Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [185,2,Mod(184,185)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(185, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("185.184");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 185 = 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 185.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: | \(1.47723243739\) |
| 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: |
\( x^{16} + 6x^{14} + 36x^{12} + 282x^{10} + 1334x^{8} + 7050x^{6} + 22500x^{4} + 93750x^{2} + 390625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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
\( x^{16} + 6x^{14} + 36x^{12} + 282x^{10} + 1334x^{8} + 7050x^{6} + 22500x^{4} + 93750x^{2} + 390625 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3 \nu^{14} - 18 \nu^{12} - 733 \nu^{10} - 4596 \nu^{8} - 10877 \nu^{6} - 103650 \nu^{4} + \cdots - 875000 ) / 875000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{14} - 75\nu^{12} - 275\nu^{10} + 687\nu^{8} - 1731\nu^{6} - 36753\nu^{4} - 77025\nu^{2} + 508125 ) / 560000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 53 \nu^{14} - 668 \nu^{12} - 3383 \nu^{10} - 23796 \nu^{8} - 162527 \nu^{6} - 539300 \nu^{4} + \cdots - 6562500 ) / 3500000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 323 \nu^{15} + 8513 \nu^{13} + 27953 \nu^{11} + 173411 \nu^{9} + 1280657 \nu^{7} + \cdots + 16765625 \nu ) / 70000000 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 207 \nu^{14} - 733 \nu^{12} + 12477 \nu^{10} + 41649 \nu^{8} + 279813 \nu^{6} + 1192825 \nu^{4} + \cdots + 35421875 ) / 7000000 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 207 \nu^{15} - 733 \nu^{13} + 12477 \nu^{11} + 41649 \nu^{9} + 279813 \nu^{7} + \cdots + 42421875 \nu ) / 35000000 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 443 \nu^{15} - 8633 \nu^{13} - 53673 \nu^{11} - 210651 \nu^{9} - 796537 \nu^{7} + \cdots - 28390625 \nu ) / 70000000 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 623 \nu^{14} + 3013 \nu^{12} + 22453 \nu^{10} + 97711 \nu^{8} + 424757 \nu^{6} + 2049375 \nu^{4} + \cdots + 20828125 ) / 14000000 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 293 \nu^{15} + 767 \nu^{13} - 1023 \nu^{11} - 9851 \nu^{9} + 134313 \nu^{7} + \cdots - 10640625 \nu ) / 35000000 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 599 \nu^{15} - 2131 \nu^{13} - 13411 \nu^{11} + 5943 \nu^{9} - 322259 \nu^{7} + \cdots - 11796875 \nu ) / 70000000 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 419 \nu^{14} - 861 \nu^{12} + 9209 \nu^{10} + 36033 \nu^{8} + 140321 \nu^{6} + 1364825 \nu^{4} + \cdots + 30171875 ) / 7000000 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 21 \nu^{14} + 19 \nu^{12} + 639 \nu^{10} + 3345 \nu^{8} + 18615 \nu^{6} + 97737 \nu^{4} + \cdots + 2011875 ) / 280000 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( -\nu^{15} - 6\nu^{13} - 36\nu^{11} - 282\nu^{9} - 1334\nu^{7} - 7050\nu^{5} - 22500\nu^{3} - 93750\nu ) / 78125 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 17 \nu^{15} + 37 \nu^{13} + 347 \nu^{11} + 79 \nu^{9} + 5723 \nu^{7} + 34015 \nu^{5} + \cdots + 578125 \nu ) / 1000000 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{13} - \beta_{12} + \beta_{4} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{14} - 3\beta_{11} - 3\beta_{10} + \beta_{8} - \beta_{7} - \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{13} + 5\beta_{12} - 6\beta_{9} - 5\beta_{6} - 2\beta_{4} - 6\beta_{3} - 3\beta_{2} - 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{15} - 10\beta_{14} - \beta_{11} + 13\beta_{10} - \beta_{8} - 5\beta_{7} - 12\beta_{5} - 2\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{13} - 4\beta_{12} - 12\beta_{9} + 17\beta_{6} - 27\beta_{4} + 4\beta_{3} + 33\beta_{2} - 65 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12\beta_{15} + 25\beta_{14} + 28\beta_{11} + 64\beta_{10} + 48\beta_{8} + 68\beta_{7} + 68\beta_{5} - 32\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -60\beta_{13} + 4\beta_{12} + 44\beta_{9} - 44\beta_{6} - 48\beta_{4} + 172\beta_{3} - 212\beta_{2} + 113 \)
|
| \(\nu^{9}\) | \(=\) |
\( -396\beta_{15} - 128\beta_{14} + 284\beta_{11} - 176\beta_{10} - 144\beta_{8} + 44\beta_{7} + 4\beta_{5} - 43\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( -327\beta_{13} - 281\beta_{12} + 660\beta_{9} + 812\beta_{6} + 565\beta_{4} - 284\beta_{3} - 20\beta_{2} - 233 \)
|
| \(\nu^{11}\) | \(=\) |
\( 316 \beta_{15} + 1147 \beta_{14} - 803 \beta_{11} - 287 \beta_{10} - 1291 \beta_{8} + 1407 \beta_{7} + \cdots - 1117 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( -314\beta_{13} - 75\beta_{12} + 2094\beta_{9} - 2037\beta_{6} - 930\beta_{4} - 1362\beta_{3} + 573\beta_{2} + 9837 \)
|
| \(\nu^{13}\) | \(=\) |
\( 3640 \beta_{15} - 1434 \beta_{14} - 193 \beta_{11} + 3193 \beta_{10} - 3197 \beta_{8} + \cdots + 10742 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 10935 \beta_{13} + 2024 \beta_{12} + 9576 \beta_{9} - 7655 \beta_{6} + 10769 \beta_{4} + 6856 \beta_{3} + \cdots - 30265 \)
|
| \(\nu^{15}\) | \(=\) |
\( 34248 \beta_{15} - 15067 \beta_{14} - 12824 \beta_{11} - 68720 \beta_{10} + 26784 \beta_{8} + \cdots - 26576 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).
| \(n\) | \(76\) | \(112\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 184.1 |
|
−2.13578 | − | 2.26867i | 2.56155 | 1.79003 | + | 1.34007i | 4.84539i | 3.87589i | −1.19935 | −2.14688 | −3.82312 | − | 2.86208i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.2 | −2.13578 | − | 1.68912i | 2.56155 | −0.122469 | − | 2.23271i | 3.60758i | − | 0.988692i | −1.19935 | 0.146883 | 0.261567 | + | 4.76858i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.3 | −2.13578 | 1.68912i | 2.56155 | −0.122469 | + | 2.23271i | − | 3.60758i | 0.988692i | −1.19935 | 0.146883 | 0.261567 | − | 4.76858i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.4 | −2.13578 | 2.26867i | 2.56155 | 1.79003 | − | 1.34007i | − | 4.84539i | − | 3.87589i | −1.19935 | −2.14688 | −3.82312 | + | 2.86208i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.5 | −0.662153 | − | 2.77476i | −1.56155 | −1.99266 | + | 1.01455i | 1.83732i | 0.382154i | 2.35829 | −4.69928 | 1.31945 | − | 0.671786i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.6 | −0.662153 | − | 0.548381i | −1.56155 | 1.14459 | + | 1.92092i | 0.363112i | − | 3.98170i | 2.35829 | 2.69928 | −0.757894 | − | 1.27194i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.7 | −0.662153 | 0.548381i | −1.56155 | 1.14459 | − | 1.92092i | − | 0.363112i | 3.98170i | 2.35829 | 2.69928 | −0.757894 | + | 1.27194i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.8 | −0.662153 | 2.77476i | −1.56155 | −1.99266 | − | 1.01455i | − | 1.83732i | − | 0.382154i | 2.35829 | −4.69928 | 1.31945 | + | 0.671786i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.9 | 0.662153 | − | 2.77476i | −1.56155 | 1.99266 | − | 1.01455i | − | 1.83732i | 0.382154i | −2.35829 | −4.69928 | 1.31945 | − | 0.671786i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.10 | 0.662153 | − | 0.548381i | −1.56155 | −1.14459 | − | 1.92092i | − | 0.363112i | − | 3.98170i | −2.35829 | 2.69928 | −0.757894 | − | 1.27194i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.11 | 0.662153 | 0.548381i | −1.56155 | −1.14459 | + | 1.92092i | 0.363112i | 3.98170i | −2.35829 | 2.69928 | −0.757894 | + | 1.27194i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.12 | 0.662153 | 2.77476i | −1.56155 | 1.99266 | + | 1.01455i | 1.83732i | − | 0.382154i | −2.35829 | −4.69928 | 1.31945 | + | 0.671786i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.13 | 2.13578 | − | 2.26867i | 2.56155 | −1.79003 | − | 1.34007i | − | 4.84539i | 3.87589i | 1.19935 | −2.14688 | −3.82312 | − | 2.86208i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.14 | 2.13578 | − | 1.68912i | 2.56155 | 0.122469 | + | 2.23271i | − | 3.60758i | − | 0.988692i | 1.19935 | 0.146883 | 0.261567 | + | 4.76858i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.15 | 2.13578 | 1.68912i | 2.56155 | 0.122469 | − | 2.23271i | 3.60758i | 0.988692i | 1.19935 | 0.146883 | 0.261567 | − | 4.76858i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 184.16 | 2.13578 | 2.26867i | 2.56155 | −1.79003 | + | 1.34007i | 4.84539i | − | 3.87589i | 1.19935 | −2.14688 | −3.82312 | + | 2.86208i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 37.b | even | 2 | 1 | inner |
| 185.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 185.2.d.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 1665.2.g.b | 16 | ||
| 5.b | even | 2 | 1 | inner | 185.2.d.a | ✓ | 16 |
| 5.c | odd | 4 | 2 | 925.2.c.f | 16 | ||
| 15.d | odd | 2 | 1 | 1665.2.g.b | 16 | ||
| 37.b | even | 2 | 1 | inner | 185.2.d.a | ✓ | 16 |
| 111.d | odd | 2 | 1 | 1665.2.g.b | 16 | ||
| 185.d | even | 2 | 1 | inner | 185.2.d.a | ✓ | 16 |
| 185.h | odd | 4 | 2 | 925.2.c.f | 16 | ||
| 555.b | odd | 2 | 1 | 1665.2.g.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 185.2.d.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 185.2.d.a | ✓ | 16 | 5.b | even | 2 | 1 | inner |
| 185.2.d.a | ✓ | 16 | 37.b | even | 2 | 1 | inner |
| 185.2.d.a | ✓ | 16 | 185.d | even | 2 | 1 | inner |
| 925.2.c.f | 16 | 5.c | odd | 4 | 2 | ||
| 925.2.c.f | 16 | 185.h | odd | 4 | 2 | ||
| 1665.2.g.b | 16 | 3.b | odd | 2 | 1 | ||
| 1665.2.g.b | 16 | 15.d | odd | 2 | 1 | ||
| 1665.2.g.b | 16 | 111.d | odd | 2 | 1 | ||
| 1665.2.g.b | 16 | 555.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(185, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 5 T^{2} + 2)^{4} \)
|
| $3$ |
\( (T^{8} + 16 T^{6} + \cdots + 34)^{2} \)
|
| $5$ |
\( T^{16} + 6 T^{14} + \cdots + 390625 \)
|
| $7$ |
\( (T^{8} + 32 T^{6} + \cdots + 34)^{2} \)
|
| $11$ |
\( (T^{4} - 21 T^{2} + 72)^{4} \)
|
| $13$ |
\( (T^{2} - 6)^{8} \)
|
| $17$ |
\( (T^{8} - 70 T^{6} + \cdots + 71824)^{2} \)
|
| $19$ |
\( (T^{8} + 54 T^{6} + \cdots + 11016)^{2} \)
|
| $23$ |
\( (T^{8} - 34 T^{6} + \cdots + 64)^{2} \)
|
| $29$ |
\( (T^{8} + 82 T^{6} + \cdots + 2176)^{2} \)
|
| $31$ |
\( (T^{8} + 210 T^{6} + \cdots + 3183624)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 3512479453921 \)
|
| $41$ |
\( (T^{4} - 69 T^{2} + 1152)^{4} \)
|
| $43$ |
\( (T^{8} - 210 T^{6} + \cdots + 876096)^{2} \)
|
| $47$ |
\( (T^{8} + 264 T^{6} + \cdots + 2754)^{2} \)
|
| $53$ |
\( (T^{8} + 168 T^{6} + \cdots + 176256)^{2} \)
|
| $59$ |
\( (T^{8} + 184 T^{6} + \cdots + 557056)^{2} \)
|
| $61$ |
\( (T^{8} + 354 T^{6} + \cdots + 2820096)^{2} \)
|
| $67$ |
\( (T^{8} + 356 T^{6} + \cdots + 27052576)^{2} \)
|
| $71$ |
\( (T^{4} + 6 T^{3} - 123 T^{2} + \cdots + 72)^{4} \)
|
| $73$ |
\( (T^{8} + 344 T^{6} + \cdots + 17430304)^{2} \)
|
| $79$ |
\( (T^{8} + 216 T^{6} + \cdots + 11016)^{2} \)
|
| $83$ |
\( (T^{8} + 144 T^{6} + \cdots + 223074)^{2} \)
|
| $89$ |
\( (T^{8} + 352 T^{6} + \cdots + 20898304)^{2} \)
|
| $97$ |
\( (T^{8} - 306 T^{6} + \cdots + 331776)^{2} \)
|
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