Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [936,4,Mod(649,936)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(936, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("936.649");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 936.c (of order \(2\), degree \(1\), 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: | \(55.2257877654\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{20} + 772 x^{18} + 254556 x^{16} + 46787036 x^{14} + 5240196398 x^{12} + 366523224996 x^{10} + \cdots + 24\!\cdots\!76 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{29}]\) |
| Coefficient ring index: | \( 2^{66} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{20} + 772 x^{18} + 254556 x^{16} + 46787036 x^{14} + 5240196398 x^{12} + 366523224996 x^{10} + \cdots + 24\!\cdots\!76 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 18\!\cdots\!33 \nu^{18} + \cdots + 57\!\cdots\!88 ) / 21\!\cdots\!90 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 12\!\cdots\!79 \nu^{18} + \cdots - 38\!\cdots\!04 ) / 14\!\cdots\!60 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 42\!\cdots\!77 \nu^{18} + \cdots - 19\!\cdots\!63 ) / 12\!\cdots\!49 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\!\cdots\!26 \nu^{18} + \cdots + 40\!\cdots\!04 ) / 21\!\cdots\!90 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 26\!\cdots\!19 \nu^{18} + \cdots - 45\!\cdots\!96 ) / 43\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11\!\cdots\!34 \nu^{18} + \cdots + 21\!\cdots\!86 ) / 19\!\cdots\!35 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 15\!\cdots\!82 \nu^{18} + \cdots + 95\!\cdots\!72 ) / 19\!\cdots\!35 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 43\!\cdots\!09 \nu^{18} + \cdots - 68\!\cdots\!36 ) / 43\!\cdots\!38 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 41\!\cdots\!80 \nu^{19} + \cdots - 35\!\cdots\!16 \nu ) / 81\!\cdots\!13 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 12\!\cdots\!60 \nu^{19} + \cdots - 13\!\cdots\!24 \nu ) / 13\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 72\!\cdots\!51 \nu^{19} + \cdots + 15\!\cdots\!48 \nu ) / 26\!\cdots\!20 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 15\!\cdots\!68 \nu^{19} + \cdots - 13\!\cdots\!92 ) / 40\!\cdots\!80 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 13\!\cdots\!51 \nu^{19} + \cdots + 14\!\cdots\!96 \nu ) / 80\!\cdots\!56 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 19\!\cdots\!85 \nu^{19} + \cdots - 72\!\cdots\!92 \nu ) / 69\!\cdots\!40 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 14\!\cdots\!37 \nu^{19} + \cdots + 13\!\cdots\!92 ) / 40\!\cdots\!80 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 14\!\cdots\!37 \nu^{19} + \cdots + 13\!\cdots\!92 ) / 40\!\cdots\!80 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 62\!\cdots\!60 \nu^{19} + \cdots - 11\!\cdots\!04 \nu ) / 13\!\cdots\!80 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 27\!\cdots\!05 \nu^{19} + \cdots - 32\!\cdots\!88 \nu ) / 28\!\cdots\!60 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 41\!\cdots\!77 \nu^{19} + \cdots + 19\!\cdots\!44 \nu ) / 40\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{19} - \beta_{16} + 2\beta_{15} + \beta_{13} + \beta_{12} - 2\beta_{11} + 2\beta_{9} ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{7} + 4\beta_{6} + \beta_{5} - \beta_{4} - 4\beta_{3} - \beta_{2} + 2\beta _1 - 1236 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 109 \beta_{19} + 12 \beta_{18} - 6 \beta_{17} + 189 \beta_{16} - 278 \beta_{15} + 84 \beta_{14} + \cdots - 452 \beta_{9} ) / 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 198 \beta_{16} + 198 \beta_{15} + 5 \beta_{8} - 284 \beta_{7} - 286 \beta_{6} - 163 \beta_{5} + \cdots + 69480 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 11821 \beta_{19} - 2780 \beta_{18} + 2380 \beta_{17} - 28825 \beta_{16} + 38606 \beta_{15} + \cdots + 82632 \beta_{9} ) / 32 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 81240 \beta_{16} - 81240 \beta_{15} - 2051 \beta_{8} + 101351 \beta_{7} + 81348 \beta_{6} + \cdots - 17441004 ) / 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1334977 \beta_{19} + 530740 \beta_{18} - 535850 \beta_{17} + 4298729 \beta_{16} + \cdots - 13934608 \beta_{9} ) / 32 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 6862170 \beta_{16} + 6862170 \beta_{15} + 82614 \beta_{8} - 8648487 \beta_{7} - 5992946 \beta_{6} + \cdots + 1172220642 ) / 8 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 156871837 \beta_{19} - 95320548 \beta_{18} + 102994812 \beta_{17} - 643601481 \beta_{16} + \cdots + 2282738144 \beta_{9} ) / 32 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 2220373296 \beta_{16} - 2220373296 \beta_{15} + 18894293 \beta_{8} + 2882933299 \beta_{7} + \cdots - 330935778396 ) / 16 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 19109563537 \beta_{19} + 16693214516 \beta_{18} - 18576634846 \beta_{17} + \cdots - 369974138448 \beta_{9} ) / 32 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 178166504406 \beta_{16} + 178166504406 \beta_{15} - 5712520865 \beta_{8} - 237652681114 \beta_{7} + \cdots + 24230091739812 ) / 8 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 2404860969769 \beta_{19} - 2884218399244 \beta_{18} + 3246716832164 \beta_{17} + \cdots + 59793831929392 \beta_{9} ) / 32 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 57290667713592 \beta_{16} - 57290667713592 \beta_{15} + 3189274613049 \beta_{8} + \cdots - 72\!\cdots\!96 ) / 16 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 311801106581893 \beta_{19} + 493787442312180 \beta_{18} - 557291258705034 \beta_{17} + \cdots - 96\!\cdots\!92 \beta_{9} ) / 32 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 46\!\cdots\!50 \beta_{16} + \cdots + 56\!\cdots\!42 ) / 8 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 41\!\cdots\!53 \beta_{19} + \cdots + 15\!\cdots\!20 \beta_{9} ) / 32 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 15\!\cdots\!64 \beta_{16} + \cdots - 17\!\cdots\!28 ) / 16 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 56\!\cdots\!77 \beta_{19} + \cdots - 25\!\cdots\!92 \beta_{9} ) / 32 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(209\) | \(469\) | \(703\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 649.1 |
|
0 | 0 | 0 | − | 18.2662i | 0 | − | 18.4571i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.2 | 0 | 0 | 0 | − | 18.2662i | 0 | 18.4571i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.3 | 0 | 0 | 0 | − | 17.2883i | 0 | − | 1.33952i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.4 | 0 | 0 | 0 | − | 17.2883i | 0 | 1.33952i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.5 | 0 | 0 | 0 | − | 7.60752i | 0 | − | 19.3712i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.6 | 0 | 0 | 0 | − | 7.60752i | 0 | 19.3712i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.7 | 0 | 0 | 0 | − | 4.89699i | 0 | − | 20.5378i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.8 | 0 | 0 | 0 | − | 4.89699i | 0 | 20.5378i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.9 | 0 | 0 | 0 | − | 3.68838i | 0 | − | 22.4610i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.10 | 0 | 0 | 0 | − | 3.68838i | 0 | 22.4610i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.11 | 0 | 0 | 0 | 3.68838i | 0 | − | 22.4610i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.12 | 0 | 0 | 0 | 3.68838i | 0 | 22.4610i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.13 | 0 | 0 | 0 | 4.89699i | 0 | − | 20.5378i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.14 | 0 | 0 | 0 | 4.89699i | 0 | 20.5378i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.15 | 0 | 0 | 0 | 7.60752i | 0 | − | 19.3712i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.16 | 0 | 0 | 0 | 7.60752i | 0 | 19.3712i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.17 | 0 | 0 | 0 | 17.2883i | 0 | − | 1.33952i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.18 | 0 | 0 | 0 | 17.2883i | 0 | 1.33952i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.19 | 0 | 0 | 0 | 18.2662i | 0 | − | 18.4571i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 649.20 | 0 | 0 | 0 | 18.2662i | 0 | 18.4571i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 39.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 936.4.c.d | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 936.4.c.d | ✓ | 20 |
| 13.b | even | 2 | 1 | inner | 936.4.c.d | ✓ | 20 |
| 39.d | odd | 2 | 1 | inner | 936.4.c.d | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 936.4.c.d | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 936.4.c.d | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 936.4.c.d | ✓ | 20 | 13.b | even | 2 | 1 | inner |
| 936.4.c.d | ✓ | 20 | 39.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + 728T_{5}^{8} + 162608T_{5}^{6} + 11120768T_{5}^{4} + 261396736T_{5}^{2} + 1882865664 \)
acting on \(S_{4}^{\mathrm{new}}(936, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( (T^{10} + 728 T^{8} + \cdots + 1882865664)^{2} \)
|
| $7$ |
\( (T^{10} + 1644 T^{8} + \cdots + 48809181184)^{2} \)
|
| $11$ |
\( (T^{10} + \cdots + 38116102782976)^{2} \)
|
| $13$ |
\( (T^{10} + \cdots + 51\!\cdots\!57)^{2} \)
|
| $17$ |
\( (T^{10} + \cdots - 10\!\cdots\!96)^{2} \)
|
| $19$ |
\( (T^{10} + \cdots + 97\!\cdots\!96)^{2} \)
|
| $23$ |
\( (T^{10} + \cdots - 83\!\cdots\!36)^{2} \)
|
| $29$ |
\( (T^{10} + \cdots - 10\!\cdots\!24)^{2} \)
|
| $31$ |
\( (T^{10} + \cdots + 35\!\cdots\!76)^{2} \)
|
| $37$ |
\( (T^{10} + \cdots + 12\!\cdots\!96)^{2} \)
|
| $41$ |
\( (T^{10} + \cdots + 34\!\cdots\!56)^{2} \)
|
| $43$ |
\( (T^{5} + 72 T^{4} + \cdots + 83965388800)^{4} \)
|
| $47$ |
\( (T^{10} + \cdots + 81\!\cdots\!00)^{2} \)
|
| $53$ |
\( (T^{10} + \cdots - 20\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{5} - 26 T^{4} + \cdots - 45217007264)^{4} \)
|
| $67$ |
\( (T^{10} + \cdots + 54\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{10} + \cdots + 12\!\cdots\!04)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots + 42\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{5} + \cdots + 75678131303424)^{4} \)
|
| $83$ |
\( (T^{10} + \cdots + 10\!\cdots\!36)^{2} \)
|
| $89$ |
\( (T^{10} + \cdots + 70\!\cdots\!84)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots + 49\!\cdots\!04)^{2} \)
|
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