Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [366,2,Mod(25,366)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(366, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([0, 22]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("366.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 366 = 2 \cdot 3 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 366.q (of order \(15\), degree \(8\), 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: | \(2.92252471398\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{15})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 20x^{14} + 154x^{12} + 580x^{10} + 1131x^{8} + 1115x^{6} + 484x^{4} + 55x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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
\( x^{16} + 20x^{14} + 154x^{12} + 580x^{10} + 1131x^{8} + 1115x^{6} + 484x^{4} + 55x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 405 \nu^{14} - 7843 \nu^{12} - 57865 \nu^{10} - 206167 \nu^{8} - 374583 \nu^{6} - 329968 \nu^{4} + \cdots - 4042 ) / 14702 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 405 \nu^{14} - 7843 \nu^{12} - 57865 \nu^{10} - 206167 \nu^{8} - 374583 \nu^{6} - 329968 \nu^{4} + \cdots - 4042 ) / 14702 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 329 \nu^{15} + 417 \nu^{14} + 8513 \nu^{13} + 8511 \nu^{12} + 87210 \nu^{11} + 66386 \nu^{10} + \cdots - 7981 ) / 14702 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 449 \nu^{15} + 1528 \nu^{14} - 7842 \nu^{13} + 28701 \nu^{12} - 47453 \nu^{11} + 200437 \nu^{10} + \cdots + 2980 ) / 14702 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1797 \nu^{15} + 417 \nu^{14} - 33874 \nu^{13} + 8511 \nu^{12} - 237691 \nu^{11} + 66386 \nu^{10} + \cdots - 630 ) / 14702 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 630 \nu^{15} + 1797 \nu^{14} - 13017 \nu^{13} + 33874 \nu^{12} - 105531 \nu^{11} + 237691 \nu^{10} + \cdots + 6064 ) / 14702 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 630 \nu^{15} + 1797 \nu^{14} + 13017 \nu^{13} + 33874 \nu^{12} + 105531 \nu^{11} + 237691 \nu^{10} + \cdots + 6064 ) / 14702 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 2350 \nu^{15} - 1348 \nu^{14} + 45055 \nu^{13} - 26032 \nu^{12} + 324688 \nu^{11} - 190238 \nu^{10} + \cdots - 21603 ) / 14702 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 630 \nu^{15} - 2202 \nu^{14} - 13017 \nu^{13} - 41717 \nu^{12} - 105531 \nu^{11} - 295556 \nu^{10} + \cdots + 19298 ) / 14702 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 2350 \nu^{15} + 1348 \nu^{14} + 45055 \nu^{13} + 26032 \nu^{12} + 324688 \nu^{11} + 190238 \nu^{10} + \cdots + 21603 ) / 14702 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 4042 \nu^{15} - 80435 \nu^{13} - 614625 \nu^{11} - 2286495 \nu^{9} - 4365335 \nu^{7} + \cdots - 7351 ) / 14702 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 1901 \nu^{15} + 3281 \nu^{14} - 37213 \nu^{13} + 62576 \nu^{12} - 277235 \nu^{11} + 448540 \nu^{10} + \cdots + 43327 ) / 14702 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 3637 \nu^{15} + 3041 \nu^{14} - 72592 \nu^{13} + 56567 \nu^{12} - 556760 \nu^{11} + 388385 \nu^{10} + \cdots - 2902 ) / 14702 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 4699 \nu^{15} + 449 \nu^{14} + 94955 \nu^{13} + 7842 \nu^{12} + 741166 \nu^{11} + 47453 \nu^{10} + \cdots - 30241 ) / 14702 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 3637 \nu^{15} - 3041 \nu^{14} - 72592 \nu^{13} - 56567 \nu^{12} - 556760 \nu^{11} - 388385 \nu^{10} + \cdots + 2902 ) / 14702 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} - \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{7} - \beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{15} - 2 \beta_{14} - \beta_{13} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{7} + \cdots - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - 6 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + \cdots + 11 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9 \beta_{15} + 16 \beta_{14} + 9 \beta_{13} + \beta_{12} + 14 \beta_{11} + 15 \beta_{10} + 8 \beta_{9} + \cdots + 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( 18 \beta_{13} + 11 \beta_{12} - 9 \beta_{11} + 10 \beta_{10} + 38 \beta_{9} + 19 \beta_{8} + 9 \beta_{7} + \cdots - 70 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 68 \beta_{15} - 108 \beta_{14} - 68 \beta_{13} - 8 \beta_{12} - 81 \beta_{11} - 98 \beta_{10} + \cdots - 44 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 3 \beta_{15} - 127 \beta_{13} - 93 \beta_{12} + 65 \beta_{11} - 73 \beta_{10} - 241 \beta_{9} + \cdots + 453 \)
|
| \(\nu^{9}\) | \(=\) |
\( 479 \beta_{15} + 694 \beta_{14} + 479 \beta_{13} + 42 \beta_{12} + 451 \beta_{11} + 622 \beta_{10} + \cdots + 284 \)
|
| \(\nu^{10}\) | \(=\) |
\( 44 \beta_{15} + 822 \beta_{13} + 710 \beta_{12} - 433 \beta_{11} + 475 \beta_{10} + 1518 \beta_{9} + \cdots - 2920 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 3255 \beta_{15} - 4382 \beta_{14} - 3255 \beta_{13} - 154 \beta_{12} - 2503 \beta_{11} - 3921 \beta_{10} + \cdots - 1888 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 428 \beta_{15} - 5118 \beta_{13} - 5143 \beta_{12} + 2773 \beta_{11} - 2927 \beta_{10} - 9523 \beta_{9} + \cdots + 18743 \)
|
| \(\nu^{13}\) | \(=\) |
\( 21693 \beta_{15} + 27512 \beta_{14} + 21693 \beta_{13} + 129 \beta_{12} + 13975 \beta_{11} + \cdots + 12742 \)
|
| \(\nu^{14}\) | \(=\) |
\( 3529 \beta_{15} + 31299 \beta_{13} + 36137 \beta_{12} - 17414 \beta_{11} + 17543 \beta_{10} + \cdots - 120090 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 143054 \beta_{15} - 172656 \beta_{14} - 143054 \beta_{13} + 4709 \beta_{12} - 78650 \beta_{11} + \cdots - 86365 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/366\mathbb{Z}\right)^\times\).
| \(n\) | \(245\) | \(307\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{7} - \beta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
0.669131 | − | 0.743145i | −0.309017 | − | 0.951057i | −0.104528 | − | 0.994522i | −1.96825 | − | 0.876321i | −0.913545 | − | 0.406737i | −2.06019 | + | 0.437908i | −0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | −1.96825 | + | 0.876321i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 25.2 | 0.669131 | − | 0.743145i | −0.309017 | − | 0.951057i | −0.104528 | − | 0.994522i | 0.936671 | + | 0.417033i | −0.913545 | − | 0.406737i | 3.80461 | − | 0.808694i | −0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | 0.936671 | − | 0.417033i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 73.1 | 0.913545 | + | 0.406737i | 0.809017 | − | 0.587785i | 0.669131 | + | 0.743145i | −0.192518 | + | 0.0409209i | 0.978148 | − | 0.207912i | 0.0489251 | − | 0.465491i | 0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | −0.192518 | − | 0.0409209i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 73.2 | 0.913545 | + | 0.406737i | 0.809017 | − | 0.587785i | 0.669131 | + | 0.743145i | 3.28870 | − | 0.699035i | 0.978148 | − | 0.207912i | −0.440618 | + | 4.19220i | 0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | 3.28870 | + | 0.699035i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 103.1 | −0.978148 | + | 0.207912i | −0.309017 | + | 0.951057i | 0.913545 | − | 0.406737i | −0.0507396 | + | 0.482755i | 0.104528 | − | 0.994522i | 3.08456 | − | 3.42575i | −0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | −0.0507396 | − | 0.482755i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 103.2 | −0.978148 | + | 0.207912i | −0.309017 | + | 0.951057i | 0.913545 | − | 0.406737i | 0.0372340 | − | 0.354258i | 0.104528 | − | 0.994522i | −0.710937 | + | 0.789575i | −0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | 0.0372340 | + | 0.354258i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 199.1 | −0.978148 | − | 0.207912i | −0.309017 | − | 0.951057i | 0.913545 | + | 0.406737i | −0.0507396 | − | 0.482755i | 0.104528 | + | 0.994522i | 3.08456 | + | 3.42575i | −0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | −0.0507396 | + | 0.482755i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 199.2 | −0.978148 | − | 0.207912i | −0.309017 | − | 0.951057i | 0.913545 | + | 0.406737i | 0.0372340 | + | 0.354258i | 0.104528 | + | 0.994522i | −0.710937 | − | 0.789575i | −0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | 0.0372340 | − | 0.354258i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 205.1 | 0.669131 | + | 0.743145i | −0.309017 | + | 0.951057i | −0.104528 | + | 0.994522i | −1.96825 | + | 0.876321i | −0.913545 | + | 0.406737i | −2.06019 | − | 0.437908i | −0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | −1.96825 | − | 0.876321i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 205.2 | 0.669131 | + | 0.743145i | −0.309017 | + | 0.951057i | −0.104528 | + | 0.994522i | 0.936671 | − | 0.417033i | −0.913545 | + | 0.406737i | 3.80461 | + | 0.808694i | −0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | 0.936671 | + | 0.417033i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 259.1 | −0.104528 | − | 0.994522i | 0.809017 | − | 0.587785i | −0.978148 | + | 0.207912i | −0.790872 | − | 0.878352i | −0.669131 | − | 0.743145i | 2.25023 | − | 1.00187i | 0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | −0.790872 | + | 0.878352i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 259.2 | −0.104528 | − | 0.994522i | 0.809017 | − | 0.587785i | −0.978148 | + | 0.207912i | 2.23978 | + | 2.48752i | −0.669131 | − | 0.743145i | 0.0234243 | − | 0.0104292i | 0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | 2.23978 | − | 2.48752i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 301.1 | −0.104528 | + | 0.994522i | 0.809017 | + | 0.587785i | −0.978148 | − | 0.207912i | −0.790872 | + | 0.878352i | −0.669131 | + | 0.743145i | 2.25023 | + | 1.00187i | 0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | −0.790872 | − | 0.878352i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 301.2 | −0.104528 | + | 0.994522i | 0.809017 | + | 0.587785i | −0.978148 | − | 0.207912i | 2.23978 | − | 2.48752i | −0.669131 | + | 0.743145i | 0.0234243 | + | 0.0104292i | 0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | 2.23978 | + | 2.48752i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 361.1 | 0.913545 | − | 0.406737i | 0.809017 | + | 0.587785i | 0.669131 | − | 0.743145i | −0.192518 | − | 0.0409209i | 0.978148 | + | 0.207912i | 0.0489251 | + | 0.465491i | 0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | −0.192518 | + | 0.0409209i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 361.2 | 0.913545 | − | 0.406737i | 0.809017 | + | 0.587785i | 0.669131 | − | 0.743145i | 3.28870 | + | 0.699035i | 0.978148 | + | 0.207912i | −0.440618 | − | 4.19220i | 0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | 3.28870 | − | 0.699035i | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 61.i | even | 15 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 366.2.q.b | ✓ | 16 |
| 61.i | even | 15 | 1 | inner | 366.2.q.b | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 366.2.q.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 366.2.q.b | ✓ | 16 | 61.i | even | 15 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} - 7 T_{5}^{15} + 12 T_{5}^{14} + 31 T_{5}^{13} - 112 T_{5}^{12} - 121 T_{5}^{11} + 584 T_{5}^{10} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(366, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $5$ |
\( T^{16} - 7 T^{15} + \cdots + 1 \)
|
| $7$ |
\( T^{16} - 12 T^{15} + \cdots + 25 \)
|
| $11$ |
\( (T^{8} - 7 T^{7} - 3 T^{6} + \cdots + 1)^{2} \)
|
| $13$ |
\( T^{16} + 4 T^{15} + \cdots + 41589601 \)
|
| $17$ |
\( T^{16} + 20 T^{15} + \cdots + 32761 \)
|
| $19$ |
\( T^{16} - T^{15} + \cdots + 130321 \)
|
| $23$ |
\( T^{16} - 6 T^{15} + \cdots + 32761 \)
|
| $29$ |
\( T^{16} + \cdots + 306915361 \)
|
| $31$ |
\( T^{16} + 5 T^{15} + \cdots + 58081 \)
|
| $37$ |
\( T^{16} + \cdots + 36097325691025 \)
|
| $41$ |
\( T^{16} + \cdots + 267025395025 \)
|
| $43$ |
\( T^{16} + \cdots + 1073283121 \)
|
| $47$ |
\( T^{16} + 13 T^{15} + \cdots + 326041 \)
|
| $53$ |
\( T^{16} + \cdots + 316448521 \)
|
| $59$ |
\( T^{16} + \cdots + 512671752121 \)
|
| $61$ |
\( T^{16} + \cdots + 191707312997281 \)
|
| $67$ |
\( T^{16} + \cdots + 9377791921 \)
|
| $71$ |
\( T^{16} + \cdots + 24\!\cdots\!25 \)
|
| $73$ |
\( T^{16} + \cdots + 428211630225625 \)
|
| $79$ |
\( T^{16} + \cdots + 2283979681 \)
|
| $83$ |
\( T^{16} + \cdots + 6182519641 \)
|
| $89$ |
\( T^{16} + \cdots + 893934321361 \)
|
| $97$ |
\( T^{16} + \cdots + 31413663121 \)
|
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