Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [150,11,Mod(101,150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(150, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("150.101");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 150.d (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: | \(95.3035879011\) |
| 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: |
\( x^{14} - 6 x^{13} - 451853 x^{12} + 31595028 x^{11} + 79693501240 x^{10} - 10455174924036 x^{9} + \cdots + 36\!\cdots\!87 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{35}\cdot 3^{29}\cdot 5^{6} \) |
| 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
\( x^{14} - 6 x^{13} - 451853 x^{12} + 31595028 x^{11} + 79693501240 x^{10} - 10455174924036 x^{9} + \cdots + 36\!\cdots\!87 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 69\!\cdots\!05 \nu^{13} + \cdots - 12\!\cdots\!03 ) / 31\!\cdots\!27 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 31\!\cdots\!75 \nu^{13} + \cdots - 60\!\cdots\!93 ) / 20\!\cdots\!40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\!\cdots\!65 \nu^{13} + \cdots - 24\!\cdots\!72 ) / 62\!\cdots\!20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 61\!\cdots\!71 \nu^{13} + \cdots - 11\!\cdots\!40 ) / 26\!\cdots\!55 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 22\!\cdots\!89 \nu^{13} + \cdots + 75\!\cdots\!15 ) / 18\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 15\!\cdots\!19 \nu^{13} + \cdots - 30\!\cdots\!24 ) / 93\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 15\!\cdots\!56 \nu^{13} + \cdots - 31\!\cdots\!07 ) / 93\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 20\!\cdots\!85 \nu^{13} + \cdots + 35\!\cdots\!68 ) / 62\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 88\!\cdots\!64 \nu^{13} + \cdots + 14\!\cdots\!22 ) / 18\!\cdots\!60 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 12\!\cdots\!24 \nu^{13} + \cdots + 21\!\cdots\!03 ) / 20\!\cdots\!40 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 12\!\cdots\!79 \nu^{13} + \cdots - 25\!\cdots\!83 ) / 18\!\cdots\!60 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 26\!\cdots\!67 \nu^{13} + \cdots + 50\!\cdots\!19 ) / 31\!\cdots\!60 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 17\!\cdots\!63 \nu^{13} + \cdots + 32\!\cdots\!43 ) / 18\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( 3 \beta_{12} - 10 \beta_{11} - 10 \beta_{10} + 10 \beta_{8} + 16 \beta_{7} + 4 \beta_{6} + \cdots + 4689 ) / 12960 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 585 \beta_{12} - 634 \beta_{11} + 1598 \beta_{10} - 288 \beta_{9} - 974 \beta_{8} + \cdots + 837184197 ) / 12960 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 216 \beta_{13} + 72480 \beta_{12} - 446120 \beta_{11} - 378968 \beta_{10} + 81936 \beta_{9} + \cdots - 40154993944 ) / 6480 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 327744 \beta_{13} - 78980145 \beta_{12} - 19786594 \beta_{11} + 227077478 \beta_{10} + \cdots + 82353367271585 ) / 12960 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 69786720 \beta_{13} + 15045947325 \beta_{12} - 91500370726 \beta_{11} - 86956482262 \beta_{10} + \cdots - 14\!\cdots\!91 ) / 12960 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 19017444636 \beta_{13} - 2293128125778 \beta_{12} + 819937486652 \beta_{11} + 7200834346928 \beta_{10} + \cdots + 23\!\cdots\!82 ) / 3240 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 12911622528672 \beta_{13} + \cdots - 20\!\cdots\!61 ) / 12960 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 13\!\cdots\!68 \beta_{13} + \cdots + 11\!\cdots\!03 ) / 12960 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 12\!\cdots\!44 \beta_{13} + \cdots - 13\!\cdots\!52 ) / 6480 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 20\!\cdots\!52 \beta_{13} + \cdots + 13\!\cdots\!11 ) / 12960 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 45\!\cdots\!56 \beta_{13} + \cdots - 36\!\cdots\!33 ) / 12960 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 38\!\cdots\!84 \beta_{13} + \cdots + 20\!\cdots\!90 ) / 1620 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 78\!\cdots\!32 \beta_{13} + \cdots - 47\!\cdots\!79 ) / 12960 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 101.1 |
|
− | 22.6274i | −236.908 | − | 54.0703i | −512.000 | 0 | −1223.47 | + | 5360.62i | 5641.65 | 11585.2i | 53201.8 | + | 25619.4i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.2 | − | 22.6274i | −149.915 | + | 191.244i | −512.000 | 0 | 4327.37 | + | 3392.19i | −18361.5 | 11585.2i | −14099.9 | − | 57340.9i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.3 | − | 22.6274i | −121.467 | − | 210.463i | −512.000 | 0 | −4762.24 | + | 2748.49i | −2558.79 | 11585.2i | −29540.5 | + | 51128.7i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.4 | − | 22.6274i | 43.8665 | + | 239.008i | −512.000 | 0 | 5408.13 | − | 992.586i | 17330.1 | 11585.2i | −55200.5 | + | 20968.9i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.5 | − | 22.6274i | 116.023 | − | 213.513i | −512.000 | 0 | −4831.24 | − | 2625.29i | 29941.9 | 11585.2i | −32126.5 | − | 49544.7i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.6 | − | 22.6274i | 137.229 | − | 200.542i | −512.000 | 0 | −4537.76 | − | 3105.13i | −22976.9 | 11585.2i | −21385.5 | − | 55040.4i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.7 | − | 22.6274i | 222.172 | + | 98.4300i | −512.000 | 0 | 2227.22 | − | 5027.19i | −23249.5 | 11585.2i | 39672.1 | + | 43736.8i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.8 | 22.6274i | −236.908 | + | 54.0703i | −512.000 | 0 | −1223.47 | − | 5360.62i | 5641.65 | − | 11585.2i | 53201.8 | − | 25619.4i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.9 | 22.6274i | −149.915 | − | 191.244i | −512.000 | 0 | 4327.37 | − | 3392.19i | −18361.5 | − | 11585.2i | −14099.9 | + | 57340.9i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.10 | 22.6274i | −121.467 | + | 210.463i | −512.000 | 0 | −4762.24 | − | 2748.49i | −2558.79 | − | 11585.2i | −29540.5 | − | 51128.7i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.11 | 22.6274i | 43.8665 | − | 239.008i | −512.000 | 0 | 5408.13 | + | 992.586i | 17330.1 | − | 11585.2i | −55200.5 | − | 20968.9i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.12 | 22.6274i | 116.023 | + | 213.513i | −512.000 | 0 | −4831.24 | + | 2625.29i | 29941.9 | − | 11585.2i | −32126.5 | + | 49544.7i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.13 | 22.6274i | 137.229 | + | 200.542i | −512.000 | 0 | −4537.76 | + | 3105.13i | −22976.9 | − | 11585.2i | −21385.5 | + | 55040.4i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.14 | 22.6274i | 222.172 | − | 98.4300i | −512.000 | 0 | 2227.22 | + | 5027.19i | −23249.5 | − | 11585.2i | 39672.1 | − | 43736.8i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 150.11.d.d | yes | 14 |
| 3.b | odd | 2 | 1 | inner | 150.11.d.d | yes | 14 |
| 5.b | even | 2 | 1 | 150.11.d.c | ✓ | 14 | |
| 5.c | odd | 4 | 2 | 150.11.b.c | 28 | ||
| 15.d | odd | 2 | 1 | 150.11.d.c | ✓ | 14 | |
| 15.e | even | 4 | 2 | 150.11.b.c | 28 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 150.11.b.c | 28 | 5.c | odd | 4 | 2 | ||
| 150.11.b.c | 28 | 15.e | even | 4 | 2 | ||
| 150.11.d.c | ✓ | 14 | 5.b | even | 2 | 1 | |
| 150.11.d.c | ✓ | 14 | 15.d | odd | 2 | 1 | |
| 150.11.d.d | yes | 14 | 1.a | even | 1 | 1 | trivial |
| 150.11.d.d | yes | 14 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{7} + 14233 T_{7}^{6} - 1219133697 T_{7}^{5} - 18754014769561 T_{7}^{4} + \cdots - 73\!\cdots\!32 \)
acting on \(S_{11}^{\mathrm{new}}(150, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 512)^{7} \)
|
| $3$ |
\( T^{14} + \cdots + 25\!\cdots\!49 \)
|
| $5$ |
\( T^{14} \)
|
| $7$ |
\( (T^{7} + \cdots - 73\!\cdots\!32)^{2} \)
|
| $11$ |
\( T^{14} + \cdots + 23\!\cdots\!68 \)
|
| $13$ |
\( (T^{7} + \cdots - 23\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{14} + \cdots + 78\!\cdots\!12 \)
|
| $19$ |
\( (T^{7} + \cdots + 12\!\cdots\!63)^{2} \)
|
| $23$ |
\( T^{14} + \cdots + 32\!\cdots\!32 \)
|
| $29$ |
\( T^{14} + \cdots + 57\!\cdots\!00 \)
|
| $31$ |
\( (T^{7} + \cdots - 54\!\cdots\!92)^{2} \)
|
| $37$ |
\( (T^{7} + \cdots - 14\!\cdots\!28)^{2} \)
|
| $41$ |
\( T^{14} + \cdots + 35\!\cdots\!28 \)
|
| $43$ |
\( (T^{7} + \cdots - 51\!\cdots\!16)^{2} \)
|
| $47$ |
\( T^{14} + \cdots + 96\!\cdots\!00 \)
|
| $53$ |
\( T^{14} + \cdots + 33\!\cdots\!72 \)
|
| $59$ |
\( T^{14} + \cdots + 35\!\cdots\!28 \)
|
| $61$ |
\( (T^{7} + \cdots - 47\!\cdots\!64)^{2} \)
|
| $67$ |
\( (T^{7} + \cdots - 11\!\cdots\!53)^{2} \)
|
| $71$ |
\( T^{14} + \cdots + 72\!\cdots\!00 \)
|
| $73$ |
\( (T^{7} + \cdots + 42\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{7} + \cdots + 12\!\cdots\!32)^{2} \)
|
| $83$ |
\( T^{14} + \cdots + 24\!\cdots\!08 \)
|
| $89$ |
\( T^{14} + \cdots + 90\!\cdots\!88 \)
|
| $97$ |
\( (T^{7} + \cdots - 19\!\cdots\!96)^{2} \)
|
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