Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,2,Mod(49,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("950.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 950.j (of order \(6\), degree \(2\), not 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: | \(7.58578819202\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} + 8 x^{14} - 36 x^{13} + 67 x^{12} + 34 x^{11} - 24 x^{10} + 182 x^{9} - 495 x^{8} + \cdots + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{16} - 4 x^{15} + 8 x^{14} - 36 x^{13} + 67 x^{12} + 34 x^{11} - 24 x^{10} + 182 x^{9} - 495 x^{8} + \cdots + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 279227424185 \nu^{15} - 1253191152168 \nu^{14} + 2149480076638 \nu^{13} + \cdots - 50\!\cdots\!04 ) / 16\!\cdots\!40 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2554774305209 \nu^{15} + 4335535617510 \nu^{14} + 5455089919862 \nu^{13} + \cdots + 10377464251808 ) / 16\!\cdots\!40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 997381935613 \nu^{15} - 1649145399090 \nu^{14} - 2330828468434 \nu^{13} + \cdots - 4027830549856 ) / 206106192978480 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2070672794 \nu^{15} + 9113657058 \nu^{14} - 20042279554 \nu^{13} + 81839364227 \nu^{12} + \cdots - 90197230528 ) / 104622432984 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 27445864585 \nu^{15} - 114495167832 \nu^{14} + 240112435262 \nu^{13} - 1033250237896 \nu^{12} + \cdots + 1092234854864 ) / 1275212330880 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 8686137725825 \nu^{15} + 42313430612454 \nu^{14} - 101803992713434 \nu^{13} + \cdots - 450005598920032 ) / 329769908765568 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 453854154643 \nu^{15} - 2148242385810 \nu^{14} + 5058579031646 \nu^{13} + \cdots + 22444040275424 ) / 13973301218880 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 312593461575 \nu^{15} + 1305052923848 \nu^{14} - 2734307093218 \nu^{13} + \cdots + 4675736035664 ) / 9315534145920 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 10900402862375 \nu^{15} - 45452419955144 \nu^{14} + 95432288434874 \nu^{13} + \cdots + 16\!\cdots\!28 ) / 274808257304640 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 21911472450815 \nu^{15} + 91435548636808 \nu^{14} - 191728937133858 \nu^{13} + \cdots + 538715661376464 ) / 549616514609280 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 35876471476807 \nu^{15} + 142013953594170 \nu^{14} - 278222669153654 \nu^{13} + \cdots - 12\!\cdots\!76 ) / 824424771913920 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 15156149970385 \nu^{15} + 63147564290984 \nu^{14} - 132518533985934 \nu^{13} + \cdots - 18\!\cdots\!28 ) / 274808257304640 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 95383263803887 \nu^{15} - 388596494728170 \nu^{14} + 786381536962454 \nu^{13} + \cdots + 35\!\cdots\!56 ) / 16\!\cdots\!40 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 107197556804581 \nu^{15} + 449995772335710 \nu^{14} - 941949725565842 \nu^{13} + \cdots - 42\!\cdots\!48 ) / 16\!\cdots\!40 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 13452207709045 \nu^{15} + 56083850298582 \nu^{14} - 117674794077602 \nu^{13} + \cdots - 11\!\cdots\!64 ) / 206106192978480 \)
|
| \(\nu\) | \(=\) |
\( ( 3 \beta_{15} - 3 \beta_{14} - 2 \beta_{13} - \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots - 3 \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} + \beta_{7} + \beta_{3} - 4\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 21 \beta_{14} + 13 \beta_{13} + \beta_{12} - \beta_{11} - 13 \beta_{10} - 13 \beta_{9} - \beta_{8} + \cdots + 33 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{15} - 10\beta_{12} - 9\beta_{10} - \beta_{9} - 9\beta_{8} - 18\beta_{5} - 12\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 171 \beta_{15} + 11 \beta_{13} - 116 \beta_{12} + 127 \beta_{11} + 11 \beta_{10} + 116 \beta_{9} + \cdots - 297 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( 113\beta_{14} + 91\beta_{13} + 12\beta_{11} + 79\beta_{7} - 79\beta_{6} + 137\beta_{4} + 113\beta_{3} - 79\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1467 \beta_{15} + 1467 \beta_{14} + 998 \beta_{13} - 1147 \beta_{12} - 998 \beta_{11} + \cdots - 1467 \beta_1 ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1008\beta_{15} - 691\beta_{12} + 115\beta_{10} + 691\beta_{9} - 806\beta_{8} - 1031\beta_{5} - 1837 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 12765 \beta_{14} + 10129 \beta_{13} + 1445 \beta_{12} + 1445 \beta_{11} + 10129 \beta_{10} + \cdots - 22791 ) / 6 \)
|
| \(\nu^{10}\) | \(=\) |
\( 8863\beta_{14} + 6041\beta_{13} - 6041\beta_{11} - 1036\beta_{7} - 7077\beta_{6} + 8851\beta_{4} + 8851\beta_{2} \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 111477 \beta_{15} - 13019 \beta_{13} - 75818 \beta_{12} - 88837 \beta_{11} + 13019 \beta_{10} + \cdots - 199185 ) / 6 \)
|
| \(\nu^{12}\) | \(=\) |
\( 9149\beta_{12} + 61962\beta_{10} + 61962\beta_{9} - 9149\beta_{8} + 9149\beta_{5} + 77616\beta _1 - 138915 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 974403 \beta_{15} + 974403 \beta_{14} + 662612 \beta_{13} + 777487 \beta_{12} - 662612 \beta_{11} + \cdots + 974403 \beta_1 ) / 6 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 80260 \beta_{13} - 541995 \beta_{11} - 541995 \beta_{7} - 80260 \beta_{6} - 80260 \beta_{4} + \cdots + 1133396 \beta_{2} \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 8518923 \beta_{14} - 6799867 \beta_{13} + 1007237 \beta_{12} - 1007237 \beta_{11} + 6799867 \beta_{10} + \cdots - 15221391 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).
| \(n\) | \(77\) | \(401\) |
| \(\chi(n)\) | \(-1\) | \(-1 - \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
−0.866025 | − | 0.500000i | −2.62877 | − | 1.51772i | 0.500000 | + | 0.866025i | 0 | 1.51772 | + | 2.62877i | 4.93155i | − | 1.00000i | 3.10694 | + | 5.38138i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | −0.866025 | − | 0.500000i | −1.47519 | − | 0.851703i | 0.500000 | + | 0.866025i | 0 | 0.851703 | + | 1.47519i | − | 3.74324i | − | 1.00000i | −0.0492032 | − | 0.0852224i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | −0.866025 | − | 0.500000i | 0.410396 | + | 0.236942i | 0.500000 | + | 0.866025i | 0 | −0.236942 | − | 0.410396i | 2.19155i | − | 1.00000i | −1.38772 | − | 2.40360i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | −0.866025 | − | 0.500000i | 2.82754 | + | 1.63248i | 0.500000 | + | 0.866025i | 0 | −1.63248 | − | 2.82754i | 2.62013i | − | 1.00000i | 3.82998 | + | 6.63372i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | 0.866025 | + | 0.500000i | −2.82754 | − | 1.63248i | 0.500000 | + | 0.866025i | 0 | −1.63248 | − | 2.82754i | − | 2.62013i | 1.00000i | 3.82998 | + | 6.63372i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 0.866025 | + | 0.500000i | −0.410396 | − | 0.236942i | 0.500000 | + | 0.866025i | 0 | −0.236942 | − | 0.410396i | − | 2.19155i | 1.00000i | −1.38772 | − | 2.40360i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 0.866025 | + | 0.500000i | 1.47519 | + | 0.851703i | 0.500000 | + | 0.866025i | 0 | 0.851703 | + | 1.47519i | 3.74324i | 1.00000i | −0.0492032 | − | 0.0852224i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.8 | 0.866025 | + | 0.500000i | 2.62877 | + | 1.51772i | 0.500000 | + | 0.866025i | 0 | 1.51772 | + | 2.62877i | − | 4.93155i | 1.00000i | 3.10694 | + | 5.38138i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.1 | −0.866025 | + | 0.500000i | −2.62877 | + | 1.51772i | 0.500000 | − | 0.866025i | 0 | 1.51772 | − | 2.62877i | − | 4.93155i | 1.00000i | 3.10694 | − | 5.38138i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.2 | −0.866025 | + | 0.500000i | −1.47519 | + | 0.851703i | 0.500000 | − | 0.866025i | 0 | 0.851703 | − | 1.47519i | 3.74324i | 1.00000i | −0.0492032 | + | 0.0852224i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.3 | −0.866025 | + | 0.500000i | 0.410396 | − | 0.236942i | 0.500000 | − | 0.866025i | 0 | −0.236942 | + | 0.410396i | − | 2.19155i | 1.00000i | −1.38772 | + | 2.40360i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.4 | −0.866025 | + | 0.500000i | 2.82754 | − | 1.63248i | 0.500000 | − | 0.866025i | 0 | −1.63248 | + | 2.82754i | − | 2.62013i | 1.00000i | 3.82998 | − | 6.63372i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.5 | 0.866025 | − | 0.500000i | −2.82754 | + | 1.63248i | 0.500000 | − | 0.866025i | 0 | −1.63248 | + | 2.82754i | 2.62013i | − | 1.00000i | 3.82998 | − | 6.63372i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.6 | 0.866025 | − | 0.500000i | −0.410396 | + | 0.236942i | 0.500000 | − | 0.866025i | 0 | −0.236942 | + | 0.410396i | 2.19155i | − | 1.00000i | −1.38772 | + | 2.40360i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.7 | 0.866025 | − | 0.500000i | 1.47519 | − | 0.851703i | 0.500000 | − | 0.866025i | 0 | 0.851703 | − | 1.47519i | − | 3.74324i | − | 1.00000i | −0.0492032 | + | 0.0852224i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 349.8 | 0.866025 | − | 0.500000i | 2.62877 | − | 1.51772i | 0.500000 | − | 0.866025i | 0 | 1.51772 | − | 2.62877i | 4.93155i | − | 1.00000i | 3.10694 | − | 5.38138i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 19.c | even | 3 | 1 | inner |
| 95.i | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 950.2.j.i | 16 | |
| 5.b | even | 2 | 1 | inner | 950.2.j.i | 16 | |
| 5.c | odd | 4 | 1 | 950.2.e.l | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 950.2.e.m | yes | 8 | |
| 19.c | even | 3 | 1 | inner | 950.2.j.i | 16 | |
| 95.i | even | 6 | 1 | inner | 950.2.j.i | 16 | |
| 95.m | odd | 12 | 1 | 950.2.e.l | ✓ | 8 | |
| 95.m | odd | 12 | 1 | 950.2.e.m | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 950.2.e.l | ✓ | 8 | 5.c | odd | 4 | 1 | |
| 950.2.e.l | ✓ | 8 | 95.m | odd | 12 | 1 | |
| 950.2.e.m | yes | 8 | 5.c | odd | 4 | 1 | |
| 950.2.e.m | yes | 8 | 95.m | odd | 12 | 1 | |
| 950.2.j.i | 16 | 1.a | even | 1 | 1 | trivial | |
| 950.2.j.i | 16 | 5.b | even | 2 | 1 | inner | |
| 950.2.j.i | 16 | 19.c | even | 3 | 1 | inner | |
| 950.2.j.i | 16 | 95.i | even | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):
|
\( T_{3}^{16} - 23T_{3}^{14} + 368T_{3}^{12} - 3063T_{3}^{10} + 18497T_{3}^{8} - 48576T_{3}^{6} + 92096T_{3}^{4} - 20480T_{3}^{2} + 4096 \)
|
|
\( T_{7}^{8} + 50T_{7}^{6} + 821T_{7}^{4} + 5240T_{7}^{2} + 11236 \)
|
|
\( T_{11}^{4} - 5T_{11}^{3} - 20T_{11}^{2} + 123T_{11} - 117 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 1)^{4} \)
|
| $3$ |
\( T^{16} - 23 T^{14} + \cdots + 4096 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( (T^{8} + 50 T^{6} + \cdots + 11236)^{2} \)
|
| $11$ |
\( (T^{4} - 5 T^{3} + \cdots - 117)^{4} \)
|
| $13$ |
\( T^{16} + \cdots + 4162314256 \)
|
| $17$ |
\( T^{16} + \cdots + 2562890625 \)
|
| $19$ |
\( (T^{8} + 35 T^{6} + \cdots + 130321)^{2} \)
|
| $23$ |
\( (T^{8} - 81 T^{6} + \cdots + 1679616)^{2} \)
|
| $29$ |
\( (T^{8} + 17 T^{7} + \cdots + 202500)^{2} \)
|
| $31$ |
\( (T^{4} - 11 T^{3} + \cdots + 1118)^{4} \)
|
| $37$ |
\( (T^{8} + 236 T^{6} + \cdots + 9585216)^{2} \)
|
| $41$ |
\( (T^{8} - 7 T^{7} + \cdots + 5948721)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 875213056 \)
|
| $47$ |
\( T^{16} + \cdots + 2998219536 \)
|
| $53$ |
\( T^{16} + \cdots + 2998219536 \)
|
| $59$ |
\( (T^{8} + 14 T^{7} + \cdots + 363609)^{2} \)
|
| $61$ |
\( (T^{8} + 9 T^{7} + \cdots + 256)^{2} \)
|
| $67$ |
\( T^{16} - 50 T^{14} + \cdots + 256 \)
|
| $71$ |
\( (T^{8} - 14 T^{7} + \cdots + 2862864)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 45137758519296 \)
|
| $79$ |
\( (T^{8} - 17 T^{7} + \cdots + 7022500)^{2} \)
|
| $83$ |
\( (T^{8} + 267 T^{6} + \cdots + 3504384)^{2} \)
|
| $89$ |
\( (T^{8} + 14 T^{7} + \cdots + 8555625)^{2} \)
|
| $97$ |
\( T^{16} - 269 T^{14} + \cdots + 6561 \)
|
| show more | |
| show less |