Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(109,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 5, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("930.109");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 930.z (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: | \(7.42608738798\) |
| 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: |
\( x^{16} - 6 x^{15} + 18 x^{14} - 44 x^{13} + 63 x^{12} - 46 x^{11} + 110 x^{10} - 120 x^{9} - 79 x^{8} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{16} - 6 x^{15} + 18 x^{14} - 44 x^{13} + 63 x^{12} - 46 x^{11} + 110 x^{10} - 120 x^{9} - 79 x^{8} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 18722 \nu^{15} + 185843 \nu^{14} - 838945 \nu^{13} + 2542453 \nu^{12} - 5698130 \nu^{11} + \cdots - 42999 ) / 782325 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 81136 \nu^{15} + 480599 \nu^{14} - 1425885 \nu^{13} + 3474934 \nu^{12} - 4895655 \nu^{11} + \cdots - 629332 ) / 782325 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 197674 \nu^{15} - 1066126 \nu^{14} + 2668400 \nu^{13} - 5438846 \nu^{12} + 3613495 \nu^{11} + \cdots + 513063 ) / 782325 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 202723 \nu^{15} + 1318367 \nu^{14} - 4308870 \nu^{13} + 11063617 \nu^{12} - 18261090 \nu^{11} + \cdots - 1446151 ) / 782325 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 208618 \nu^{15} + 1270047 \nu^{14} - 3845335 \nu^{13} + 9367137 \nu^{12} - 13443350 \nu^{11} + \cdots - 482471 ) / 782325 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 18068 \nu^{15} + 114030 \nu^{14} - 361140 \nu^{13} + 910990 \nu^{12} - 1436140 \nu^{11} + \cdots - 38786 ) / 52155 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 116478 \nu^{15} + 715962 \nu^{14} - 2204690 \nu^{13} + 5456387 \nu^{12} - 8121210 \nu^{11} + \cdots - 878996 ) / 260775 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 38786 \nu^{15} - 250784 \nu^{14} + 812178 \nu^{13} - 2067724 \nu^{12} + 3354508 \nu^{11} + \cdots + 45616 ) / 52155 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 629332 \nu^{15} - 3857128 \nu^{14} + 11808575 \nu^{13} - 29116493 \nu^{12} + 43122850 \nu^{11} + \cdots + 1934304 ) / 782325 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 41958 \nu^{15} - 268474 \nu^{14} + 860612 \nu^{13} - 2177219 \nu^{12} + 3468822 \nu^{11} + \cdots + 145107 ) / 52155 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 227516 \nu^{15} + 1457174 \nu^{14} - 4702060 \nu^{13} + 12019889 \nu^{12} - 19527595 \nu^{11} + \cdots - 683537 ) / 260775 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 45616 \nu^{15} + 312482 \nu^{14} - 1071872 \nu^{13} + 2819282 \nu^{12} - 4941532 \nu^{11} + \cdots - 12088 ) / 52155 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 381541 \nu^{15} + 2389864 \nu^{14} - 7495250 \nu^{13} + 18750324 \nu^{12} - 28945995 \nu^{11} + \cdots - 1113587 ) / 260775 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 230968 \nu^{15} - 1481454 \nu^{14} + 4764706 \nu^{13} - 12092649 \nu^{12} + 19409306 \nu^{11} + \cdots + 555224 ) / 156465 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( - \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{6} + \cdots - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -6\beta_{15} - 6\beta_{14} - 5\beta_{13} - 6\beta_{9} - 2\beta_{5} + 2\beta_{3} + 6\beta_{2} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 8 \beta_{15} - 6 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} + 8 \beta_{11} - 6 \beta_{10} + \cdots + 6 \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8 \beta_{14} + 8 \beta_{12} + 6 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} - 33 \beta_{7} - 18 \beta_{6} + \cdots + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 39 \beta_{15} - 21 \beta_{14} - 39 \beta_{13} + 72 \beta_{12} - 59 \beta_{11} + 57 \beta_{10} + \cdots - 59 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 288 \beta_{15} - 229 \beta_{14} - 229 \beta_{13} + 39 \beta_{12} - 136 \beta_{11} + 39 \beta_{10} + \cdots - 268 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 653 \beta_{15} - 424 \beta_{14} - 424 \beta_{13} - 424 \beta_{12} + 93 \beta_{11} - 424 \beta_{10} + \cdots - 361 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 268 \beta_{15} + 424 \beta_{14} - 716 \beta_{12} + 268 \beta_{11} - 984 \beta_{10} - 424 \beta_{9} + \cdots + 984 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1613 \beta_{14} + 1613 \beta_{12} - 3021 \beta_{11} + 629 \beta_{10} + 3021 \beta_{9} - 3021 \beta_{8} + \cdots - 1881 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 8907 \beta_{15} - 5145 \beta_{14} - 7026 \beta_{13} + 3021 \beta_{12} - 12557 \beta_{11} + \cdots - 13298 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 32881 \beta_{15} - 21464 \beta_{14} - 21464 \beta_{13} - 21464 \beta_{12} - 13298 \beta_{11} + \cdots - 28490 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 21464 \beta_{15} + 13298 \beta_{14} - 80941 \beta_{12} + 8166 \beta_{11} - 89107 \beta_{10} + \cdots - 8166 \)
|
| \(\nu^{14}\) | \(=\) |
\( 94239 \beta_{15} + 175180 \beta_{14} + 94239 \beta_{13} - 44285 \beta_{12} - 94239 \beta_{11} + \cdots + 94239 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( 152359 \beta_{14} + 152359 \beta_{12} - 668376 \beta_{11} + 58120 \beta_{10} + 668376 \beta_{9} + \cdots - 412792 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(311\) | \(871\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1 - \beta_{9} + \beta_{11} - \beta_{15}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 109.1 |
|
−0.587785 | − | 0.809017i | 0.587785 | − | 0.809017i | −0.309017 | + | 0.951057i | 0.898602 | + | 2.04756i | −1.00000 | −2.04378 | − | 0.664066i | 0.951057 | − | 0.309017i | −0.309017 | − | 0.951057i | 1.12833 | − | 1.93051i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.2 | −0.587785 | − | 0.809017i | 0.587785 | − | 0.809017i | −0.309017 | + | 0.951057i | 1.71943 | − | 1.42953i | −1.00000 | 2.04378 | + | 0.664066i | 0.951057 | − | 0.309017i | −0.309017 | − | 0.951057i | −2.16717 | − | 0.550794i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.3 | 0.587785 | + | 0.809017i | −0.587785 | + | 0.809017i | −0.309017 | + | 0.951057i | 0.898602 | − | 2.04756i | −1.00000 | 2.04378 | + | 0.664066i | −0.951057 | + | 0.309017i | −0.309017 | − | 0.951057i | 2.18470 | − | 0.476543i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.4 | 0.587785 | + | 0.809017i | −0.587785 | + | 0.809017i | −0.309017 | + | 0.951057i | 1.71943 | + | 1.42953i | −1.00000 | −2.04378 | − | 0.664066i | −0.951057 | + | 0.309017i | −0.309017 | − | 0.951057i | −0.145857 | + | 2.23131i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.1 | −0.951057 | − | 0.309017i | 0.951057 | − | 0.309017i | 0.809017 | + | 0.587785i | −1.82930 | + | 1.28594i | −1.00000 | 0.907165 | − | 1.24861i | −0.587785 | − | 0.809017i | 0.809017 | − | 0.587785i | 2.13715 | − | 0.657718i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.2 | −0.951057 | − | 0.309017i | 0.951057 | − | 0.309017i | 0.809017 | + | 0.587785i | 2.21127 | + | 0.332092i | −1.00000 | −0.907165 | + | 1.24861i | −0.587785 | − | 0.809017i | 0.809017 | − | 0.587785i | −2.00042 | − | 0.999158i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.3 | 0.951057 | + | 0.309017i | −0.951057 | + | 0.309017i | 0.809017 | + | 0.587785i | −1.82930 | − | 1.28594i | −1.00000 | −0.907165 | + | 1.24861i | 0.587785 | + | 0.809017i | 0.809017 | − | 0.587785i | −1.34239 | − | 1.78829i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.4 | 0.951057 | + | 0.309017i | −0.951057 | + | 0.309017i | 0.809017 | + | 0.587785i | 2.21127 | − | 0.332092i | −1.00000 | 0.907165 | − | 1.24861i | 0.587785 | + | 0.809017i | 0.809017 | − | 0.587785i | 2.20566 | + | 0.367482i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 469.1 | −0.951057 | + | 0.309017i | 0.951057 | + | 0.309017i | 0.809017 | − | 0.587785i | −1.82930 | − | 1.28594i | −1.00000 | 0.907165 | + | 1.24861i | −0.587785 | + | 0.809017i | 0.809017 | + | 0.587785i | 2.13715 | + | 0.657718i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 469.2 | −0.951057 | + | 0.309017i | 0.951057 | + | 0.309017i | 0.809017 | − | 0.587785i | 2.21127 | − | 0.332092i | −1.00000 | −0.907165 | − | 1.24861i | −0.587785 | + | 0.809017i | 0.809017 | + | 0.587785i | −2.00042 | + | 0.999158i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 469.3 | 0.951057 | − | 0.309017i | −0.951057 | − | 0.309017i | 0.809017 | − | 0.587785i | −1.82930 | + | 1.28594i | −1.00000 | −0.907165 | − | 1.24861i | 0.587785 | − | 0.809017i | 0.809017 | + | 0.587785i | −1.34239 | + | 1.78829i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 469.4 | 0.951057 | − | 0.309017i | −0.951057 | − | 0.309017i | 0.809017 | − | 0.587785i | 2.21127 | + | 0.332092i | −1.00000 | 0.907165 | + | 1.24861i | 0.587785 | − | 0.809017i | 0.809017 | + | 0.587785i | 2.20566 | − | 0.367482i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 529.1 | −0.587785 | + | 0.809017i | 0.587785 | + | 0.809017i | −0.309017 | − | 0.951057i | 0.898602 | − | 2.04756i | −1.00000 | −2.04378 | + | 0.664066i | 0.951057 | + | 0.309017i | −0.309017 | + | 0.951057i | 1.12833 | + | 1.93051i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 529.2 | −0.587785 | + | 0.809017i | 0.587785 | + | 0.809017i | −0.309017 | − | 0.951057i | 1.71943 | + | 1.42953i | −1.00000 | 2.04378 | − | 0.664066i | 0.951057 | + | 0.309017i | −0.309017 | + | 0.951057i | −2.16717 | + | 0.550794i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 529.3 | 0.587785 | − | 0.809017i | −0.587785 | − | 0.809017i | −0.309017 | − | 0.951057i | 0.898602 | + | 2.04756i | −1.00000 | 2.04378 | − | 0.664066i | −0.951057 | − | 0.309017i | −0.309017 | + | 0.951057i | 2.18470 | + | 0.476543i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 529.4 | 0.587785 | − | 0.809017i | −0.587785 | − | 0.809017i | −0.309017 | − | 0.951057i | 1.71943 | − | 1.42953i | −1.00000 | −2.04378 | + | 0.664066i | −0.951057 | − | 0.309017i | −0.309017 | + | 0.951057i | −0.145857 | − | 2.23131i | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 31.d | even | 5 | 1 | inner |
| 155.n | even | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 930.2.z.b | ✓ | 16 |
| 5.b | even | 2 | 1 | inner | 930.2.z.b | ✓ | 16 |
| 31.d | even | 5 | 1 | inner | 930.2.z.b | ✓ | 16 |
| 155.n | even | 10 | 1 | inner | 930.2.z.b | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 930.2.z.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 930.2.z.b | ✓ | 16 | 5.b | even | 2 | 1 | inner |
| 930.2.z.b | ✓ | 16 | 31.d | even | 5 | 1 | inner |
| 930.2.z.b | ✓ | 16 | 155.n | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 6T_{7}^{6} + 16T_{7}^{4} - 11T_{7}^{2} + 121 \)
acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \)
|
| $5$ |
\( (T^{8} - 6 T^{7} + \cdots + 625)^{2} \)
|
| $7$ |
\( (T^{8} - 6 T^{6} + \cdots + 121)^{2} \)
|
| $11$ |
\( (T^{8} - 4 T^{7} + 18 T^{6} + \cdots + 16)^{2} \)
|
| $13$ |
\( (T^{8} + 4 T^{6} + 46 T^{4} + \cdots + 1)^{2} \)
|
| $17$ |
\( T^{16} - 56 T^{14} + \cdots + 40960000 \)
|
| $19$ |
\( (T^{8} - 4 T^{7} + \cdots + 93025)^{2} \)
|
| $23$ |
\( (T^{8} - 20 T^{6} + \cdots + 160000)^{2} \)
|
| $29$ |
\( (T^{8} - 18 T^{7} + \cdots + 13456)^{2} \)
|
| $31$ |
\( (T^{8} - 20 T^{7} + \cdots + 923521)^{2} \)
|
| $37$ |
\( (T^{8} + 62 T^{6} + \cdots + 21025)^{2} \)
|
| $41$ |
\( (T^{8} - 16 T^{7} + \cdots + 400)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 324928500625 \)
|
| $47$ |
\( T^{16} + \cdots + 83181696160000 \)
|
| $53$ |
\( T^{16} + \cdots + 100000000 \)
|
| $59$ |
\( (T^{8} + 86 T^{6} + \cdots + 839056)^{2} \)
|
| $61$ |
\( (T^{4} + 4 T^{3} + \cdots + 1525)^{4} \)
|
| $67$ |
\( (T^{8} + 166 T^{6} + \cdots + 201601)^{2} \)
|
| $71$ |
\( (T^{8} + 6 T^{7} + \cdots + 336400)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 19858796855041 \)
|
| $79$ |
\( (T^{8} + 4 T^{7} + \cdots + 36481)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 2342560000 \)
|
| $89$ |
\( (T^{8} - 10 T^{7} + \cdots + 16)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 65\!\cdots\!25 \)
|
| show more | |
| show less |