Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,11,Mod(28,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.28");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.g (of order \(4\), degree \(2\), 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: | \(28.5910763703\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 4 x^{19} + 8 x^{18} - 57344 x^{17} + 18664853 x^{16} - 274248412 x^{15} + 2591841992 x^{14} + \cdots + 11\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{29}\cdot 3^{38}\cdot 5^{14} \) |
| Twist minimal: | no (minimal twist has level 15) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{20} - 4 x^{19} + 8 x^{18} - 57344 x^{17} + 18664853 x^{16} - 274248412 x^{15} + 2591841992 x^{14} + \cdots + 11\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 50\!\cdots\!97 \nu^{19} + \cdots + 24\!\cdots\!00 ) / 84\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 73\!\cdots\!39 \nu^{19} + \cdots - 29\!\cdots\!00 ) / 30\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 34\!\cdots\!17 \nu^{19} + \cdots + 61\!\cdots\!00 ) / 42\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 46\!\cdots\!79 \nu^{19} + \cdots - 24\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 44\!\cdots\!99 \nu^{19} + \cdots - 98\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 45\!\cdots\!24 \nu^{19} + \cdots + 16\!\cdots\!00 ) / 28\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 53\!\cdots\!57 \nu^{19} + \cdots - 21\!\cdots\!00 ) / 24\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 14\!\cdots\!73 \nu^{19} + \cdots + 15\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 10\!\cdots\!03 \nu^{19} + \cdots + 43\!\cdots\!00 ) / 30\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 47\!\cdots\!67 \nu^{19} + \cdots + 58\!\cdots\!00 ) / 98\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 29\!\cdots\!35 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 52\!\cdots\!52 \nu^{19} + \cdots - 40\!\cdots\!00 ) / 98\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 25\!\cdots\!46 \nu^{19} + \cdots - 63\!\cdots\!00 ) / 28\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 43\!\cdots\!03 \nu^{19} + \cdots + 27\!\cdots\!00 ) / 28\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 44\!\cdots\!33 \nu^{19} + \cdots + 20\!\cdots\!00 ) / 28\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 15\!\cdots\!99 \nu^{19} + \cdots + 95\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 16\!\cdots\!11 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 37\!\cdots\!61 \nu^{19} + \cdots + 14\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} + 1457\beta_{3} - 6\beta_{2} - 6\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{19} - \beta_{18} - \beta_{17} + 3 \beta_{16} + \beta_{14} + 4 \beta_{12} + \beta_{11} + \cdots + 9108 \)
|
| \(\nu^{4}\) | \(=\) |
\( 33 \beta_{19} + 66 \beta_{18} - 7 \beta_{16} - 53 \beta_{15} - 87 \beta_{14} - 109 \beta_{13} + \cdots - 3700754 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 1018 \beta_{19} - 5429 \beta_{18} + 5429 \beta_{17} - 1118 \beta_{16} + 7561 \beta_{15} + \cdots + 51586117 \)
|
| \(\nu^{6}\) | \(=\) |
\( 249862 \beta_{19} - 375346 \beta_{17} + 347804 \beta_{16} - 493223 \beta_{15} + 356149 \beta_{14} + \cdots + 47343 \)
|
| \(\nu^{7}\) | \(=\) |
\( 16306841 \beta_{19} + 24304763 \beta_{18} + 24304763 \beta_{17} - 40752823 \beta_{16} + \cdots - 246573919056 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 864694051 \beta_{19} - 1729388102 \beta_{18} + 276904613 \beta_{16} + 1754837255 \beta_{15} + \cdots + 40433330015344 \)
|
| \(\nu^{9}\) | \(=\) |
\( 44303324842 \beta_{19} + 102195307761 \beta_{18} - 102195307761 \beta_{17} + 41493684846 \beta_{16} + \cdots - 11\!\cdots\!33 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 3599609835846 \beta_{19} + 7567177772218 \beta_{17} - 7155884084772 \beta_{16} + 9506555508179 \beta_{15} + \cdots - 1444726352299 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 205187685012537 \beta_{19} - 420098821942923 \beta_{18} - 420098821942923 \beta_{17} + \cdots + 48\!\cdots\!80 \)
|
| \(\nu^{12}\) | \(=\) |
\( 16\!\cdots\!95 \beta_{19} + \cdots - 58\!\cdots\!48 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 97\!\cdots\!58 \beta_{19} + \cdots + 20\!\cdots\!97 \)
|
| \(\nu^{14}\) | \(=\) |
\( 49\!\cdots\!46 \beta_{19} + \cdots + 32\!\cdots\!39 \)
|
| \(\nu^{15}\) | \(=\) |
\( 27\!\cdots\!65 \beta_{19} + \cdots - 88\!\cdots\!84 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 29\!\cdots\!79 \beta_{19} + \cdots + 92\!\cdots\!20 \)
|
| \(\nu^{17}\) | \(=\) |
\( 18\!\cdots\!82 \beta_{19} + \cdots - 37\!\cdots\!93 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 70\!\cdots\!62 \beta_{19} + \cdots - 62\!\cdots\!43 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 39\!\cdots\!53 \beta_{19} + \cdots + 15\!\cdots\!52 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
−42.4637 | + | 42.4637i | 0 | − | 2582.33i | −3082.93 | + | 511.056i | 0 | −3582.02 | + | 3582.02i | 66172.3 | + | 66172.3i | 0 | 109211. | − | 152614.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.2 | −29.9381 | + | 29.9381i | 0 | − | 768.579i | −2095.22 | − | 2318.55i | 0 | 15970.0 | − | 15970.0i | −7646.81 | − | 7646.81i | 0 | 132140. | + | 6686.35i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.3 | −20.8144 | + | 20.8144i | 0 | 157.526i | 1515.02 | − | 2733.19i | 0 | −6932.43 | + | 6932.43i | −24592.7 | − | 24592.7i | 0 | 25355.6 | + | 88423.8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.4 | −11.6866 | + | 11.6866i | 0 | 750.847i | −2257.04 | + | 2161.34i | 0 | −21416.5 | + | 21416.5i | −20741.9 | − | 20741.9i | 0 | 1118.39 | − | 51635.8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.5 | 2.85456 | − | 2.85456i | 0 | 1007.70i | 3074.01 | + | 562.191i | 0 | 5862.00 | − | 5862.00i | 5799.62 | + | 5799.62i | 0 | 10379.8 | − | 7170.15i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.6 | 12.5555 | − | 12.5555i | 0 | 708.721i | −3105.25 | + | 350.813i | 0 | −8580.16 | + | 8580.16i | 21755.1 | + | 21755.1i | 0 | −34583.2 | + | 43392.4i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.7 | 19.3738 | − | 19.3738i | 0 | 273.309i | 2998.53 | + | 880.025i | 0 | 3127.99 | − | 3127.99i | 25133.9 | + | 25133.9i | 0 | 75142.5 | − | 41043.6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.8 | 28.0945 | − | 28.0945i | 0 | − | 554.607i | −2550.39 | + | 1805.86i | 0 | 20119.5 | − | 20119.5i | 13187.4 | + | 13187.4i | 0 | −20917.3 | + | 122387.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.9 | 32.7993 | − | 32.7993i | 0 | − | 1127.59i | 1119.22 | − | 2917.70i | 0 | −16059.1 | + | 16059.1i | −3397.64 | − | 3397.64i | 0 | −58989.1 | − | 132408.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.10 | 41.2250 | − | 41.2250i | 0 | − | 2375.00i | −953.956 | − | 2975.83i | 0 | 16792.7 | − | 16792.7i | −55695.2 | − | 55695.2i | 0 | −162006. | − | 83352.0i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.1 | −42.4637 | − | 42.4637i | 0 | 2582.33i | −3082.93 | − | 511.056i | 0 | −3582.02 | − | 3582.02i | 66172.3 | − | 66172.3i | 0 | 109211. | + | 152614.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.2 | −29.9381 | − | 29.9381i | 0 | 768.579i | −2095.22 | + | 2318.55i | 0 | 15970.0 | + | 15970.0i | −7646.81 | + | 7646.81i | 0 | 132140. | − | 6686.35i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.3 | −20.8144 | − | 20.8144i | 0 | − | 157.526i | 1515.02 | + | 2733.19i | 0 | −6932.43 | − | 6932.43i | −24592.7 | + | 24592.7i | 0 | 25355.6 | − | 88423.8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.4 | −11.6866 | − | 11.6866i | 0 | − | 750.847i | −2257.04 | − | 2161.34i | 0 | −21416.5 | − | 21416.5i | −20741.9 | + | 20741.9i | 0 | 1118.39 | + | 51635.8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.5 | 2.85456 | + | 2.85456i | 0 | − | 1007.70i | 3074.01 | − | 562.191i | 0 | 5862.00 | + | 5862.00i | 5799.62 | − | 5799.62i | 0 | 10379.8 | + | 7170.15i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.6 | 12.5555 | + | 12.5555i | 0 | − | 708.721i | −3105.25 | − | 350.813i | 0 | −8580.16 | − | 8580.16i | 21755.1 | − | 21755.1i | 0 | −34583.2 | − | 43392.4i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.7 | 19.3738 | + | 19.3738i | 0 | − | 273.309i | 2998.53 | − | 880.025i | 0 | 3127.99 | + | 3127.99i | 25133.9 | − | 25133.9i | 0 | 75142.5 | + | 41043.6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.8 | 28.0945 | + | 28.0945i | 0 | 554.607i | −2550.39 | − | 1805.86i | 0 | 20119.5 | + | 20119.5i | 13187.4 | − | 13187.4i | 0 | −20917.3 | − | 122387.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.9 | 32.7993 | + | 32.7993i | 0 | 1127.59i | 1119.22 | + | 2917.70i | 0 | −16059.1 | − | 16059.1i | −3397.64 | + | 3397.64i | 0 | −58989.1 | + | 132408.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.10 | 41.2250 | + | 41.2250i | 0 | 2375.00i | −953.956 | + | 2975.83i | 0 | 16792.7 | + | 16792.7i | −55695.2 | + | 55695.2i | 0 | −162006. | + | 83352.0i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 45.11.g.c | 20 | |
| 3.b | odd | 2 | 1 | 15.11.f.a | ✓ | 20 | |
| 5.c | odd | 4 | 1 | inner | 45.11.g.c | 20 | |
| 15.d | odd | 2 | 1 | 75.11.f.d | 20 | ||
| 15.e | even | 4 | 1 | 15.11.f.a | ✓ | 20 | |
| 15.e | even | 4 | 1 | 75.11.f.d | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.11.f.a | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 15.11.f.a | ✓ | 20 | 15.e | even | 4 | 1 | |
| 45.11.g.c | 20 | 1.a | even | 1 | 1 | trivial | |
| 45.11.g.c | 20 | 5.c | odd | 4 | 1 | inner | |
| 75.11.f.d | 20 | 15.d | odd | 2 | 1 | ||
| 75.11.f.d | 20 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} - 64 T_{2}^{19} + 2048 T_{2}^{18} - 13316 T_{2}^{17} + 16946117 T_{2}^{16} + \cdots + 68\!\cdots\!76 \)
acting on \(S_{11}^{\mathrm{new}}(45, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 68\!\cdots\!76 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + \cdots + 78\!\cdots\!25 \)
|
| $7$ |
\( T^{20} + \cdots + 53\!\cdots\!00 \)
|
| $11$ |
\( (T^{10} + \cdots + 18\!\cdots\!32)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 19\!\cdots\!76 \)
|
| $17$ |
\( T^{20} + \cdots + 21\!\cdots\!24 \)
|
| $19$ |
\( T^{20} + \cdots + 16\!\cdots\!00 \)
|
| $23$ |
\( T^{20} + \cdots + 11\!\cdots\!00 \)
|
| $29$ |
\( T^{20} + \cdots + 17\!\cdots\!00 \)
|
| $31$ |
\( (T^{10} + \cdots - 37\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $41$ |
\( (T^{10} + \cdots + 24\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 98\!\cdots\!24 \)
|
| $47$ |
\( T^{20} + \cdots + 71\!\cdots\!24 \)
|
| $53$ |
\( T^{20} + \cdots + 50\!\cdots\!00 \)
|
| $59$ |
\( T^{20} + \cdots + 30\!\cdots\!00 \)
|
| $61$ |
\( (T^{10} + \cdots - 79\!\cdots\!24)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 16\!\cdots\!24 \)
|
| $71$ |
\( (T^{10} + \cdots + 23\!\cdots\!24)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 61\!\cdots\!00 \)
|
| $79$ |
\( T^{20} + \cdots + 31\!\cdots\!00 \)
|
| $83$ |
\( T^{20} + \cdots + 68\!\cdots\!76 \)
|
| $89$ |
\( T^{20} + \cdots + 70\!\cdots\!00 \)
|
| $97$ |
\( T^{20} + \cdots + 23\!\cdots\!24 \)
|
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