Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [121,12,Mod(1,121)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("121.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.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: | \(92.9695248477\) |
| Analytic rank: | \(1\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 8 x^{19} - 30183 x^{18} + 292414 x^{17} + 381222296 x^{16} - 4139834960 x^{15} + \cdots + 11\!\cdots\!24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{4}\cdot 5\cdot 11^{16} \) |
| Twist minimal: | no (minimal twist has level 11) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 8 x^{19} - 30183 x^{18} + 292414 x^{17} + 381222296 x^{16} - 4139834960 x^{15} + \cdots + 11\!\cdots\!24 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 13\!\cdots\!19 \nu^{19} + \cdots + 56\!\cdots\!52 ) / 21\!\cdots\!20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 20\!\cdots\!91 \nu^{19} + \cdots - 23\!\cdots\!32 ) / 85\!\cdots\!08 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\!\cdots\!19 \nu^{19} + \cdots + 56\!\cdots\!32 ) / 35\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 55\!\cdots\!08 \nu^{19} + \cdots - 19\!\cdots\!44 ) / 42\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 95\!\cdots\!07 \nu^{19} + \cdots - 44\!\cdots\!36 ) / 66\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 60\!\cdots\!72 \nu^{19} + \cdots + 28\!\cdots\!76 ) / 33\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 49\!\cdots\!13 \nu^{19} + \cdots - 23\!\cdots\!64 ) / 22\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 25\!\cdots\!57 \nu^{19} + \cdots - 12\!\cdots\!12 ) / 95\!\cdots\!28 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 89\!\cdots\!72 \nu^{19} + \cdots + 39\!\cdots\!96 ) / 33\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 27\!\cdots\!27 \nu^{19} + \cdots - 12\!\cdots\!72 ) / 66\!\cdots\!96 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 14\!\cdots\!93 \nu^{19} + \cdots + 68\!\cdots\!64 ) / 11\!\cdots\!60 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 77\!\cdots\!78 \nu^{19} + \cdots + 36\!\cdots\!64 ) / 33\!\cdots\!80 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 10\!\cdots\!19 \nu^{19} + \cdots + 48\!\cdots\!72 ) / 39\!\cdots\!80 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 43\!\cdots\!91 \nu^{19} + \cdots + 21\!\cdots\!28 ) / 66\!\cdots\!60 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 91\!\cdots\!71 \nu^{19} + \cdots - 41\!\cdots\!36 ) / 13\!\cdots\!92 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 11\!\cdots\!53 \nu^{19} + \cdots - 52\!\cdots\!36 ) / 13\!\cdots\!92 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 12\!\cdots\!46 \nu^{19} + \cdots + 57\!\cdots\!32 ) / 95\!\cdots\!28 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 13\!\cdots\!21 \nu^{19} + \cdots - 64\!\cdots\!88 ) / 66\!\cdots\!60 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 24\!\cdots\!29 \nu^{19} + \cdots - 11\!\cdots\!92 ) / 66\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{3} + 121\beta _1 - 1 ) / 121 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -4\beta_{4} - 125\beta_{3} + 121\beta_{2} - 329\beta _1 + 365645 ) / 121 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 66 \beta_{9} - 121 \beta_{8} + 363 \beta_{7} + 418 \beta_{6} + 3267 \beta_{5} - 200 \beta_{4} + \cdots - 1163871 ) / 121 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 154 \beta_{19} - 242 \beta_{18} - 275 \beta_{17} + 759 \beta_{16} - 396 \beta_{15} + \cdots + 1788982851 ) / 121 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 18194 \beta_{19} - 9152 \beta_{18} + 26906 \beta_{17} - 29502 \beta_{16} + 286 \beta_{15} + \cdots - 14997862493 ) / 121 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 1378894 \beta_{19} - 1882584 \beta_{18} - 2950200 \beta_{17} + 8036622 \beta_{16} + \cdots + 9932578994513 ) / 121 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 220807246 \beta_{19} - 109306252 \beta_{18} + 390611386 \beta_{17} - 345474316 \beta_{16} + \cdots - 130021990377835 ) / 121 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 10585073708 \beta_{19} - 11392215362 \beta_{18} - 25876239499 \beta_{17} + 66147952541 \beta_{16} + \cdots + 58\!\cdots\!55 ) / 121 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 1956317826804 \beta_{19} - 937537296916 \beta_{18} + 3851371617016 \beta_{17} - 3044666991410 \beta_{16} + \cdots - 10\!\cdots\!57 ) / 121 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 80156194569044 \beta_{19} - 62416377049136 \beta_{18} - 214387684031708 \beta_{17} + \cdots + 35\!\cdots\!09 ) / 121 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 15\!\cdots\!16 \beta_{19} + \cdots - 74\!\cdots\!79 ) / 121 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 61\!\cdots\!74 \beta_{19} + \cdots + 21\!\cdots\!35 ) / 121 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 11\!\cdots\!58 \beta_{19} + \cdots - 53\!\cdots\!33 ) / 121 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 46\!\cdots\!10 \beta_{19} + \cdots + 13\!\cdots\!37 ) / 121 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 86\!\cdots\!70 \beta_{19} + \cdots - 38\!\cdots\!91 ) / 121 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 35\!\cdots\!56 \beta_{19} + \cdots + 87\!\cdots\!63 ) / 121 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 62\!\cdots\!52 \beta_{19} + \cdots - 28\!\cdots\!77 ) / 121 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 27\!\cdots\!44 \beta_{19} + \cdots + 56\!\cdots\!89 ) / 121 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 44\!\cdots\!32 \beta_{19} + \cdots - 20\!\cdots\!75 ) / 121 \)
|
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 |
|
−82.1256 | 416.278 | 4696.61 | 10582.7 | −34187.1 | −18856.9 | −217518. | −3859.34 | −869114. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −81.9171 | 502.915 | 4662.40 | 4830.79 | −41197.3 | −48621.6 | −214164. | 75776.7 | −395724. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −75.7015 | −672.682 | 3682.72 | −3311.85 | 50923.1 | 75002.2 | −123751. | 275354. | 250712. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −68.2184 | −800.227 | 2605.74 | 3885.02 | 54590.2 | −51424.0 | −38048.3 | 463217. | −265029. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −60.6289 | −26.2681 | 1627.86 | −12838.7 | 1592.61 | 43792.6 | 25472.4 | −176457. | 778394. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −52.0653 | 0.0841659 | 662.794 | −6220.46 | −4.38212 | −74633.1 | 72121.1 | −177147. | 323870. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −40.8153 | 595.914 | −382.113 | −7376.15 | −24322.4 | 75091.1 | 99185.7 | 177967. | 301059. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −39.9087 | −40.8434 | −455.293 | 6458.51 | 1630.01 | 33726.1 | 99903.3 | −175479. | −257751. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −24.0805 | −337.620 | −1468.13 | 8145.04 | 8130.06 | 6687.83 | 84670.2 | −63159.7 | −196137. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −8.62173 | 607.657 | −1973.67 | 571.261 | −5239.06 | −18434.7 | 34673.7 | 192100. | −4925.26 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 6.06963 | −351.198 | −2011.16 | −7528.95 | −2131.64 | −64727.7 | −24637.6 | −53806.7 | −45697.9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 18.9977 | 721.657 | −1687.09 | 9120.80 | 13709.8 | −50885.0 | −70958.0 | 343642. | 173274. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 19.1521 | 67.8628 | −1681.20 | −1990.17 | 1299.71 | 24598.4 | −71421.9 | −172542. | −38116.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 26.2209 | −464.904 | −1360.46 | 6214.88 | −12190.2 | 43684.1 | −89373.0 | 38989.0 | 162960. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 28.0632 | −728.012 | −1260.45 | −10635.5 | −20430.4 | −41160.3 | −92846.0 | 352855. | −298466. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 54.7866 | 396.768 | 953.574 | −4938.72 | 21737.6 | 40493.9 | −59959.9 | −19722.2 | −270576. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 69.2448 | −461.319 | 2746.84 | 9386.70 | −31943.9 | 18718.8 | 48391.2 | 35668.0 | 649980. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 70.0963 | 132.153 | 2865.50 | 10462.0 | 9263.41 | −74803.0 | 57303.6 | −159683. | 733345. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 70.7336 | 308.998 | 2955.24 | −4585.55 | 21856.5 | −7403.48 | 64172.7 | −81667.4 | −324352. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.20 | 82.7180 | −635.212 | 4794.27 | −9657.71 | −52543.5 | 25354.6 | 227166. | 226348. | −798867. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
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 | 121.12.a.h | 20 | |
| 11.b | odd | 2 | 1 | 121.12.a.j | 20 | ||
| 11.d | odd | 10 | 2 | 11.12.c.a | ✓ | 40 | |
| 33.f | even | 10 | 2 | 99.12.f.a | 40 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.12.c.a | ✓ | 40 | 11.d | odd | 10 | 2 | |
| 99.12.f.a | 40 | 33.f | even | 10 | 2 | ||
| 121.12.a.h | 20 | 1.a | even | 1 | 1 | trivial | |
| 121.12.a.j | 20 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} + 88 T_{2}^{19} - 26595 T_{2}^{18} - 2384162 T_{2}^{17} + 289564208 T_{2}^{16} + \cdots + 93\!\cdots\!04 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(121))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 93\!\cdots\!04 \)
|
| $3$ |
\( T^{20} + \cdots + 34\!\cdots\!45 \)
|
| $5$ |
\( T^{20} + \cdots + 22\!\cdots\!00 \)
|
| $7$ |
\( T^{20} + \cdots + 55\!\cdots\!80 \)
|
| $11$ |
\( T^{20} \)
|
| $13$ |
\( T^{20} + \cdots + 27\!\cdots\!00 \)
|
| $17$ |
\( T^{20} + \cdots - 16\!\cdots\!19 \)
|
| $19$ |
\( T^{20} + \cdots + 23\!\cdots\!25 \)
|
| $23$ |
\( T^{20} + \cdots + 15\!\cdots\!24 \)
|
| $29$ |
\( T^{20} + \cdots - 87\!\cdots\!00 \)
|
| $31$ |
\( T^{20} + \cdots - 22\!\cdots\!00 \)
|
| $37$ |
\( T^{20} + \cdots - 33\!\cdots\!56 \)
|
| $41$ |
\( T^{20} + \cdots - 58\!\cdots\!79 \)
|
| $43$ |
\( T^{20} + \cdots + 19\!\cdots\!56 \)
|
| $47$ |
\( T^{20} + \cdots + 67\!\cdots\!04 \)
|
| $53$ |
\( T^{20} + \cdots + 15\!\cdots\!84 \)
|
| $59$ |
\( T^{20} + \cdots - 31\!\cdots\!75 \)
|
| $61$ |
\( T^{20} + \cdots - 48\!\cdots\!20 \)
|
| $67$ |
\( T^{20} + \cdots - 29\!\cdots\!16 \)
|
| $71$ |
\( T^{20} + \cdots - 19\!\cdots\!16 \)
|
| $73$ |
\( T^{20} + \cdots - 22\!\cdots\!51 \)
|
| $79$ |
\( T^{20} + \cdots - 32\!\cdots\!00 \)
|
| $83$ |
\( T^{20} + \cdots + 40\!\cdots\!21 \)
|
| $89$ |
\( T^{20} + \cdots + 75\!\cdots\!00 \)
|
| $97$ |
\( T^{20} + \cdots - 82\!\cdots\!75 \)
|
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