Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [150,4,Mod(31,150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(150, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("150.31");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 150.g (of order \(5\), 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: | \(8.85028650086\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} + 66 x^{14} + 116 x^{13} - 15174 x^{12} + 66830 x^{11} - 253085 x^{10} + \cdots + 48733788520125 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{16} - 4 x^{15} + 66 x^{14} + 116 x^{13} - 15174 x^{12} + 66830 x^{11} - 253085 x^{10} + \cdots + 48733788520125 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 10\!\cdots\!81 \nu^{15} + \cdots + 20\!\cdots\!75 ) / 22\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 30\!\cdots\!41 \nu^{15} + \cdots - 10\!\cdots\!25 ) / 61\!\cdots\!40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 40\!\cdots\!01 \nu^{15} + \cdots + 10\!\cdots\!25 ) / 61\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 85\!\cdots\!44 \nu^{15} + \cdots + 34\!\cdots\!75 ) / 91\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 26\!\cdots\!35 \nu^{15} + \cdots - 90\!\cdots\!25 ) / 18\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 15\!\cdots\!92 \nu^{15} + \cdots + 10\!\cdots\!75 ) / 10\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 16\!\cdots\!93 \nu^{15} + \cdots + 77\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 13\!\cdots\!90 \nu^{15} + \cdots + 82\!\cdots\!25 ) / 38\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 45\!\cdots\!16 \nu^{15} + \cdots - 14\!\cdots\!75 ) / 11\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 41\!\cdots\!47 \nu^{15} + \cdots - 46\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 17\!\cdots\!39 \nu^{15} + \cdots + 27\!\cdots\!75 ) / 38\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 13\!\cdots\!89 \nu^{15} + \cdots + 45\!\cdots\!00 ) / 22\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 26\!\cdots\!32 \nu^{15} + \cdots - 41\!\cdots\!75 ) / 25\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 45\!\cdots\!47 \nu^{15} + \cdots + 18\!\cdots\!00 ) / 22\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 13\!\cdots\!85 \nu^{15} + \cdots - 10\!\cdots\!25 ) / 63\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{10} + 3\beta_{7} - 2\beta_{3} - \beta_{2} ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 15 \beta_{15} + 5 \beta_{14} - 15 \beta_{13} + 5 \beta_{11} - 9 \beta_{10} - 5 \beta_{9} + 5 \beta_{8} + \cdots - 35 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10 \beta_{15} - 15 \beta_{14} + 100 \beta_{13} - 10 \beta_{12} + 110 \beta_{11} + \beta_{10} + \cdots - 675 ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 280 \beta_{15} - 200 \beta_{14} + 315 \beta_{13} + 10 \beta_{12} + 45 \beta_{11} + 1481 \beta_{10} + \cdots + 26610 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 4195 \beta_{15} - 7460 \beta_{14} - 3100 \beta_{13} - 4270 \beta_{12} - 9275 \beta_{11} + \cdots + 35730 ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 75565 \beta_{15} + 38305 \beta_{14} - 165060 \beta_{13} - 5550 \beta_{12} + 8160 \beta_{11} + \cdots - 1795475 ) / 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 207025 \beta_{15} + 248870 \beta_{14} + 475385 \beta_{13} + 101970 \beta_{12} + 1206100 \beta_{11} + \cdots - 15405855 ) / 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 7835715 \beta_{15} - 3778065 \beta_{14} + 11920060 \beta_{13} + 893340 \beta_{12} + 2169895 \beta_{11} + \cdots + 236792440 ) / 5 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 69454905 \beta_{15} - 111437175 \beta_{14} - 38429235 \beta_{13} - 67390690 \beta_{12} + \cdots + 1391163585 ) / 5 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 697760905 \beta_{15} + 252311815 \beta_{14} - 2151958150 \beta_{13} - 173187820 \beta_{12} + \cdots - 32088989220 ) / 5 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 6597997540 \beta_{15} + 9611569455 \beta_{14} + 2383843690 \beta_{13} + 5476132900 \beta_{12} + \cdots - 259224447650 ) / 5 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 118247695100 \beta_{15} - 38886251005 \beta_{14} + 253469004735 \beta_{13} + 33670782470 \beta_{12} + \cdots + 3281798731670 ) / 5 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 1094068646715 \beta_{15} - 1557206583190 \beta_{14} - 226950738890 \beta_{13} - 878491189660 \beta_{12} + \cdots + 33433860439940 ) / 5 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 10728765764470 \beta_{15} + 2132910564075 \beta_{14} - 33182869647685 \beta_{13} + \cdots - 470812076760790 ) / 5 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 146259104445530 \beta_{15} + 197726667011940 \beta_{14} - 11684313381100 \beta_{13} + \cdots - 47\!\cdots\!95 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{4} + \beta_{5} + \beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0.618034 | + | 1.90211i | −2.42705 | + | 1.76336i | −3.23607 | + | 2.35114i | −9.30538 | + | 6.19757i | −4.85410 | − | 3.52671i | −2.27951 | −6.47214 | − | 4.70228i | 2.78115 | − | 8.55951i | −17.5395 | − | 13.8696i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.2 | 0.618034 | + | 1.90211i | −2.42705 | + | 1.76336i | −3.23607 | + | 2.35114i | −2.17298 | − | 10.9671i | −4.85410 | − | 3.52671i | 31.0680 | −6.47214 | − | 4.70228i | 2.78115 | − | 8.55951i | 19.5178 | − | 10.9113i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.3 | 0.618034 | + | 1.90211i | −2.42705 | + | 1.76336i | −3.23607 | + | 2.35114i | 8.48827 | − | 7.27663i | −4.85410 | − | 3.52671i | −25.3959 | −6.47214 | − | 4.70228i | 2.78115 | − | 8.55951i | 19.0870 | + | 11.6484i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.4 | 0.618034 | + | 1.90211i | −2.42705 | + | 1.76336i | −3.23607 | + | 2.35114i | 9.53518 | + | 5.83783i | −4.85410 | − | 3.52671i | 13.6975 | −6.47214 | − | 4.70228i | 2.78115 | − | 8.55951i | −5.21115 | + | 21.7450i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.1 | −1.61803 | + | 1.17557i | 0.927051 | − | 2.85317i | 1.23607 | − | 3.80423i | −7.40404 | − | 8.37736i | 1.85410 | + | 5.70634i | 24.2141 | 2.47214 | + | 7.60845i | −7.28115 | − | 5.29007i | 21.8282 | + | 4.85088i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.2 | −1.61803 | + | 1.17557i | 0.927051 | − | 2.85317i | 1.23607 | − | 3.80423i | −3.72153 | + | 10.5428i | 1.85410 | + | 5.70634i | 7.73387 | 2.47214 | + | 7.60845i | −7.28115 | − | 5.29007i | −6.37221 | − | 21.4335i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.3 | −1.61803 | + | 1.17557i | 0.927051 | − | 2.85317i | 1.23607 | − | 3.80423i | 0.900748 | − | 11.1440i | 1.85410 | + | 5.70634i | −31.6588 | 2.47214 | + | 7.60845i | −7.28115 | − | 5.29007i | 11.6431 | + | 19.0903i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.4 | −1.61803 | + | 1.17557i | 0.927051 | − | 2.85317i | 1.23607 | − | 3.80423i | 11.1797 | − | 0.115716i | 1.85410 | + | 5.70634i | 5.62070 | 2.47214 | + | 7.60845i | −7.28115 | − | 5.29007i | −17.9532 | + | 13.3298i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.1 | −1.61803 | − | 1.17557i | 0.927051 | + | 2.85317i | 1.23607 | + | 3.80423i | −7.40404 | + | 8.37736i | 1.85410 | − | 5.70634i | 24.2141 | 2.47214 | − | 7.60845i | −7.28115 | + | 5.29007i | 21.8282 | − | 4.85088i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.2 | −1.61803 | − | 1.17557i | 0.927051 | + | 2.85317i | 1.23607 | + | 3.80423i | −3.72153 | − | 10.5428i | 1.85410 | − | 5.70634i | 7.73387 | 2.47214 | − | 7.60845i | −7.28115 | + | 5.29007i | −6.37221 | + | 21.4335i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.3 | −1.61803 | − | 1.17557i | 0.927051 | + | 2.85317i | 1.23607 | + | 3.80423i | 0.900748 | + | 11.1440i | 1.85410 | − | 5.70634i | −31.6588 | 2.47214 | − | 7.60845i | −7.28115 | + | 5.29007i | 11.6431 | − | 19.0903i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.4 | −1.61803 | − | 1.17557i | 0.927051 | + | 2.85317i | 1.23607 | + | 3.80423i | 11.1797 | + | 0.115716i | 1.85410 | − | 5.70634i | 5.62070 | 2.47214 | − | 7.60845i | −7.28115 | + | 5.29007i | −17.9532 | − | 13.3298i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.1 | 0.618034 | − | 1.90211i | −2.42705 | − | 1.76336i | −3.23607 | − | 2.35114i | −9.30538 | − | 6.19757i | −4.85410 | + | 3.52671i | −2.27951 | −6.47214 | + | 4.70228i | 2.78115 | + | 8.55951i | −17.5395 | + | 13.8696i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.2 | 0.618034 | − | 1.90211i | −2.42705 | − | 1.76336i | −3.23607 | − | 2.35114i | −2.17298 | + | 10.9671i | −4.85410 | + | 3.52671i | 31.0680 | −6.47214 | + | 4.70228i | 2.78115 | + | 8.55951i | 19.5178 | + | 10.9113i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.3 | 0.618034 | − | 1.90211i | −2.42705 | − | 1.76336i | −3.23607 | − | 2.35114i | 8.48827 | + | 7.27663i | −4.85410 | + | 3.52671i | −25.3959 | −6.47214 | + | 4.70228i | 2.78115 | + | 8.55951i | 19.0870 | − | 11.6484i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.4 | 0.618034 | − | 1.90211i | −2.42705 | − | 1.76336i | −3.23607 | − | 2.35114i | 9.53518 | − | 5.83783i | −4.85410 | + | 3.52671i | 13.6975 | −6.47214 | + | 4.70228i | 2.78115 | + | 8.55951i | −5.21115 | − | 21.7450i | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 25.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 150.4.g.b | ✓ | 16 |
| 25.d | even | 5 | 1 | inner | 150.4.g.b | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 150.4.g.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 150.4.g.b | ✓ | 16 | 25.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 23 T_{7}^{7} - 1477 T_{7}^{6} + 38269 T_{7}^{5} + 377930 T_{7}^{4} - 15110346 T_{7}^{3} + \cdots - 820937664 \)
acting on \(S_{4}^{\mathrm{new}}(150, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{4} \)
|
| $3$ |
\( (T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{4} \)
|
| $5$ |
\( T^{16} + \cdots + 59\!\cdots\!25 \)
|
| $7$ |
\( (T^{8} - 23 T^{7} + \cdots - 820937664)^{2} \)
|
| $11$ |
\( T^{16} + \cdots + 58\!\cdots\!96 \)
|
| $13$ |
\( T^{16} + \cdots + 10\!\cdots\!96 \)
|
| $17$ |
\( T^{16} + \cdots + 43\!\cdots\!56 \)
|
| $19$ |
\( T^{16} + \cdots + 44\!\cdots\!00 \)
|
| $23$ |
\( T^{16} + \cdots + 35\!\cdots\!36 \)
|
| $29$ |
\( T^{16} + \cdots + 45\!\cdots\!00 \)
|
| $31$ |
\( T^{16} + \cdots + 16\!\cdots\!96 \)
|
| $37$ |
\( T^{16} + \cdots + 37\!\cdots\!96 \)
|
| $41$ |
\( T^{16} + \cdots + 13\!\cdots\!56 \)
|
| $43$ |
\( (T^{8} + \cdots + 33\!\cdots\!76)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 24\!\cdots\!96 \)
|
| $53$ |
\( T^{16} + \cdots + 21\!\cdots\!21 \)
|
| $59$ |
\( T^{16} + \cdots + 47\!\cdots\!00 \)
|
| $61$ |
\( T^{16} + \cdots + 70\!\cdots\!76 \)
|
| $67$ |
\( T^{16} + \cdots + 39\!\cdots\!36 \)
|
| $71$ |
\( T^{16} + \cdots + 10\!\cdots\!16 \)
|
| $73$ |
\( T^{16} + \cdots + 11\!\cdots\!76 \)
|
| $79$ |
\( T^{16} + \cdots + 28\!\cdots\!00 \)
|
| $83$ |
\( T^{16} + \cdots + 76\!\cdots\!76 \)
|
| $89$ |
\( T^{16} + \cdots + 32\!\cdots\!00 \)
|
| $97$ |
\( T^{16} + \cdots + 53\!\cdots\!61 \)
|
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