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: | \(1\) |
| 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: |
\( x^{12} - 279 x^{10} - 35 x^{9} + 28301 x^{8} + 8817 x^{7} - 1268995 x^{6} - 693685 x^{5} + \cdots + 469284840 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 279 x^{10} - 35 x^{9} + 28301 x^{8} + 8817 x^{7} - 1268995 x^{6} - 693685 x^{5} + \cdots + 469284840 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 47 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 76\nu - 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 61 \nu^{11} - 7666 \nu^{10} - 7161 \nu^{9} + 1486189 \nu^{8} + 995257 \nu^{7} + \cdots - 138326291340 ) / 551620608 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 9 \nu^{11} + 1459 \nu^{10} - 25036 \nu^{9} - 402835 \nu^{8} + 5904100 \nu^{7} + \cdots + 30959432352 ) / 137905152 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 691 \nu^{11} + 10512 \nu^{10} + 171215 \nu^{9} - 2869893 \nu^{8} - 15132223 \nu^{7} + \cdots - 735685797900 ) / 275810304 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4989 \nu^{11} - 16864 \nu^{10} - 1326545 \nu^{9} + 4366603 \nu^{8} + 124506977 \nu^{7} + \cdots + 699455452020 ) / 551620608 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 313 \nu^{11} + 1439 \nu^{10} + 85824 \nu^{9} - 361523 \nu^{8} - 8339624 \nu^{7} + \cdots - 43486059024 ) / 25073664 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 4247 \nu^{11} - 11882 \nu^{10} - 1137653 \nu^{9} + 3088985 \nu^{8} + 108778485 \nu^{7} + \cdots + 531106203684 ) / 275810304 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 479 \nu^{11} + 1618 \nu^{10} + 128997 \nu^{9} - 407857 \nu^{8} - 12295141 \nu^{7} + \cdots - 61483180644 ) / 10407936 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 9199 \nu^{11} + 35250 \nu^{10} + 2444629 \nu^{9} - 9016353 \nu^{8} - 229145109 \nu^{7} + \cdots - 1273146037540 ) / 91936768 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 47 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 76\beta _1 + 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} - \beta_{10} + \beta_{9} + 4\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 103\beta_{2} - 9\beta _1 + 3592 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{11} - 9 \beta_{10} - 7 \beta_{9} - 2 \beta_{8} - 26 \beta_{7} - \beta_{6} - 5 \beta_{5} + \cdots + 415 \)
|
| \(\nu^{6}\) | \(=\) |
\( 145 \beta_{11} - 121 \beta_{10} + 169 \beta_{9} - 36 \beta_{8} + 616 \beta_{7} - 129 \beta_{6} + \cdots + 311395 \)
|
| \(\nu^{7}\) | \(=\) |
\( 64 \beta_{11} - 1440 \beta_{10} - 960 \beta_{9} - 222 \beta_{8} - 5362 \beta_{7} - 168 \beta_{6} + \cdots - 17928 \)
|
| \(\nu^{8}\) | \(=\) |
\( 16850 \beta_{11} - 11546 \beta_{10} + 20218 \beta_{9} - 7878 \beta_{8} + 78686 \beta_{7} + \cdots + 28202423 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 3334 \beta_{11} - 172154 \beta_{10} - 98566 \beta_{9} - 9786 \beta_{8} - 767294 \beta_{7} + \cdots - 7301207 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1832179 \beta_{11} - 1011003 \beta_{10} + 2148939 \beta_{9} - 1176070 \beta_{8} + 9481154 \beta_{7} + \cdots + 2611326516 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 1531257 \beta_{11} - 18599727 \beta_{10} - 9117841 \beta_{9} + 1101700 \beta_{8} - 95376840 \beta_{7} + \cdots - 1244940257 \)
|
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 |
|
−11.0778 | 0 | 90.7179 | 110.200 | 0 | 49.0000 | −650.466 | 0 | −1220.78 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −10.2995 | 0 | 74.0794 | −81.5391 | 0 | 49.0000 | −433.397 | 0 | 839.811 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −8.14167 | 0 | 34.2868 | −2.94932 | 0 | 49.0000 | −18.6184 | 0 | 24.0124 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.81524 | 0 | −8.81347 | 38.9524 | 0 | 49.0000 | 196.527 | 0 | −187.565 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −4.50289 | 0 | −11.7240 | −89.0380 | 0 | 49.0000 | 196.884 | 0 | 400.928 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −3.92693 | 0 | −16.5792 | 33.1169 | 0 | 49.0000 | 190.767 | 0 | −130.048 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.605328 | 0 | −31.6336 | 49.4462 | 0 | 49.0000 | −38.5192 | 0 | 29.9311 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.04965 | 0 | −27.7989 | −17.1508 | 0 | 49.0000 | −122.567 | 0 | −35.1532 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.09906 | 0 | −15.1977 | −94.3884 | 0 | 49.0000 | −193.466 | 0 | −386.903 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 6.72044 | 0 | 13.1644 | 64.6787 | 0 | 49.0000 | −126.584 | 0 | 434.670 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 8.51421 | 0 | 40.4917 | −62.3391 | 0 | 49.0000 | 72.3002 | 0 | −530.768 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 8.77535 | 0 | 45.0067 | 1.01020 | 0 | 49.0000 | 114.138 | 0 | 8.86489 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.o | ✓ | 12 |
| 3.b | odd | 2 | 1 | 693.6.a.r | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 693.6.a.o | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 693.6.a.r | yes | 12 | 3.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(693))\):
|
\( T_{2}^{12} + 12 T_{2}^{11} - 213 T_{2}^{10} - 2605 T_{2}^{9} + 15926 T_{2}^{8} + 201277 T_{2}^{7} + \cdots + 201979008 \)
|
|
\( T_{5}^{12} + 50 T_{5}^{11} - 23281 T_{5}^{10} - 1035360 T_{5}^{9} + 184979755 T_{5}^{8} + \cdots + 99\!\cdots\!16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 201979008 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots + 99\!\cdots\!16 \)
|
| $7$ |
\( (T - 49)^{12} \)
|
| $11$ |
\( (T - 121)^{12} \)
|
| $13$ |
\( T^{12} + \cdots + 13\!\cdots\!52 \)
|
| $17$ |
\( T^{12} + \cdots + 37\!\cdots\!80 \)
|
| $19$ |
\( T^{12} + \cdots - 79\!\cdots\!64 \)
|
| $23$ |
\( T^{12} + \cdots - 11\!\cdots\!16 \)
|
| $29$ |
\( T^{12} + \cdots - 64\!\cdots\!80 \)
|
| $31$ |
\( T^{12} + \cdots + 20\!\cdots\!20 \)
|
| $37$ |
\( T^{12} + \cdots - 14\!\cdots\!24 \)
|
| $41$ |
\( T^{12} + \cdots - 31\!\cdots\!88 \)
|
| $43$ |
\( T^{12} + \cdots + 91\!\cdots\!84 \)
|
| $47$ |
\( T^{12} + \cdots - 35\!\cdots\!12 \)
|
| $53$ |
\( T^{12} + \cdots + 93\!\cdots\!00 \)
|
| $59$ |
\( T^{12} + \cdots - 33\!\cdots\!72 \)
|
| $61$ |
\( T^{12} + \cdots + 34\!\cdots\!40 \)
|
| $67$ |
\( T^{12} + \cdots - 45\!\cdots\!16 \)
|
| $71$ |
\( T^{12} + \cdots - 74\!\cdots\!40 \)
|
| $73$ |
\( T^{12} + \cdots + 20\!\cdots\!40 \)
|
| $79$ |
\( T^{12} + \cdots - 15\!\cdots\!20 \)
|
| $83$ |
\( T^{12} + \cdots - 39\!\cdots\!08 \)
|
| $89$ |
\( T^{12} + \cdots - 39\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots + 67\!\cdots\!24 \)
|
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