Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1950,2,Mod(751,1950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1950, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1950.751");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1950.bc (of order \(6\), degree \(2\), 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: | \(15.5708283941\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 6 x^{11} + 87 x^{10} - 380 x^{9} + 2556 x^{8} - 8010 x^{7} + 29687 x^{6} - 62556 x^{5} + \cdots + 14089 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 6 x^{11} + 87 x^{10} - 380 x^{9} + 2556 x^{8} - 8010 x^{7} + 29687 x^{6} - 62556 x^{5} + \cdots + 14089 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 176 \nu^{10} + 880 \nu^{9} - 13279 \nu^{8} + 47836 \nu^{7} - 335904 \nu^{6} + 843982 \nu^{5} + \cdots - 3607897 ) / 737586 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1070 \nu^{11} - 9796 \nu^{10} + 107747 \nu^{9} - 581520 \nu^{8} + 3242004 \nu^{7} + \cdots - 30150460 ) / 33157458 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1070 \nu^{11} + 1974 \nu^{10} - 68637 \nu^{9} + 123933 \nu^{8} - 1646316 \nu^{7} + \cdots + 8385745 ) / 33157458 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2140 \nu^{11} - 11770 \nu^{10} + 176384 \nu^{9} - 705453 \nu^{8} + 4888320 \nu^{7} + \cdots - 38536205 ) / 33157458 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 196298 \nu^{11} - 1423807 \nu^{10} + 17298796 \nu^{9} - 87970594 \nu^{8} + 505040292 \nu^{7} + \cdots - 8896864806 ) / 2884698846 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 199579 \nu^{11} + 43670 \nu^{10} - 10853189 \nu^{9} - 20210311 \nu^{8} - 114973215 \nu^{7} + \cdots - 34891536081 ) / 2884698846 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 292669 \nu^{11} - 3011773 \nu^{10} + 30806433 \nu^{9} - 211570737 \nu^{8} + 1058307265 \nu^{7} + \cdots - 39899132821 ) / 2884698846 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 370049 \nu^{11} - 2416592 \nu^{10} + 32929734 \nu^{9} - 154475736 \nu^{8} + 981881669 \nu^{7} + \cdots - 13717956122 ) / 2884698846 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 370049 \nu^{11} + 1653947 \nu^{10} - 29116509 \nu^{9} + 94203315 \nu^{8} - 763671335 \nu^{7} + \cdots - 3683244445 ) / 2884698846 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 1663873 \nu^{11} + 8187240 \nu^{10} - 136318210 \nu^{9} + 487832491 \nu^{8} + \cdots + 14213570375 ) / 2884698846 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 2230220 \nu^{11} - 13193117 \nu^{10} + 192374130 \nu^{9} - 827318553 \nu^{8} + \cdots - 52508450723 ) / 2884698846 \)
|
| \(\nu\) | \(=\) |
\( -\beta_{4} - \beta_{3} + \beta_{2} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} + \beta_{10} + 4 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots - 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2 \beta_{11} + \beta_{10} + 12 \beta_{9} - \beta_{8} - 2 \beta_{7} - 4 \beta_{6} - 6 \beta_{5} + \cdots - 15 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 16 \beta_{11} - 18 \beta_{10} - 89 \beta_{9} - 95 \beta_{8} + 45 \beta_{7} + 41 \beta_{6} + \cdots + 242 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 44 \beta_{11} - 46 \beta_{10} - 321 \beta_{9} - 158 \beta_{8} + 92 \beta_{7} + 133 \beta_{6} + \cdots + 592 \)
|
| \(\nu^{6}\) | \(=\) |
\( 329 \beta_{11} + 328 \beta_{10} + 2130 \beta_{9} + 2213 \beta_{8} - 971 \beta_{7} - 838 \beta_{6} + \cdots - 4964 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1001 \beta_{11} + 1617 \beta_{10} + 8482 \beta_{9} + 8101 \beta_{8} - 3297 \beta_{7} - 3829 \beta_{6} + \cdots - 19054 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 7402 \beta_{11} - 4938 \beta_{10} - 51821 \beta_{9} - 43913 \beta_{8} + 19371 \beta_{7} + \cdots + 96085 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 23870 \beta_{11} - 47752 \beta_{10} - 233457 \beta_{9} - 283244 \beta_{8} + 100886 \beta_{7} + \cdots + 551212 \)
|
| \(\nu^{10}\) | \(=\) |
\( 175288 \beta_{11} + 37401 \beta_{10} + 1241888 \beta_{9} + 697514 \beta_{8} - 348537 \beta_{7} + \cdots - 1683590 \)
|
| \(\nu^{11}\) | \(=\) |
\( 602019 \beta_{11} + 1254110 \beta_{10} + 6570934 \beta_{9} + 8427566 \beta_{8} - 2812501 \beta_{7} + \cdots - 14885497 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).
| \(n\) | \(301\) | \(1301\) | \(1327\) |
| \(\chi(n)\) | \(1 - \beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 751.1 |
|
−0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | −0.866025 | − | 0.500000i | −4.44290 | − | 2.56511i | 1.00000i | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 751.2 | −0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | −0.866025 | − | 0.500000i | 0.242731 | + | 0.140141i | 1.00000i | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 751.3 | −0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | −0.866025 | − | 0.500000i | 3.96812 | + | 2.29099i | 1.00000i | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 751.4 | 0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | 0.866025 | + | 0.500000i | −2.20583 | − | 1.27354i | − | 1.00000i | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 751.5 | 0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | 0.866025 | + | 0.500000i | 2.32854 | + | 1.34438i | − | 1.00000i | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 751.6 | 0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | 0.866025 | + | 0.500000i | 3.10934 | + | 1.79518i | − | 1.00000i | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 901.1 | −0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | −0.866025 | + | 0.500000i | −4.44290 | + | 2.56511i | − | 1.00000i | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 901.2 | −0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | −0.866025 | + | 0.500000i | 0.242731 | − | 0.140141i | − | 1.00000i | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 901.3 | −0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | −0.866025 | + | 0.500000i | 3.96812 | − | 2.29099i | − | 1.00000i | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 901.4 | 0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | 0.866025 | − | 0.500000i | −2.20583 | + | 1.27354i | 1.00000i | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 901.5 | 0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | 0.866025 | − | 0.500000i | 2.32854 | − | 1.34438i | 1.00000i | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 901.6 | 0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | 0.866025 | − | 0.500000i | 3.10934 | − | 1.79518i | 1.00000i | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.e | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1950.2.bc.k | yes | 12 |
| 5.b | even | 2 | 1 | 1950.2.bc.h | ✓ | 12 | |
| 5.c | odd | 4 | 1 | 1950.2.y.m | 12 | ||
| 5.c | odd | 4 | 1 | 1950.2.y.n | 12 | ||
| 13.e | even | 6 | 1 | inner | 1950.2.bc.k | yes | 12 |
| 65.l | even | 6 | 1 | 1950.2.bc.h | ✓ | 12 | |
| 65.r | odd | 12 | 1 | 1950.2.y.m | 12 | ||
| 65.r | odd | 12 | 1 | 1950.2.y.n | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1950.2.y.m | 12 | 5.c | odd | 4 | 1 | ||
| 1950.2.y.m | 12 | 65.r | odd | 12 | 1 | ||
| 1950.2.y.n | 12 | 5.c | odd | 4 | 1 | ||
| 1950.2.y.n | 12 | 65.r | odd | 12 | 1 | ||
| 1950.2.bc.h | ✓ | 12 | 5.b | even | 2 | 1 | |
| 1950.2.bc.h | ✓ | 12 | 65.l | even | 6 | 1 | |
| 1950.2.bc.k | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 1950.2.bc.k | yes | 12 | 13.e | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{12} - 6 T_{7}^{11} - 19 T_{7}^{10} + 186 T_{7}^{9} + 421 T_{7}^{8} - 5244 T_{7}^{7} + \cdots + 26244 \)
acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 1)^{3} \)
|
| $3$ |
\( (T^{2} - T + 1)^{6} \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} - 6 T^{11} + \cdots + 26244 \)
|
| $11$ |
\( T^{12} + 12 T^{11} + \cdots + 11664 \)
|
| $13$ |
\( T^{12} + 10 T^{10} + \cdots + 4826809 \)
|
| $17$ |
\( T^{12} + 50 T^{10} + \cdots + 876096 \)
|
| $19$ |
\( T^{12} + 6 T^{11} + \cdots + 73513476 \)
|
| $23$ |
\( T^{12} + 4 T^{11} + \cdots + 46656 \)
|
| $29$ |
\( T^{12} + 92 T^{10} + \cdots + 56070144 \)
|
| $31$ |
\( T^{12} + \cdots + 248629824 \)
|
| $37$ |
\( T^{12} - 12 T^{11} + \cdots + 56070144 \)
|
| $41$ |
\( T^{12} - 50 T^{10} + \cdots + 876096 \)
|
| $43$ |
\( T^{12} - 10 T^{11} + \cdots + 45077796 \)
|
| $47$ |
\( T^{12} + 200 T^{10} + \cdots + 2985984 \)
|
| $53$ |
\( (T^{6} - 8 T^{5} + \cdots + 15768)^{2} \)
|
| $59$ |
\( T^{12} - 104 T^{10} + \cdots + 746496 \)
|
| $61$ |
\( T^{12} - 24 T^{11} + \cdots + 82955664 \)
|
| $67$ |
\( T^{12} + \cdots + 753831936 \)
|
| $71$ |
\( T^{12} + \cdots + 438439973904 \)
|
| $73$ |
\( T^{12} + 458 T^{10} + \cdots + 2653641 \)
|
| $79$ |
\( (T^{6} - 26 T^{5} + \cdots + 320086)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 900388436544 \)
|
| $89$ |
\( T^{12} - 24 T^{11} + \cdots + 77158656 \)
|
| $97$ |
\( T^{12} + 12 T^{11} + \cdots + 21233664 \)
|
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