Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [693,6,Mod(1,693)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(693, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("693.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 693 = 3^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 693.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: | \(111.145987130\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{12} - 4 x^{11} - 277 x^{10} + 1059 x^{9} + 27146 x^{8} - 95119 x^{7} - 1117110 x^{6} + \cdots + 125048448 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4 x^{11} - 277 x^{10} + 1059 x^{9} + 27146 x^{8} - 95119 x^{7} - 1117110 x^{6} + \cdots + 125048448 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 48 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 81\nu + 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 119863 \nu^{11} + 6509930 \nu^{10} + 49599471 \nu^{9} - 1534814339 \nu^{8} + \cdots - 40403952826048 ) / 41725100032 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 192733 \nu^{11} - 1046898 \nu^{10} + 33707461 \nu^{9} + 163655135 \nu^{8} + \cdots - 23975436433984 ) / 41725100032 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 777431 \nu^{11} - 2691882 \nu^{10} - 198510479 \nu^{9} + 773587363 \nu^{8} + \cdots + 24245669101248 ) / 125175300096 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 478863 \nu^{11} + 6348026 \nu^{10} + 119342343 \nu^{9} - 1660380731 \nu^{8} + \cdots - 34238128284864 ) / 62587650048 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 297057 \nu^{11} - 3433958 \nu^{10} - 79715817 \nu^{9} + 838407989 \nu^{8} + \cdots + 28917853581120 ) / 31293825024 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 2736421 \nu^{11} + 32755262 \nu^{10} + 767157325 \nu^{9} - 7990399753 \nu^{8} + \cdots - 145961837732928 ) / 125175300096 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 233111 \nu^{11} + 1742026 \nu^{10} + 57222575 \nu^{9} - 465980003 \nu^{8} + \cdots - 8007249931968 ) / 7823456256 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 702789 \nu^{11} + 3605918 \nu^{10} + 178718445 \nu^{9} - 995285705 \nu^{8} + \cdots - 44675145635904 ) / 15646912512 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 48 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 81\beta _1 - 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{11} + \beta_{7} - 6\beta_{6} + \beta_{3} + 105\beta_{2} - 21\beta _1 + 3885 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{11} + \beta_{10} - 5 \beta_{9} - 7 \beta_{8} + 2 \beta_{7} + 6 \beta_{6} - 10 \beta_{5} + \cdots - 1923 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 158 \beta_{11} - 15 \beta_{10} + 9 \beta_{9} + \beta_{8} + 160 \beta_{7} - 954 \beta_{6} + \cdots + 350024 \)
|
| \(\nu^{7}\) | \(=\) |
\( 361 \beta_{11} + 233 \beta_{10} - 939 \beta_{9} - 1207 \beta_{8} + 157 \beta_{7} + 1040 \beta_{6} + \cdots - 236560 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 19162 \beta_{11} - 3062 \beta_{10} + 3264 \beta_{9} + 978 \beta_{8} + 19620 \beta_{7} + \cdots + 33001728 \)
|
| \(\nu^{9}\) | \(=\) |
\( 49012 \beta_{11} + 36816 \beta_{10} - 132560 \beta_{9} - 155456 \beta_{8} - 196 \beta_{7} + \cdots - 26888109 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 2118553 \beta_{11} - 454020 \beta_{10} + 638508 \beta_{9} + 232876 \beta_{8} + 2213457 \beta_{7} + \cdots + 3192673629 \)
|
| \(\nu^{11}\) | \(=\) |
\( 5997694 \beta_{11} + 5010557 \beta_{10} - 16845401 \beta_{9} - 18097195 \beta_{8} - 2037394 \beta_{7} + \cdots - 3080635799 \)
|
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 |
|
−10.3159 | 0 | 74.4169 | 4.71172 | 0 | −49.0000 | −437.567 | 0 | −48.6055 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.30104 | 0 | 54.5093 | 38.7594 | 0 | −49.0000 | −209.360 | 0 | −360.503 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −7.66494 | 0 | 26.7513 | −83.5606 | 0 | −49.0000 | 40.2310 | 0 | 640.487 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.86870 | 0 | −23.7705 | −7.51580 | 0 | −49.0000 | 159.989 | 0 | 21.5606 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.56002 | 0 | −25.4463 | 107.748 | 0 | −49.0000 | 147.064 | 0 | −275.837 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.32667 | 0 | −30.2400 | −65.5903 | 0 | −49.0000 | 82.5717 | 0 | 87.0164 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.06319 | 0 | −30.8696 | 78.7831 | 0 | −49.0000 | −66.8423 | 0 | 83.7613 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.69241 | 0 | −18.3661 | −6.89646 | 0 | −49.0000 | −185.972 | 0 | −25.4646 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 6.39272 | 0 | 8.86689 | 44.6421 | 0 | −49.0000 | −147.884 | 0 | 285.385 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 7.11124 | 0 | 18.5697 | −75.8581 | 0 | −49.0000 | −95.5060 | 0 | −539.445 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 9.87858 | 0 | 65.5863 | 79.7631 | 0 | −49.0000 | 331.785 | 0 | 787.946 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 9.89909 | 0 | 65.9920 | −64.9859 | 0 | −49.0000 | 336.490 | 0 | −643.301 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 693.6.a.q | yes | 12 |
| 3.b | odd | 2 | 1 | 693.6.a.p | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 693.6.a.p | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 693.6.a.q | yes | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(693))\):
|
\( T_{2}^{12} - 4 T_{2}^{11} - 277 T_{2}^{10} + 1059 T_{2}^{9} + 27146 T_{2}^{8} - 95119 T_{2}^{7} + \cdots + 125048448 \)
|
|
\( T_{5}^{12} - 50 T_{5}^{11} - 23281 T_{5}^{10} + 933280 T_{5}^{9} + 199546635 T_{5}^{8} + \cdots + 77\!\cdots\!96 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 125048448 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots + 77\!\cdots\!96 \)
|
| $7$ |
\( (T + 49)^{12} \)
|
| $11$ |
\( (T - 121)^{12} \)
|
| $13$ |
\( T^{12} + \cdots + 58\!\cdots\!36 \)
|
| $17$ |
\( T^{12} + \cdots - 16\!\cdots\!24 \)
|
| $19$ |
\( T^{12} + \cdots + 11\!\cdots\!44 \)
|
| $23$ |
\( T^{12} + \cdots + 67\!\cdots\!72 \)
|
| $29$ |
\( T^{12} + \cdots + 23\!\cdots\!16 \)
|
| $31$ |
\( T^{12} + \cdots + 27\!\cdots\!96 \)
|
| $37$ |
\( T^{12} + \cdots + 19\!\cdots\!64 \)
|
| $41$ |
\( T^{12} + \cdots + 50\!\cdots\!60 \)
|
| $43$ |
\( T^{12} + \cdots + 63\!\cdots\!92 \)
|
| $47$ |
\( T^{12} + \cdots - 48\!\cdots\!48 \)
|
| $53$ |
\( T^{12} + \cdots + 16\!\cdots\!40 \)
|
| $59$ |
\( T^{12} + \cdots - 17\!\cdots\!72 \)
|
| $61$ |
\( T^{12} + \cdots - 11\!\cdots\!92 \)
|
| $67$ |
\( T^{12} + \cdots - 54\!\cdots\!32 \)
|
| $71$ |
\( T^{12} + \cdots - 11\!\cdots\!60 \)
|
| $73$ |
\( T^{12} + \cdots + 10\!\cdots\!56 \)
|
| $79$ |
\( T^{12} + \cdots - 18\!\cdots\!60 \)
|
| $83$ |
\( T^{12} + \cdots - 42\!\cdots\!00 \)
|
| $89$ |
\( T^{12} + \cdots + 45\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots + 31\!\cdots\!00 \)
|
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