Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [165,4,Mod(34,165)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(165, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("165.34");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 165.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: | \(9.73531515095\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} + 66x^{12} + 1705x^{10} + 22060x^{8} + 151880x^{6} + 537860x^{4} + 825344x^{2} + 262144 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4}\cdot 5 \) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} + 66x^{12} + 1705x^{10} + 22060x^{8} + 151880x^{6} + 537860x^{4} + 825344x^{2} + 262144 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 13137 \nu^{12} - 712162 \nu^{10} - 13476473 \nu^{8} - 100603116 \nu^{6} - 186755016 \nu^{4} + \cdots + 542747648 ) / 37402624 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 1221 \nu^{12} - 72330 \nu^{10} - 1598477 \nu^{8} - 16399708 \nu^{6} - 79067688 \nu^{4} + \cdots - 29614336 ) / 2337664 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1221 \nu^{12} - 72330 \nu^{10} - 1598477 \nu^{8} - 16399708 \nu^{6} - 79067688 \nu^{4} + \cdots - 63510464 ) / 1168832 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 241963 \nu^{13} - 2468784 \nu^{12} - 18454550 \nu^{11} - 142340960 \nu^{10} + \cdots - 11130454016 ) / 5984419840 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 6365 \nu^{13} + 427514 \nu^{11} + 10924517 \nu^{9} + 129108220 \nu^{7} + 653010728 \nu^{5} + \cdots - 1630262272 \nu ) / 85491712 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 51749 \nu^{13} - 3102858 \nu^{11} - 69715565 \nu^{9} - 732372828 \nu^{7} + \cdots - 2918278144 \nu ) / 598441984 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 737029 \nu^{13} - 1877968 \nu^{12} + 46201930 \nu^{11} - 124343200 \nu^{10} + \cdots - 77504233472 ) / 5984419840 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 155247 \nu^{13} + 9308574 \nu^{11} + 209146695 \nu^{9} + 2197118484 \nu^{7} + \cdots + 10550160384 \nu ) / 598441984 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 1772009 \nu^{13} - 1877968 \nu^{12} + 108259090 \nu^{11} - 124343200 \nu^{10} + \cdots - 77504233472 ) / 5984419840 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 1254519 \nu^{13} + 1858288 \nu^{12} - 77230510 \nu^{11} + 101581280 \nu^{10} + \cdots + 4719460352 ) / 2992209920 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 55801 \nu^{13} + 3314994 \nu^{11} + 73138529 \nu^{9} + 739391436 \nu^{7} + 3369032456 \nu^{5} + \cdots - 2673449984 \nu ) / 85491712 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 4008547 \nu^{13} + 15002416 \nu^{12} + 238708870 \nu^{11} + 901354080 \nu^{10} + \cdots + 671570673664 ) / 5984419840 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 4008547 \nu^{13} + 10655664 \nu^{12} - 238708870 \nu^{11} + 634669920 \nu^{10} + \cdots + 582935986176 ) / 5984419840 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{8} + 3\beta_{6} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} - 29 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{9} - 16\beta_{8} - 3\beta_{7} - 42\beta_{6} ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5 \beta_{13} - 3 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} - 4 \beta_{6} - 4 \beta_{5} + \cdots + 421 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 6 \beta_{13} + 6 \beta_{12} - 15 \beta_{11} + 3 \beta_{10} - 90 \beta_{9} + 302 \beta_{8} + \cdots + 3 \beta_{2} ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 139 \beta_{13} + 105 \beta_{12} - 122 \beta_{11} + 134 \beta_{10} - 110 \beta_{9} + 12 \beta_{7} + \cdots - 7209 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 210 \beta_{13} - 210 \beta_{12} + 537 \beta_{11} - 143 \beta_{10} + 2180 \beta_{9} + \cdots - 143 \beta_{2} ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 3023 \beta_{13} - 2865 \beta_{12} + 2944 \beta_{11} - 3436 \beta_{10} + 2452 \beta_{9} - 492 \beta_{7} + \cdots + 133657 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 5262 \beta_{13} + 5262 \beta_{12} - 14349 \beta_{11} + 4635 \beta_{10} - 49248 \beta_{9} + \cdots + 4635 \beta_{2} ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 60991 \beta_{13} + 71025 \beta_{12} - 66008 \beta_{11} + 79952 \beta_{10} - 52064 \beta_{9} + \cdots - 2578069 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 117378 \beta_{13} - 117378 \beta_{12} + 344397 \beta_{11} - 129011 \beta_{10} + 1079324 \beta_{9} + \cdots - 129011 \beta_{2} ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( 399537 \beta_{13} - 557567 \beta_{12} + 478552 \beta_{11} - 592916 \beta_{10} + 364188 \beta_{9} + \cdots + 16938411 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 2496558 \beta_{13} + 2496558 \beta_{12} - 7857753 \beta_{11} + 3302799 \beta_{10} + \cdots + 3302799 \beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).
| \(n\) | \(46\) | \(56\) | \(67\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 34.1 |
|
− | 5.15324i | − | 3.00000i | −18.5558 | −7.01850 | − | 8.70291i | −15.4597 | 17.2148i | 54.3968i | −9.00000 | −44.8482 | + | 36.1680i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.2 | − | 3.70507i | − | 3.00000i | −5.72754 | −10.6368 | + | 3.44353i | −11.1152 | − | 30.9104i | − | 8.41961i | −9.00000 | 12.7585 | + | 39.4102i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.3 | − | 3.57884i | 3.00000i | −4.80807 | −0.925309 | − | 11.1420i | 10.7365 | 7.85216i | − | 11.4234i | −9.00000 | −39.8753 | + | 3.31153i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.4 | − | 2.98683i | − | 3.00000i | −0.921158 | 7.35339 | + | 8.42185i | −8.96049 | − | 6.37827i | − | 21.1433i | −9.00000 | 25.1547 | − | 21.9633i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.5 | − | 2.20690i | 3.00000i | 3.12958 | 4.62093 | + | 10.1807i | 6.62071 | 1.50972i | − | 24.5619i | −9.00000 | 22.4678 | − | 10.1979i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.6 | − | 1.41169i | 3.00000i | 6.00714 | −11.1339 | + | 1.01772i | 4.23506 | − | 15.2844i | − | 19.7737i | −9.00000 | 1.43670 | + | 15.7176i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.7 | − | 0.352288i | − | 3.00000i | 7.87589 | 10.7402 | − | 3.10602i | −1.05686 | − | 19.8486i | − | 5.59288i | −9.00000 | −1.09421 | − | 3.78365i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.8 | 0.352288i | 3.00000i | 7.87589 | 10.7402 | + | 3.10602i | −1.05686 | 19.8486i | 5.59288i | −9.00000 | −1.09421 | + | 3.78365i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.9 | 1.41169i | − | 3.00000i | 6.00714 | −11.1339 | − | 1.01772i | 4.23506 | 15.2844i | 19.7737i | −9.00000 | 1.43670 | − | 15.7176i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.10 | 2.20690i | − | 3.00000i | 3.12958 | 4.62093 | − | 10.1807i | 6.62071 | − | 1.50972i | 24.5619i | −9.00000 | 22.4678 | + | 10.1979i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.11 | 2.98683i | 3.00000i | −0.921158 | 7.35339 | − | 8.42185i | −8.96049 | 6.37827i | 21.1433i | −9.00000 | 25.1547 | + | 21.9633i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.12 | 3.57884i | − | 3.00000i | −4.80807 | −0.925309 | + | 11.1420i | 10.7365 | − | 7.85216i | 11.4234i | −9.00000 | −39.8753 | − | 3.31153i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.13 | 3.70507i | 3.00000i | −5.72754 | −10.6368 | − | 3.44353i | −11.1152 | 30.9104i | 8.41961i | −9.00000 | 12.7585 | − | 39.4102i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.14 | 5.15324i | 3.00000i | −18.5558 | −7.01850 | + | 8.70291i | −15.4597 | − | 17.2148i | − | 54.3968i | −9.00000 | −44.8482 | − | 36.1680i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 165.4.c.b | ✓ | 14 |
| 3.b | odd | 2 | 1 | 495.4.c.d | 14 | ||
| 5.b | even | 2 | 1 | inner | 165.4.c.b | ✓ | 14 |
| 5.c | odd | 4 | 1 | 825.4.a.ba | 7 | ||
| 5.c | odd | 4 | 1 | 825.4.a.bd | 7 | ||
| 15.d | odd | 2 | 1 | 495.4.c.d | 14 | ||
| 15.e | even | 4 | 1 | 2475.4.a.bo | 7 | ||
| 15.e | even | 4 | 1 | 2475.4.a.bs | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.4.c.b | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 165.4.c.b | ✓ | 14 | 5.b | even | 2 | 1 | inner |
| 495.4.c.d | 14 | 3.b | odd | 2 | 1 | ||
| 495.4.c.d | 14 | 15.d | odd | 2 | 1 | ||
| 825.4.a.ba | 7 | 5.c | odd | 4 | 1 | ||
| 825.4.a.bd | 7 | 5.c | odd | 4 | 1 | ||
| 2475.4.a.bo | 7 | 15.e | even | 4 | 1 | ||
| 2475.4.a.bs | 7 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} + 69T_{2}^{12} + 1798T_{2}^{10} + 22642T_{2}^{8} + 143537T_{2}^{6} + 424913T_{2}^{4} + 454864T_{2}^{2} + 50176 \)
acting on \(S_{4}^{\mathrm{new}}(165, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} + 69 T^{12} + \cdots + 50176 \)
|
| $3$ |
\( (T^{2} + 9)^{7} \)
|
| $5$ |
\( T^{14} + \cdots + 476837158203125 \)
|
| $7$ |
\( T^{14} + \cdots + 148986826592256 \)
|
| $11$ |
\( (T - 11)^{14} \)
|
| $13$ |
\( T^{14} + \cdots + 24\!\cdots\!76 \)
|
| $17$ |
\( T^{14} + \cdots + 46\!\cdots\!64 \)
|
| $19$ |
\( (T^{7} + \cdots - 46691778560000)^{2} \)
|
| $23$ |
\( T^{14} + \cdots + 40\!\cdots\!36 \)
|
| $29$ |
\( (T^{7} + \cdots + 38672535237120)^{2} \)
|
| $31$ |
\( (T^{7} + \cdots + 197687760429056)^{2} \)
|
| $37$ |
\( T^{14} + \cdots + 94\!\cdots\!96 \)
|
| $41$ |
\( (T^{7} + \cdots + 26\!\cdots\!92)^{2} \)
|
| $43$ |
\( T^{14} + \cdots + 24\!\cdots\!00 \)
|
| $47$ |
\( T^{14} + \cdots + 74\!\cdots\!96 \)
|
| $53$ |
\( T^{14} + \cdots + 61\!\cdots\!56 \)
|
| $59$ |
\( (T^{7} + \cdots - 30\!\cdots\!20)^{2} \)
|
| $61$ |
\( (T^{7} + \cdots + 39\!\cdots\!12)^{2} \)
|
| $67$ |
\( T^{14} + \cdots + 17\!\cdots\!56 \)
|
| $71$ |
\( (T^{7} + \cdots + 765697934131200)^{2} \)
|
| $73$ |
\( T^{14} + \cdots + 13\!\cdots\!56 \)
|
| $79$ |
\( (T^{7} + \cdots + 55\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{14} + \cdots + 58\!\cdots\!44 \)
|
| $89$ |
\( (T^{7} + \cdots - 70\!\cdots\!40)^{2} \)
|
| $97$ |
\( T^{14} + \cdots + 40\!\cdots\!76 \)
|
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