 \)
|
| $73$ |
\( T^{12} + \cdots + 83\!\cdots\!64 \)
|
| $79$ |
\( T^{12} + \cdots - 42\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots + 10\!\cdots\!96 \)
|
| $89$ |
\( T^{12} + \cdots - 60\!\cdots\!75 \)
|
| $97$ |
\( T^{12} + \cdots + 13\!\cdots\!00 \)
|
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