Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [22,5,Mod(7,22)]
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([7]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22.7");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 22 = 2 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 22.d (of order \(10\), 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: | \(2.27413918784\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} + 138 x^{14} - 428 x^{13} + 7783 x^{12} - 18620 x^{11} + 235604 x^{10} + \cdots + 1499670491 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{8}\cdot 11^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 4 x^{15} + 138 x^{14} - 428 x^{13} + 7783 x^{12} - 18620 x^{11} + 235604 x^{10} + \cdots + 1499670491 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 24\!\cdots\!52 \nu^{15} + \cdots + 64\!\cdots\!13 ) / 58\!\cdots\!31 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 20\!\cdots\!00 \nu^{15} + \cdots + 54\!\cdots\!14 ) / 31\!\cdots\!22 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 63\!\cdots\!72 \nu^{15} + \cdots + 61\!\cdots\!46 ) / 31\!\cdots\!22 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 22\!\cdots\!65 \nu^{15} + \cdots + 27\!\cdots\!63 ) / 96\!\cdots\!71 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 11\!\cdots\!56 \nu^{15} + \cdots + 14\!\cdots\!38 ) / 31\!\cdots\!22 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 36\!\cdots\!14 \nu^{15} + \cdots + 27\!\cdots\!44 ) / 96\!\cdots\!71 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 76\!\cdots\!50 \nu^{15} + \cdots - 23\!\cdots\!38 ) / 19\!\cdots\!42 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 14\!\cdots\!82 \nu^{15} + \cdots - 31\!\cdots\!49 ) / 17\!\cdots\!22 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 15\!\cdots\!18 \nu^{15} + \cdots + 17\!\cdots\!85 ) / 17\!\cdots\!22 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 14\!\cdots\!60 \nu^{15} + \cdots + 16\!\cdots\!66 ) / 96\!\cdots\!71 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 13\!\cdots\!40 \nu^{15} + \cdots - 16\!\cdots\!43 ) / 47\!\cdots\!62 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 63\!\cdots\!06 \nu^{15} + \cdots + 20\!\cdots\!26 ) / 19\!\cdots\!42 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 10\!\cdots\!20 \nu^{15} + \cdots + 22\!\cdots\!82 ) / 19\!\cdots\!42 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 13\!\cdots\!84 \nu^{15} + \cdots + 11\!\cdots\!23 ) / 19\!\cdots\!42 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 20\!\cdots\!92 \nu^{15} + \cdots - 84\!\cdots\!16 ) / 19\!\cdots\!42 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{15} - 2 \beta_{13} + \beta_{12} - 4 \beta_{11} + 3 \beta_{10} - 8 \beta_{9} + 2 \beta_{7} + \cdots + 8 ) / 22 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2 \beta_{15} - 5 \beta_{14} + 3 \beta_{13} - 2 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + \beta_{8} + \cdots - 204 ) / 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 13 \beta_{15} - 21 \beta_{14} + 87 \beta_{13} - 2 \beta_{12} + 59 \beta_{11} - 242 \beta_{10} + \cdots - 644 ) / 22 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 70 \beta_{15} + 202 \beta_{14} - 68 \beta_{13} + 2 \beta_{12} + 97 \beta_{11} - 333 \beta_{10} + \cdots + 4357 ) / 11 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 1426 \beta_{15} + 1465 \beta_{14} - 2664 \beta_{13} - 356 \beta_{12} - 663 \beta_{11} + 7326 \beta_{10} + \cdots + 30337 ) / 22 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1186 \beta_{15} - 5716 \beta_{14} - 259 \beta_{13} - 788 \beta_{12} - 3255 \beta_{11} + 18109 \beta_{10} + \cdots - 92504 ) / 11 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 66328 \beta_{15} - 68005 \beta_{14} + 69680 \beta_{13} + 17335 \beta_{12} - 4648 \beta_{11} + \cdots - 1240266 ) / 22 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 18107 \beta_{15} + 127200 \beta_{14} + 87194 \beta_{13} + 58295 \beta_{12} + 93976 \beta_{11} + \cdots + 1600827 ) / 11 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 2438043 \beta_{15} + 2661543 \beta_{14} - 1490270 \beta_{13} - 563322 \beta_{12} + 752714 \beta_{11} + \cdots + 45597572 ) / 22 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 2717076 \beta_{15} - 1786399 \beta_{14} - 4663669 \beta_{13} - 2957353 \beta_{12} - 2373319 \beta_{11} + \cdots - 7547092 ) / 11 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 77811612 \beta_{15} - 93939692 \beta_{14} + 19463149 \beta_{13} + 13065524 \beta_{12} + \cdots - 1526859144 ) / 22 \)
|
| \(\nu^{12}\) | \(=\) |
\( 14322956 \beta_{15} - 2170156 \beta_{14} + 16325990 \beta_{13} + 11323158 \beta_{12} + 4305643 \beta_{11} + \cdots - 104590147 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 2179600762 \beta_{15} + 3051330737 \beta_{14} + 326168257 \beta_{13} - 113166782 \beta_{12} + \cdots + 46663603783 ) / 22 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 7130399021 \beta_{15} + 3431668430 \beta_{14} - 5699114548 \beta_{13} - 4644317402 \beta_{12} + \cdots + 78938670464 ) / 11 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 51112645034 \beta_{15} - 91072961036 \beta_{14} - 39024604999 \beta_{13} - 9812286729 \beta_{12} + \cdots - 1283826177161 ) / 22 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) |
| \(\chi(n)\) | \(1 + \beta_{2} + \beta_{3} - \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−1.66251 | + | 2.28825i | −1.86966 | + | 5.75422i | −2.47214 | − | 7.60845i | −23.9466 | + | 17.3982i | −10.0587 | − | 13.8447i | −38.7210 | + | 12.5812i | 21.5200 | + | 6.99226i | 35.9150 | + | 26.0937i | − | 83.7204i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | −1.66251 | + | 2.28825i | 3.38726 | − | 10.4249i | −2.47214 | − | 7.60845i | 17.3145 | − | 12.5798i | 18.2234 | + | 25.0824i | 27.3556 | − | 8.88838i | 21.5200 | + | 6.99226i | −31.6748 | − | 23.0131i | 60.5339i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | 1.66251 | − | 2.28825i | 0.309288 | − | 0.951890i | −2.47214 | − | 7.60845i | 35.4031 | − | 25.7219i | −1.66396 | − | 2.29025i | −65.6073 | + | 21.3171i | −21.5200 | − | 6.99226i | 64.7199 | + | 47.0218i | − | 123.774i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | 1.66251 | − | 2.28825i | 3.26328 | − | 10.0434i | −2.47214 | − | 7.60845i | −24.6252 | + | 17.8912i | −17.5564 | − | 24.1643i | 86.5218 | − | 28.1126i | −21.5200 | − | 6.99226i | −24.6896 | − | 17.9380i | 86.0928i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.1 | −2.68999 | − | 0.874032i | −3.20187 | + | 2.32629i | 6.47214 | + | 4.70228i | 13.3838 | + | 41.1910i | 10.6463 | − | 3.45918i | −7.80030 | + | 10.7362i | −13.3001 | − | 18.3060i | −20.1901 | + | 62.1386i | − | 122.501i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.2 | −2.68999 | − | 0.874032i | 4.50928 | − | 3.27619i | 6.47214 | + | 4.70228i | −10.0117 | − | 30.8128i | −14.9934 | + | 4.87166i | 50.5008 | − | 69.5083i | −13.3001 | − | 18.3060i | −15.4301 | + | 47.4891i | 91.6368i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.3 | 2.68999 | + | 0.874032i | −13.1019 | + | 9.51912i | 6.47214 | + | 4.70228i | 8.38512 | + | 25.8068i | −43.5642 | + | 14.1549i | 26.6976 | − | 36.7461i | 13.3001 | + | 18.3060i | 56.0169 | − | 172.402i | 76.7489i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.4 | 2.68999 | + | 0.874032i | 5.70436 | − | 4.14446i | 6.47214 | + | 4.70228i | −0.903101 | − | 2.77946i | 18.9671 | − | 6.16277i | −3.94724 | + | 5.43292i | 13.3001 | + | 18.3060i | −9.66723 | + | 29.7527i | − | 8.26607i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.1 | −2.68999 | + | 0.874032i | −3.20187 | − | 2.32629i | 6.47214 | − | 4.70228i | 13.3838 | − | 41.1910i | 10.6463 | + | 3.45918i | −7.80030 | − | 10.7362i | −13.3001 | + | 18.3060i | −20.1901 | − | 62.1386i | 122.501i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.2 | −2.68999 | + | 0.874032i | 4.50928 | + | 3.27619i | 6.47214 | − | 4.70228i | −10.0117 | + | 30.8128i | −14.9934 | − | 4.87166i | 50.5008 | + | 69.5083i | −13.3001 | + | 18.3060i | −15.4301 | − | 47.4891i | − | 91.6368i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.3 | 2.68999 | − | 0.874032i | −13.1019 | − | 9.51912i | 6.47214 | − | 4.70228i | 8.38512 | − | 25.8068i | −43.5642 | − | 14.1549i | 26.6976 | + | 36.7461i | 13.3001 | − | 18.3060i | 56.0169 | + | 172.402i | − | 76.7489i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.4 | 2.68999 | − | 0.874032i | 5.70436 | + | 4.14446i | 6.47214 | − | 4.70228i | −0.903101 | + | 2.77946i | 18.9671 | + | 6.16277i | −3.94724 | − | 5.43292i | 13.3001 | − | 18.3060i | −9.66723 | − | 29.7527i | 8.26607i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.1 | −1.66251 | − | 2.28825i | −1.86966 | − | 5.75422i | −2.47214 | + | 7.60845i | −23.9466 | − | 17.3982i | −10.0587 | + | 13.8447i | −38.7210 | − | 12.5812i | 21.5200 | − | 6.99226i | 35.9150 | − | 26.0937i | 83.7204i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.2 | −1.66251 | − | 2.28825i | 3.38726 | + | 10.4249i | −2.47214 | + | 7.60845i | 17.3145 | + | 12.5798i | 18.2234 | − | 25.0824i | 27.3556 | + | 8.88838i | 21.5200 | − | 6.99226i | −31.6748 | + | 23.0131i | − | 60.5339i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.3 | 1.66251 | + | 2.28825i | 0.309288 | + | 0.951890i | −2.47214 | + | 7.60845i | 35.4031 | + | 25.7219i | −1.66396 | + | 2.29025i | −65.6073 | − | 21.3171i | −21.5200 | + | 6.99226i | 64.7199 | − | 47.0218i | 123.774i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.4 | 1.66251 | + | 2.28825i | 3.26328 | + | 10.0434i | −2.47214 | + | 7.60845i | −24.6252 | − | 17.8912i | −17.5564 | + | 24.1643i | 86.5218 | + | 28.1126i | −21.5200 | + | 6.99226i | −24.6896 | + | 17.9380i | − | 86.0928i | ||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.d | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 22.5.d.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 198.5.j.a | 16 | ||
| 4.b | odd | 2 | 1 | 176.5.n.c | 16 | ||
| 11.c | even | 5 | 1 | 242.5.b.e | 16 | ||
| 11.d | odd | 10 | 1 | inner | 22.5.d.a | ✓ | 16 |
| 11.d | odd | 10 | 1 | 242.5.b.e | 16 | ||
| 33.f | even | 10 | 1 | 198.5.j.a | 16 | ||
| 44.g | even | 10 | 1 | 176.5.n.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.5.d.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 22.5.d.a | ✓ | 16 | 11.d | odd | 10 | 1 | inner |
| 176.5.n.c | 16 | 4.b | odd | 2 | 1 | ||
| 176.5.n.c | 16 | 44.g | even | 10 | 1 | ||
| 198.5.j.a | 16 | 3.b | odd | 2 | 1 | ||
| 198.5.j.a | 16 | 33.f | even | 10 | 1 | ||
| 242.5.b.e | 16 | 11.c | even | 5 | 1 | ||
| 242.5.b.e | 16 | 11.d | odd | 10 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(22, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} - 8 T^{6} + \cdots + 4096)^{2} \)
|
| $3$ |
\( T^{16} + \cdots + 3117763586961 \)
|
| $5$ |
\( T^{16} + \cdots + 88\!\cdots\!96 \)
|
| $7$ |
\( T^{16} + \cdots + 65\!\cdots\!76 \)
|
| $11$ |
\( T^{16} + \cdots + 21\!\cdots\!21 \)
|
| $13$ |
\( T^{16} + \cdots + 48\!\cdots\!36 \)
|
| $17$ |
\( T^{16} + \cdots + 61\!\cdots\!81 \)
|
| $19$ |
\( T^{16} + \cdots + 23\!\cdots\!21 \)
|
| $23$ |
\( (T^{8} + \cdots + 70\!\cdots\!76)^{2} \)
|
| $29$ |
\( T^{16} + \cdots + 39\!\cdots\!76 \)
|
| $31$ |
\( T^{16} + \cdots + 25\!\cdots\!36 \)
|
| $37$ |
\( T^{16} + \cdots + 14\!\cdots\!16 \)
|
| $41$ |
\( T^{16} + \cdots + 98\!\cdots\!01 \)
|
| $43$ |
\( T^{16} + \cdots + 10\!\cdots\!76 \)
|
| $47$ |
\( T^{16} + \cdots + 39\!\cdots\!36 \)
|
| $53$ |
\( T^{16} + \cdots + 16\!\cdots\!16 \)
|
| $59$ |
\( T^{16} + \cdots + 47\!\cdots\!01 \)
|
| $61$ |
\( T^{16} + \cdots + 95\!\cdots\!36 \)
|
| $67$ |
\( (T^{8} + \cdots - 18\!\cdots\!24)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 52\!\cdots\!96 \)
|
| $73$ |
\( T^{16} + \cdots + 87\!\cdots\!41 \)
|
| $79$ |
\( T^{16} + \cdots + 81\!\cdots\!96 \)
|
| $83$ |
\( T^{16} + \cdots + 80\!\cdots\!81 \)
|
| $89$ |
\( (T^{8} + \cdots - 37\!\cdots\!64)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 18\!\cdots\!21 \)
|
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