Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [156,2,Mod(155,156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(156, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("156.155");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 156 = 2^{2} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 156.h (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: | \(1.24566627153\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 43x^{12} + 517x^{8} + 1804x^{4} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{10} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 43x^{12} + 517x^{8} + 1804x^{4} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -383\nu^{14} - 16513\nu^{10} - 200087\nu^{6} - 728592\nu^{2} ) / 34268 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 85\nu^{14} + 3651\nu^{10} + 43277\nu^{6} + 142248\nu^{2} ) / 5272 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2069 \nu^{15} - 3460 \nu^{14} + 62 \nu^{13} + 89831 \nu^{11} - 150788 \nu^{10} - 190 \nu^{9} + \cdots - 363776 \nu ) / 274144 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 667 \nu^{15} - 185 \nu^{13} + 28355 \nu^{11} - 7171 \nu^{9} + 332573 \nu^{7} - 64885 \nu^{5} + \cdots - 105076 \nu ) / 68536 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2199 \nu^{15} + 797 \nu^{14} + 229 \nu^{13} - 94407 \nu^{11} + 32931 \nu^{10} + \cdots + 373092 \nu ) / 137072 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2199 \nu^{15} + 229 \nu^{13} - 220 \nu^{12} + 94407 \nu^{11} + 9247 \nu^{9} - 10380 \nu^{8} + \cdots - 174968 ) / 137072 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2199 \nu^{15} + 229 \nu^{13} + 220 \nu^{12} + 94407 \nu^{11} + 9247 \nu^{9} + 10380 \nu^{8} + \cdots + 174968 ) / 137072 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 4873 \nu^{15} + 644 \nu^{13} + 1604 \nu^{12} - 212783 \nu^{11} + 17924 \nu^{9} + 56988 \nu^{8} + \cdots - 161088 ) / 274144 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2199 \nu^{15} + 797 \nu^{14} - 229 \nu^{13} + 94407 \nu^{11} + 32931 \nu^{10} - 9247 \nu^{9} + \cdots - 373092 \nu ) / 137072 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 4873 \nu^{15} + 644 \nu^{13} - 1604 \nu^{12} - 212783 \nu^{11} + 17924 \nu^{9} - 56988 \nu^{8} + \cdots + 161088 ) / 274144 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 6467 \nu^{15} + 1198 \nu^{14} + 396 \nu^{13} - 278645 \nu^{11} + 47178 \nu^{10} + \cdots + 1109960 \nu ) / 274144 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 3533 \nu^{15} + 599 \nu^{13} - 520 \nu^{12} + 151117 \nu^{11} + 23589 \nu^{9} - 18304 \nu^{8} + \cdots - 108264 ) / 137072 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 3601 \nu^{15} + 62 \nu^{13} + 155883 \nu^{11} - 190 \nu^{9} + 1898325 \nu^{7} - 65314 \nu^{5} + \cdots - 637920 \nu ) / 137072 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 3533 \nu^{15} - 599 \nu^{13} - 520 \nu^{12} - 151117 \nu^{11} - 23589 \nu^{9} - 18304 \nu^{8} + \cdots - 108264 ) / 137072 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 5602 \nu^{15} - 537 \nu^{13} - 520 \nu^{12} - 240948 \nu^{11} - 23779 \nu^{9} - 18304 \nu^{8} + \cdots - 108264 ) / 137072 \)
|
| \(\nu\) | \(=\) |
\( ( 3 \beta_{15} - 3 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + \cdots - 2 \beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{11} - \beta_{9} - 3\beta_{5} + 2\beta_{3} - \beta_{2} - 5\beta_1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6 \beta_{15} - 7 \beta_{14} + 4 \beta_{13} + \beta_{12} + 4 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \cdots + 4 \beta_1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -15\beta_{14} - 15\beta_{12} + 10\beta_{10} - 10\beta_{8} + \beta_{7} - \beta_{6} - 38 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 49 \beta_{15} + 67 \beta_{14} + 38 \beta_{13} - 18 \beta_{12} + 30 \beta_{11} + 19 \beta_{10} + \cdots + 30 \beta_1 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -24\beta_{11} + 14\beta_{9} + 38\beta_{5} - 24\beta_{3} - 4\beta_{2} + 39\beta_1 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 213 \beta_{15} + 325 \beta_{14} - 190 \beta_{13} - 112 \beta_{12} - 118 \beta_{11} + 95 \beta_{10} + \cdots - 118 \beta_1 ) / 8 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 389\beta_{14} + 389\beta_{12} - 230\beta_{10} + 230\beta_{8} + 81\beta_{7} - 81\beta_{6} + 678 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 486 \beta_{15} - 797 \beta_{14} - 484 \beta_{13} + 311 \beta_{12} - 244 \beta_{11} - 242 \beta_{10} + \cdots - 244 \beta_1 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 1110\beta_{11} - 883\beta_{9} - 1993\beta_{5} + 1110\beta_{3} + 537\beta_{2} - 1541\beta_1 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 4591 \beta_{15} - 7885 \beta_{14} + 4954 \beta_{13} + 3294 \beta_{12} + 2114 \beta_{11} + \cdots + 2114 \beta_1 ) / 8 \)
|
| \(\nu^{12}\) | \(=\) |
\( -2547\beta_{14} - 2547\beta_{12} + 1352\beta_{10} - 1352\beta_{8} - 780\beta_{7} + 780\beta_{6} - 3621 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 22195 \beta_{15} + 39251 \beta_{14} + 25330 \beta_{13} - 17056 \beta_{12} + 9530 \beta_{11} + \cdots + 9530 \beta_1 ) / 8 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( -26586\beta_{11} + 25345\beta_{9} + 51931\beta_{5} - 26586\beta_{3} - 17071\beta_{2} + 34845\beta_1 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 54478 \beta_{15} + 98135 \beta_{14} - 64596 \beta_{13} - 43657 \beta_{12} - 22180 \beta_{11} + \cdots - 22180 \beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 155.1 |
|
−1.17915 | − | 0.780776i | −1.26870 | − | 1.17915i | 0.780776 | + | 1.84130i | −0.662153 | 0.575339 | + | 2.38096i | −3.83206 | 0.516994 | − | 2.78078i | 0.219224 | + | 2.99198i | 0.780776 | + | 0.516994i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.2 | −1.17915 | − | 0.780776i | 1.26870 | − | 1.17915i | 0.780776 | + | 1.84130i | −0.662153 | −2.41664 | + | 0.399813i | 3.83206 | 0.516994 | − | 2.78078i | 0.219224 | − | 2.99198i | 0.780776 | + | 0.516994i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.3 | −1.17915 | + | 0.780776i | −1.26870 | + | 1.17915i | 0.780776 | − | 1.84130i | −0.662153 | 0.575339 | − | 2.38096i | −3.83206 | 0.516994 | + | 2.78078i | 0.219224 | − | 2.99198i | 0.780776 | − | 0.516994i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.4 | −1.17915 | + | 0.780776i | 1.26870 | + | 1.17915i | 0.780776 | − | 1.84130i | −0.662153 | −2.41664 | − | 0.399813i | 3.83206 | 0.516994 | + | 2.78078i | 0.219224 | + | 2.99198i | 0.780776 | − | 0.516994i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.5 | −0.599676 | − | 1.28078i | −1.62493 | + | 0.599676i | −1.28078 | + | 1.53610i | 2.13578 | 1.74248 | + | 1.72156i | 1.52162 | 2.73546 | + | 0.719224i | 2.28078 | − | 1.94886i | −1.28078 | − | 2.73546i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.6 | −0.599676 | − | 1.28078i | 1.62493 | + | 0.599676i | −1.28078 | + | 1.53610i | 2.13578 | −0.206379 | − | 2.44078i | −1.52162 | 2.73546 | + | 0.719224i | 2.28078 | + | 1.94886i | −1.28078 | − | 2.73546i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.7 | −0.599676 | + | 1.28078i | −1.62493 | − | 0.599676i | −1.28078 | − | 1.53610i | 2.13578 | 1.74248 | − | 1.72156i | 1.52162 | 2.73546 | − | 0.719224i | 2.28078 | + | 1.94886i | −1.28078 | + | 2.73546i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.8 | −0.599676 | + | 1.28078i | 1.62493 | − | 0.599676i | −1.28078 | − | 1.53610i | 2.13578 | −0.206379 | + | 2.44078i | −1.52162 | 2.73546 | − | 0.719224i | 2.28078 | − | 1.94886i | −1.28078 | + | 2.73546i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.9 | 0.599676 | − | 1.28078i | −1.62493 | − | 0.599676i | −1.28078 | − | 1.53610i | −2.13578 | −1.74248 | + | 1.72156i | −1.52162 | −2.73546 | + | 0.719224i | 2.28078 | + | 1.94886i | −1.28078 | + | 2.73546i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.10 | 0.599676 | − | 1.28078i | 1.62493 | − | 0.599676i | −1.28078 | − | 1.53610i | −2.13578 | 0.206379 | − | 2.44078i | 1.52162 | −2.73546 | + | 0.719224i | 2.28078 | − | 1.94886i | −1.28078 | + | 2.73546i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.11 | 0.599676 | + | 1.28078i | −1.62493 | + | 0.599676i | −1.28078 | + | 1.53610i | −2.13578 | −1.74248 | − | 1.72156i | −1.52162 | −2.73546 | − | 0.719224i | 2.28078 | − | 1.94886i | −1.28078 | − | 2.73546i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.12 | 0.599676 | + | 1.28078i | 1.62493 | + | 0.599676i | −1.28078 | + | 1.53610i | −2.13578 | 0.206379 | + | 2.44078i | 1.52162 | −2.73546 | − | 0.719224i | 2.28078 | + | 1.94886i | −1.28078 | − | 2.73546i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.13 | 1.17915 | − | 0.780776i | −1.26870 | + | 1.17915i | 0.780776 | − | 1.84130i | 0.662153 | −0.575339 | + | 2.38096i | 3.83206 | −0.516994 | − | 2.78078i | 0.219224 | − | 2.99198i | 0.780776 | − | 0.516994i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.14 | 1.17915 | − | 0.780776i | 1.26870 | + | 1.17915i | 0.780776 | − | 1.84130i | 0.662153 | 2.41664 | + | 0.399813i | −3.83206 | −0.516994 | − | 2.78078i | 0.219224 | + | 2.99198i | 0.780776 | − | 0.516994i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.15 | 1.17915 | + | 0.780776i | −1.26870 | − | 1.17915i | 0.780776 | + | 1.84130i | 0.662153 | −0.575339 | − | 2.38096i | 3.83206 | −0.516994 | + | 2.78078i | 0.219224 | + | 2.99198i | 0.780776 | + | 0.516994i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.16 | 1.17915 | + | 0.780776i | 1.26870 | − | 1.17915i | 0.780776 | + | 1.84130i | 0.662153 | 2.41664 | − | 0.399813i | −3.83206 | −0.516994 | + | 2.78078i | 0.219224 | − | 2.99198i | 0.780776 | + | 0.516994i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 39.d | odd | 2 | 1 | inner |
| 52.b | odd | 2 | 1 | inner |
| 156.h | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 156.2.h.b | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 156.2.h.b | ✓ | 16 |
| 4.b | odd | 2 | 1 | inner | 156.2.h.b | ✓ | 16 |
| 12.b | even | 2 | 1 | inner | 156.2.h.b | ✓ | 16 |
| 13.b | even | 2 | 1 | inner | 156.2.h.b | ✓ | 16 |
| 39.d | odd | 2 | 1 | inner | 156.2.h.b | ✓ | 16 |
| 52.b | odd | 2 | 1 | inner | 156.2.h.b | ✓ | 16 |
| 156.h | even | 2 | 1 | inner | 156.2.h.b | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 156.2.h.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 156.2.h.b | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 156.2.h.b | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
| 156.2.h.b | ✓ | 16 | 12.b | even | 2 | 1 | inner |
| 156.2.h.b | ✓ | 16 | 13.b | even | 2 | 1 | inner |
| 156.2.h.b | ✓ | 16 | 39.d | odd | 2 | 1 | inner |
| 156.2.h.b | ✓ | 16 | 52.b | odd | 2 | 1 | inner |
| 156.2.h.b | ✓ | 16 | 156.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 5T_{5}^{2} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} + T^{6} + 4 T^{4} + \cdots + 16)^{2} \)
|
| $3$ |
\( (T^{8} - 5 T^{6} + 20 T^{4} + \cdots + 81)^{2} \)
|
| $5$ |
\( (T^{4} - 5 T^{2} + 2)^{4} \)
|
| $7$ |
\( (T^{4} - 17 T^{2} + 34)^{4} \)
|
| $11$ |
\( (T^{2} + 4)^{8} \)
|
| $13$ |
\( (T^{4} + 2 T^{3} + \cdots + 169)^{4} \)
|
| $17$ |
\( (T^{4} + 51 T^{2} + 136)^{4} \)
|
| $19$ |
\( (T^{4} - 34 T^{2} + 136)^{4} \)
|
| $23$ |
\( (T^{4} - 68 T^{2} + 1088)^{4} \)
|
| $29$ |
\( (T^{4} + 68 T^{2} + 544)^{4} \)
|
| $31$ |
\( (T^{4} - 34 T^{2} + 136)^{4} \)
|
| $37$ |
\( (T^{4} + 85 T^{2} + 272)^{4} \)
|
| $41$ |
\( (T^{4} - 74 T^{2} + 1352)^{4} \)
|
| $43$ |
\( (T^{4} + 23 T^{2} + 128)^{4} \)
|
| $47$ |
\( (T^{4} + 77 T^{2} + 1444)^{4} \)
|
| $53$ |
\( T^{16} \)
|
| $59$ |
\( (T^{4} + 132 T^{2} + 1024)^{4} \)
|
| $61$ |
\( (T^{2} + 2 T - 16)^{8} \)
|
| $67$ |
\( (T^{4} - 102 T^{2} + 2176)^{4} \)
|
| $71$ |
\( (T^{4} + 221 T^{2} + 1156)^{4} \)
|
| $73$ |
\( (T^{4} + 272 T^{2} + 17408)^{4} \)
|
| $79$ |
\( (T^{4} + 148 T^{2} + 5408)^{4} \)
|
| $83$ |
\( (T^{2} + 36)^{8} \)
|
| $89$ |
\( (T^{4} - 266 T^{2} + 17672)^{4} \)
|
| $97$ |
\( (T^{4} + 340 T^{2} + 17408)^{4} \)
|
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