Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [22,6,Mod(3,22)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([8]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22.3");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 22 = 2 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 22.c (of order \(5\), degree \(4\), minimal) |
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: | no |
| Analytic conductor: | \(3.52844403589\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} + 329 x^{10} + 1365 x^{9} + 52729 x^{8} + 242031 x^{7} + 7017903 x^{6} + \cdots + 93740681241 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 11^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{12} - 2 x^{11} + 329 x^{10} + 1365 x^{9} + 52729 x^{8} + 242031 x^{7} + 7017903 x^{6} + \cdots + 93740681241 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 25\!\cdots\!10 \nu^{11} + \cdots + 12\!\cdots\!99 ) / 28\!\cdots\!87 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 87\!\cdots\!90 \nu^{11} + \cdots + 59\!\cdots\!36 ) / 28\!\cdots\!87 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 99\!\cdots\!27 \nu^{11} + \cdots + 14\!\cdots\!39 ) / 28\!\cdots\!87 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 25\!\cdots\!34 \nu^{11} + \cdots + 30\!\cdots\!57 ) / 62\!\cdots\!14 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 26\!\cdots\!64 \nu^{11} + \cdots - 15\!\cdots\!61 ) / 61\!\cdots\!02 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 96\!\cdots\!09 \nu^{11} + \cdots - 27\!\cdots\!15 ) / 20\!\cdots\!38 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 55\!\cdots\!37 \nu^{11} + \cdots + 37\!\cdots\!10 ) / 11\!\cdots\!02 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 25\!\cdots\!04 \nu^{11} + \cdots + 11\!\cdots\!55 ) / 29\!\cdots\!78 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 62\!\cdots\!54 \nu^{11} + \cdots - 92\!\cdots\!25 ) / 62\!\cdots\!14 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 32\!\cdots\!21 \nu^{11} + \cdots - 22\!\cdots\!76 ) / 20\!\cdots\!38 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 18\!\cdots\!83 \nu^{11} + \cdots - 78\!\cdots\!46 ) / 98\!\cdots\!26 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{10} + \beta_{8} - 12\beta_{7} + \beta_{6} - 7\beta_{5} + 7\beta_{4} - 4\beta_{2} - 6\beta _1 + 6 ) / 22 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -23\beta_{11} - 39\beta_{9} + 29\beta_{8} - 39\beta_{7} - 18\beta_{4} - 618\beta_{3} - 2940\beta _1 - 618 ) / 22 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 28 \beta_{11} + 370 \beta_{10} - 668 \beta_{9} + 370 \beta_{8} + 944 \beta_{7} - 28 \beta_{6} + \cdots - 5535 ) / 11 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 9263 \beta_{11} + 9263 \beta_{10} - 5747 \beta_{9} + 8898 \beta_{7} - 1522 \beta_{6} + \cdots + 282342 \beta_1 ) / 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 214817 \beta_{11} - 31281 \beta_{10} + 504305 \beta_{9} - 214817 \beta_{8} - 183536 \beta_{6} + \cdots + 5765838 ) / 22 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 1335072 \beta_{10} - 1391448 \beta_{8} - 2966180 \beta_{7} - 1391448 \beta_{6} - 2685288 \beta_{5} + \cdots + 51141116 ) / 11 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 65518455 \beta_{11} - 145596419 \beta_{9} + 19432867 \beta_{8} - 145596419 \beta_{7} + \cdots - 2132821838 ) / 22 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 836060169 \beta_{11} + 918801931 \beta_{10} - 635348235 \beta_{9} + 918801931 \beta_{8} + \cdots - 75956936510 ) / 22 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 4096336324 \beta_{11} + 4096336324 \beta_{10} + 16758126568 \beta_{9} + 38899513584 \beta_{7} + \cdots + 354164978028 \beta_1 ) / 11 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 312802064249 \beta_{11} - 254920103939 \beta_{10} + 663038391897 \beta_{9} - 312802064249 \beta_{8} + \cdots + 23171526831924 ) / 22 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 595045952311 \beta_{10} - 278771150471 \beta_{8} - 2177065400452 \beta_{7} - 278771150471 \beta_{6} + \cdots + 21120734276566 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) |
| \(\chi(n)\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−3.23607 | + | 2.35114i | −1.93335 | − | 5.95024i | 4.94427 | − | 15.2169i | 69.5734 | + | 50.5481i | 20.2463 | + | 14.7098i | −63.1076 | + | 194.225i | 19.7771 | + | 60.8676i | 164.924 | − | 119.824i | −343.990 | ||||||||||||||||||||||||||||||||||||||
| 3.2 | −3.23607 | + | 2.35114i | −1.32178 | − | 4.06802i | 4.94427 | − | 15.2169i | −21.5825 | − | 15.6806i | 13.8418 | + | 10.0567i | 76.6137 | − | 235.793i | 19.7771 | + | 60.8676i | 181.789 | − | 132.078i | 106.710 | |||||||||||||||||||||||||||||||||||||||
| 3.3 | −3.23607 | + | 2.35114i | 8.84530 | + | 27.2230i | 4.94427 | − | 15.2169i | −34.7549 | − | 25.2509i | −92.6292 | − | 67.2990i | −31.2782 | + | 96.2643i | 19.7771 | + | 60.8676i | −466.263 | + | 338.760i | 171.838 | |||||||||||||||||||||||||||||||||||||||
| 5.1 | 1.23607 | + | 3.80423i | −15.7161 | − | 11.4184i | −12.9443 | + | 9.40456i | 32.8187 | − | 101.006i | 24.0121 | − | 73.9017i | −39.7423 | + | 28.8745i | −51.7771 | − | 37.6183i | 41.5249 | + | 127.801i | 424.814 | |||||||||||||||||||||||||||||||||||||||
| 5.2 | 1.23607 | + | 3.80423i | −8.33499 | − | 6.05573i | −12.9443 | + | 9.40456i | −30.5739 | + | 94.0969i | 12.7347 | − | 39.1935i | −73.6937 | + | 53.5416i | −51.7771 | − | 37.6183i | −42.2909 | − | 130.158i | −395.757 | |||||||||||||||||||||||||||||||||||||||
| 5.3 | 1.23607 | + | 3.80423i | 18.4610 | + | 13.4127i | −12.9443 | + | 9.40456i | 6.51915 | − | 20.0639i | −28.2058 | + | 86.8086i | −31.7919 | + | 23.0982i | −51.7771 | − | 37.6183i | 85.8160 | + | 264.115i | 84.3857 | |||||||||||||||||||||||||||||||||||||||
| 9.1 | 1.23607 | − | 3.80423i | −15.7161 | + | 11.4184i | −12.9443 | − | 9.40456i | 32.8187 | + | 101.006i | 24.0121 | + | 73.9017i | −39.7423 | − | 28.8745i | −51.7771 | + | 37.6183i | 41.5249 | − | 127.801i | 424.814 | |||||||||||||||||||||||||||||||||||||||
| 9.2 | 1.23607 | − | 3.80423i | −8.33499 | + | 6.05573i | −12.9443 | − | 9.40456i | −30.5739 | − | 94.0969i | 12.7347 | + | 39.1935i | −73.6937 | − | 53.5416i | −51.7771 | + | 37.6183i | −42.2909 | + | 130.158i | −395.757 | |||||||||||||||||||||||||||||||||||||||
| 9.3 | 1.23607 | − | 3.80423i | 18.4610 | − | 13.4127i | −12.9443 | − | 9.40456i | 6.51915 | + | 20.0639i | −28.2058 | − | 86.8086i | −31.7919 | − | 23.0982i | −51.7771 | + | 37.6183i | 85.8160 | − | 264.115i | 84.3857 | |||||||||||||||||||||||||||||||||||||||
| 15.1 | −3.23607 | − | 2.35114i | −1.93335 | + | 5.95024i | 4.94427 | + | 15.2169i | 69.5734 | − | 50.5481i | 20.2463 | − | 14.7098i | −63.1076 | − | 194.225i | 19.7771 | − | 60.8676i | 164.924 | + | 119.824i | −343.990 | |||||||||||||||||||||||||||||||||||||||
| 15.2 | −3.23607 | − | 2.35114i | −1.32178 | + | 4.06802i | 4.94427 | + | 15.2169i | −21.5825 | + | 15.6806i | 13.8418 | − | 10.0567i | 76.6137 | + | 235.793i | 19.7771 | − | 60.8676i | 181.789 | + | 132.078i | 106.710 | |||||||||||||||||||||||||||||||||||||||
| 15.3 | −3.23607 | − | 2.35114i | 8.84530 | − | 27.2230i | 4.94427 | + | 15.2169i | −34.7549 | + | 25.2509i | −92.6292 | + | 67.2990i | −31.2782 | − | 96.2643i | 19.7771 | − | 60.8676i | −466.263 | − | 338.760i | 171.838 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 22.6.c.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 198.6.f.e | 12 | ||
| 11.c | even | 5 | 1 | inner | 22.6.c.b | ✓ | 12 |
| 11.c | even | 5 | 1 | 242.6.a.p | 6 | ||
| 11.d | odd | 10 | 1 | 242.6.a.o | 6 | ||
| 33.h | odd | 10 | 1 | 198.6.f.e | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.6.c.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 22.6.c.b | ✓ | 12 | 11.c | even | 5 | 1 | inner |
| 198.6.f.e | 12 | 3.b | odd | 2 | 1 | ||
| 198.6.f.e | 12 | 33.h | odd | 10 | 1 | ||
| 242.6.a.o | 6 | 11.d | odd | 10 | 1 | ||
| 242.6.a.p | 6 | 11.c | even | 5 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 399 T_{3}^{10} + 13187 T_{3}^{9} + 147431 T_{3}^{8} - 1098409 T_{3}^{7} + \cdots + 12238669621161 \)
acting on \(S_{6}^{\mathrm{new}}(22, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 4 T^{3} + \cdots + 256)^{3} \)
|
| $3$ |
\( T^{12} + \cdots + 12238669621161 \)
|
| $5$ |
\( T^{12} + \cdots + 47\!\cdots\!36 \)
|
| $7$ |
\( T^{12} + \cdots + 81\!\cdots\!00 \)
|
| $11$ |
\( T^{12} + \cdots + 17\!\cdots\!01 \)
|
| $13$ |
\( T^{12} + \cdots + 17\!\cdots\!76 \)
|
| $17$ |
\( T^{12} + \cdots + 33\!\cdots\!41 \)
|
| $19$ |
\( T^{12} + \cdots + 87\!\cdots\!41 \)
|
| $23$ |
\( (T^{6} + \cdots + 28\!\cdots\!16)^{2} \)
|
| $29$ |
\( T^{12} + \cdots + 13\!\cdots\!56 \)
|
| $31$ |
\( T^{12} + \cdots + 93\!\cdots\!96 \)
|
| $37$ |
\( T^{12} + \cdots + 12\!\cdots\!36 \)
|
| $41$ |
\( T^{12} + \cdots + 10\!\cdots\!21 \)
|
| $43$ |
\( (T^{6} + \cdots + 64\!\cdots\!64)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 68\!\cdots\!76 \)
|
| $53$ |
\( T^{12} + \cdots + 12\!\cdots\!16 \)
|
| $59$ |
\( T^{12} + \cdots + 45\!\cdots\!01 \)
|
| $61$ |
\( T^{12} + \cdots + 54\!\cdots\!16 \)
|
| $67$ |
\( (T^{6} + \cdots - 19\!\cdots\!76)^{2} \)
|
| $71$ |
\( T^{12} + \cdots + 38\!\cdots\!76 \)
|
| $73$ |
\( T^{12} + \cdots + 13\!\cdots\!41 \)
|
| $79$ |
\( T^{12} + \cdots + 34\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots + 26\!\cdots\!41 \)
|
| $89$ |
\( (T^{6} + \cdots + 61\!\cdots\!16)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 30\!\cdots\!41 \)
|
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