Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [252,4,Mod(55,252)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(252, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("252.55");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 252.b (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: | \(14.8684813214\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{11} - 2x^{10} - 6x^{9} + 56x^{7} - 448x^{6} + 448x^{5} - 3072x^{3} - 8192x^{2} - 32768x + 262144 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{13} \) |
| Twist minimal: | no (minimal twist has level 84) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - x^{11} - 2x^{10} - 6x^{9} + 56x^{7} - 448x^{6} + 448x^{5} - 3072x^{3} - 8192x^{2} - 32768x + 262144 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{11} + \nu^{10} + 2 \nu^{9} + 6 \nu^{8} - 56 \nu^{6} + 448 \nu^{5} - 448 \nu^{4} + \cdots + 32768 ) / 32768 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{11} - 7 \nu^{10} + 10 \nu^{9} + 22 \nu^{8} + 48 \nu^{7} - 56 \nu^{6} + 3136 \nu^{4} + \cdots + 98304 ) / 32768 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3 \nu^{11} - 5 \nu^{10} - 50 \nu^{9} + 98 \nu^{8} + 176 \nu^{7} + 216 \nu^{6} + 896 \nu^{5} + \cdots + 294912 ) / 32768 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11 \nu^{11} + 69 \nu^{10} + 26 \nu^{9} - 610 \nu^{8} + 1568 \nu^{7} + 360 \nu^{6} + 3136 \nu^{5} + \cdots - 1310720 ) / 98304 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 29 \nu^{11} + 91 \nu^{10} + 142 \nu^{9} + 34 \nu^{8} - 80 \nu^{7} + 2264 \nu^{6} - 12928 \nu^{5} + \cdots - 3440640 ) / 98304 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 31 \nu^{11} + 41 \nu^{10} + 122 \nu^{9} + 182 \nu^{8} - 688 \nu^{7} + 3784 \nu^{6} - 6272 \nu^{5} + \cdots - 3244032 ) / 98304 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 45 \nu^{11} + 91 \nu^{10} + 158 \nu^{9} + 98 \nu^{8} + 464 \nu^{7} + 5336 \nu^{6} - 10496 \nu^{5} + \cdots - 4685824 ) / 98304 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 65 \nu^{11} + 111 \nu^{10} + 206 \nu^{9} + 26 \nu^{8} + 1760 \nu^{7} + 1080 \nu^{6} - 12608 \nu^{5} + \cdots - 5046272 ) / 98304 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 37 \nu^{11} + 79 \nu^{10} + 66 \nu^{9} + 186 \nu^{8} + 456 \nu^{7} + 1304 \nu^{6} - 9312 \nu^{5} + \cdots - 3031040 ) / 49152 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 59 \nu^{11} + 101 \nu^{10} + 138 \nu^{9} + 318 \nu^{8} + 384 \nu^{7} + 4648 \nu^{6} - 10560 \nu^{5} + \cdots - 5128192 ) / 49152 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{2} + 2 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + 3 \beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + \cdots + 10 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{11} + \beta_{10} - 2 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + 3 \beta_{6} + 4 \beta_{5} + \cdots + 10 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 9 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + 9 \beta_{7} - 33 \beta_{6} + 4 \beta_{5} + \cdots - 46 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 11 \beta_{11} - 23 \beta_{10} - 22 \beta_{9} + 89 \beta_{8} + 21 \beta_{7} - 5 \beta_{6} + 12 \beta_{5} + \cdots + 898 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 5 \beta_{11} - 15 \beta_{10} + 34 \beta_{9} + 113 \beta_{8} - 179 \beta_{7} + 3 \beta_{6} + 44 \beta_{5} + \cdots - 334 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 99 \beta_{11} + 425 \beta_{10} - 110 \beta_{9} + 9 \beta_{8} + 197 \beta_{7} - 197 \beta_{6} + \cdots + 1010 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 211 \beta_{11} - 695 \beta_{10} + 834 \beta_{9} + 265 \beta_{8} + 581 \beta_{7} + 507 \beta_{6} + \cdots + 16946 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 13 \beta_{11} + 1129 \beta_{10} - 1086 \beta_{9} - 1559 \beta_{8} + 293 \beta_{7} + 2875 \beta_{6} + \cdots + 52306 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 125 \beta_{11} + 4857 \beta_{10} + 2946 \beta_{9} - 1095 \beta_{8} + 6325 \beta_{7} - 16213 \beta_{6} + \cdots + 154098 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
−2.82801 | − | 0.0488466i | 0 | 7.99523 | + | 0.276277i | 16.6517i | 0 | 15.0420 | + | 10.8045i | −22.5971 | − | 1.17185i | 0 | 0.813380 | − | 47.0912i | ||||||||||||||||||||||||||||||||||||||||||||
| 55.2 | −2.82801 | + | 0.0488466i | 0 | 7.99523 | − | 0.276277i | − | 16.6517i | 0 | 15.0420 | − | 10.8045i | −22.5971 | + | 1.17185i | 0 | 0.813380 | + | 47.0912i | ||||||||||||||||||||||||||||||||||||||||||||
| 55.3 | −2.16644 | − | 1.81839i | 0 | 1.38692 | + | 7.87886i | − | 4.47531i | 0 | −14.9825 | + | 10.8869i | 11.3222 | − | 19.5910i | 0 | −8.13786 | + | 9.69549i | ||||||||||||||||||||||||||||||||||||||||||||
| 55.4 | −2.16644 | + | 1.81839i | 0 | 1.38692 | − | 7.87886i | 4.47531i | 0 | −14.9825 | − | 10.8869i | 11.3222 | + | 19.5910i | 0 | −8.13786 | − | 9.69549i | |||||||||||||||||||||||||||||||||||||||||||||
| 55.5 | −0.965027 | − | 2.65871i | 0 | −6.13744 | + | 5.13145i | 19.4608i | 0 | 1.95109 | − | 18.4172i | 19.5658 | + | 11.3657i | 0 | 51.7405 | − | 18.7802i | |||||||||||||||||||||||||||||||||||||||||||||
| 55.6 | −0.965027 | + | 2.65871i | 0 | −6.13744 | − | 5.13145i | − | 19.4608i | 0 | 1.95109 | + | 18.4172i | 19.5658 | − | 11.3657i | 0 | 51.7405 | + | 18.7802i | ||||||||||||||||||||||||||||||||||||||||||||
| 55.7 | 0.951271 | − | 2.66366i | 0 | −6.19017 | − | 5.06772i | − | 10.8462i | 0 | −15.1344 | − | 10.6748i | −19.3872 | + | 11.6677i | 0 | −28.8905 | − | 10.3176i | ||||||||||||||||||||||||||||||||||||||||||||
| 55.8 | 0.951271 | + | 2.66366i | 0 | −6.19017 | + | 5.06772i | 10.8462i | 0 | −15.1344 | + | 10.6748i | −19.3872 | − | 11.6677i | 0 | −28.8905 | + | 10.3176i | |||||||||||||||||||||||||||||||||||||||||||||
| 55.9 | 1.72458 | − | 2.24184i | 0 | −2.05167 | − | 7.73244i | 6.58775i | 0 | 15.1925 | + | 10.5918i | −20.8731 | − | 8.73568i | 0 | 14.7687 | + | 11.3611i | |||||||||||||||||||||||||||||||||||||||||||||
| 55.10 | 1.72458 | + | 2.24184i | 0 | −2.05167 | + | 7.73244i | − | 6.58775i | 0 | 15.1925 | − | 10.5918i | −20.8731 | + | 8.73568i | 0 | 14.7687 | − | 11.3611i | ||||||||||||||||||||||||||||||||||||||||||||
| 55.11 | 2.78362 | − | 0.501431i | 0 | 7.49713 | − | 2.79159i | − | 4.57514i | 0 | 2.93118 | − | 18.2868i | 19.4694 | − | 11.5300i | 0 | −2.29411 | − | 12.7355i | ||||||||||||||||||||||||||||||||||||||||||||
| 55.12 | 2.78362 | + | 0.501431i | 0 | 7.49713 | + | 2.79159i | 4.57514i | 0 | 2.93118 | + | 18.2868i | 19.4694 | + | 11.5300i | 0 | −2.29411 | + | 12.7355i | |||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 28.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 252.4.b.f | 12 | |
| 3.b | odd | 2 | 1 | 84.4.b.a | ✓ | 12 | |
| 4.b | odd | 2 | 1 | 252.4.b.e | 12 | ||
| 7.b | odd | 2 | 1 | 252.4.b.e | 12 | ||
| 12.b | even | 2 | 1 | 84.4.b.b | yes | 12 | |
| 21.c | even | 2 | 1 | 84.4.b.b | yes | 12 | |
| 24.f | even | 2 | 1 | 1344.4.b.g | 12 | ||
| 24.h | odd | 2 | 1 | 1344.4.b.h | 12 | ||
| 28.d | even | 2 | 1 | inner | 252.4.b.f | 12 | |
| 84.h | odd | 2 | 1 | 84.4.b.a | ✓ | 12 | |
| 168.e | odd | 2 | 1 | 1344.4.b.h | 12 | ||
| 168.i | even | 2 | 1 | 1344.4.b.g | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.4.b.a | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 84.4.b.a | ✓ | 12 | 84.h | odd | 2 | 1 | |
| 84.4.b.b | yes | 12 | 12.b | even | 2 | 1 | |
| 84.4.b.b | yes | 12 | 21.c | even | 2 | 1 | |
| 252.4.b.e | 12 | 4.b | odd | 2 | 1 | ||
| 252.4.b.e | 12 | 7.b | odd | 2 | 1 | ||
| 252.4.b.f | 12 | 1.a | even | 1 | 1 | trivial | |
| 252.4.b.f | 12 | 28.d | even | 2 | 1 | inner | |
| 1344.4.b.g | 12 | 24.f | even | 2 | 1 | ||
| 1344.4.b.g | 12 | 168.i | even | 2 | 1 | ||
| 1344.4.b.h | 12 | 24.h | odd | 2 | 1 | ||
| 1344.4.b.h | 12 | 168.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(252, [\chi])\):
|
\( T_{5}^{12} + 858T_{5}^{10} + 249644T_{5}^{8} + 29440120T_{5}^{6} + 1456436096T_{5}^{4} + 30453516288T_{5}^{2} + 224760987648 \)
|
|
\( T_{11}^{12} + 10414 T_{11}^{10} + 39015876 T_{11}^{8} + 64285126792 T_{11}^{6} + 45887475687456 T_{11}^{4} + \cdots + 58\!\cdots\!00 \)
|
|
\( T_{19}^{6} + 42T_{19}^{5} - 27084T_{19}^{4} - 1835880T_{19}^{3} + 99824480T_{19}^{2} + 8618904192T_{19} + 111463534080 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + T^{11} + \cdots + 262144 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots + 224760987648 \)
|
| $7$ |
\( T^{12} + \cdots + 16\!\cdots\!49 \)
|
| $11$ |
\( T^{12} + \cdots + 58\!\cdots\!00 \)
|
| $13$ |
\( T^{12} + \cdots + 62\!\cdots\!08 \)
|
| $17$ |
\( T^{12} + \cdots + 59\!\cdots\!12 \)
|
| $19$ |
\( (T^{6} + 42 T^{5} + \cdots + 111463534080)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 17\!\cdots\!12 \)
|
| $29$ |
\( (T^{6} + \cdots - 3698977199040)^{2} \)
|
| $31$ |
\( (T^{6} + \cdots + 34547086393344)^{2} \)
|
| $37$ |
\( (T^{6} + 122 T^{5} + \cdots + 11485380096)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 18\!\cdots\!00 \)
|
| $43$ |
\( T^{12} + \cdots + 47\!\cdots\!12 \)
|
| $47$ |
\( (T^{6} + \cdots + 118618190512128)^{2} \)
|
| $53$ |
\( (T^{6} + \cdots - 24896048217408)^{2} \)
|
| $59$ |
\( (T^{6} + \cdots - 19\!\cdots\!20)^{2} \)
|
| $61$ |
\( T^{12} + \cdots + 19\!\cdots\!00 \)
|
| $67$ |
\( T^{12} + \cdots + 64\!\cdots\!68 \)
|
| $71$ |
\( T^{12} + \cdots + 62\!\cdots\!00 \)
|
| $73$ |
\( T^{12} + \cdots + 22\!\cdots\!32 \)
|
| $79$ |
\( T^{12} + \cdots + 96\!\cdots\!88 \)
|
| $83$ |
\( (T^{6} + \cdots - 17\!\cdots\!56)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 18\!\cdots\!72 \)
|
| $97$ |
\( T^{12} + \cdots + 29\!\cdots\!88 \)
|
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