Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [693,2,Mod(64,693)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(693, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("693.64");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 693 = 3^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 693.m (of order \(5\), degree \(4\), 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: | \(5.53363286007\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 7x^{14} + 25x^{12} + 52x^{10} + 309x^{8} + 218x^{6} + 60x^{4} - 2x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{16} + 7x^{14} + 25x^{12} + 52x^{10} + 309x^{8} + 218x^{6} + 60x^{4} - 2x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7128 \nu^{14} - 1725314 \nu^{12} - 11742192 \nu^{10} - 40515400 \nu^{8} - 77407495 \nu^{6} + \cdots - 39693802 ) / 52533933 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 786069 \nu^{14} + 5321221 \nu^{12} + 18430659 \nu^{10} + 35945132 \nu^{8} + 230662226 \nu^{6} + \cdots - 97249690 ) / 52533933 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3404768 \nu^{14} + 24230081 \nu^{12} + 88225360 \nu^{10} + 188972384 \nu^{8} + 1079046400 \nu^{6} + \cdots + 13037152 ) / 52533933 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3404768 \nu^{15} + 24230081 \nu^{13} + 88225360 \nu^{11} + 188972384 \nu^{9} + \cdots + 13037152 \nu ) / 52533933 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4406280 \nu^{14} + 31436396 \nu^{12} + 113650011 \nu^{10} + 239657509 \nu^{8} + 1378232347 \nu^{6} + \cdots - 35869442 ) / 52533933 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 6809536 \nu^{14} + 48460162 \nu^{12} + 176450720 \nu^{10} + 377944768 \nu^{8} + 2158092800 \nu^{6} + \cdots + 26074304 ) / 52533933 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 7811048 \nu^{15} - 55666477 \nu^{13} - 201875371 \nu^{11} - 428629893 \nu^{9} + \cdots - 29701643 \nu ) / 52533933 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 7818176 \nu^{15} - 53941163 \nu^{13} - 190133179 \nu^{11} - 388114493 \nu^{9} + \cdots + 9992159 \nu ) / 52533933 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 12840131 \nu^{15} + 89873789 \nu^{13} + 322728589 \nu^{11} + 679429004 \nu^{9} + \cdots + 214772362 \nu ) / 52533933 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 13861142 \nu^{14} + 95961934 \nu^{12} + 339380302 \nu^{10} + 696618939 \nu^{8} + 4235451080 \nu^{6} + \cdots - 19991446 ) / 52533933 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 14439322 \nu^{14} + 102905573 \nu^{12} + 373395635 \nu^{10} + 794341566 \nu^{8} + 4549850740 \nu^{6} + \cdots + 54989878 ) / 52533933 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 247627 \nu^{15} + 1780845 \nu^{13} + 6510149 \nu^{11} + 13972387 \nu^{9} + 78658056 \nu^{7} + \cdots - 837981 \nu ) / 564881 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 25877283 \nu^{15} + 177729085 \nu^{13} + 624427308 \nu^{11} + 1269135548 \nu^{9} + \cdots - 206423878 \nu ) / 52533933 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 26656650 \nu^{15} - 190008446 \nu^{13} - 688921017 \nu^{11} - 1462628968 \nu^{9} + \cdots - 101356012 \nu ) / 52533933 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} + 2\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{15} - \beta_{13} - \beta_{10} + \beta_{8} + 5\beta_{5} \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{12} + \beta_{11} + 5\beta_{7} + 7\beta_{6} - \beta_{4} - 4\beta_{3} - \beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( 5\beta_{15} + \beta_{14} + 5\beta_{10} + 8\beta_{9} - 24\beta_{8} - 24\beta_{5} - 16\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 24\beta_{12} - 24\beta_{11} - 8\beta_{7} + 10\beta_{4} + 8\beta_{3} + 37\beta_{2} + 10 \)
|
| \(\nu^{7}\) | \(=\) |
\( -24\beta_{14} + 24\beta_{13} - 8\beta_{10} - 117\beta_{9} + 117\beta_{8} + 50\beta_{5} + 50\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -50\beta_{12} + 117\beta_{11} - 110\beta_{6} - 110\beta_{4} - 50\beta_{3} - 178\beta_{2} - 178 \)
|
| \(\nu^{9}\) | \(=\) |
\( -117\beta_{15} + 117\beta_{14} - 167\beta_{13} + 579\beta_{9} - 285\beta_{8} - 285\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( -285\beta_{11} + 285\beta_{7} + 403\beta_{6} + 294\beta_{3} + 403\beta_{2} + 874 \)
|
| \(\nu^{11}\) | \(=\) |
\( 864\beta_{15} - 285\beta_{14} + 864\beta_{13} + 285\beta_{10} - 1552\beta_{9} - 1552\beta_{5} + 1344\beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 1552\beta_{12} - 2896\beta_{7} - 2240\beta_{6} + 2109\beta_{4} - 2240 \)
|
| \(\nu^{13}\) | \(=\) |
\( -4448\beta_{15} - 2896\beta_{13} - 2896\beta_{10} + 8240\beta_{8} + 14589\beta_{5} \)
|
| \(\nu^{14}\) | \(=\) |
\( - 14589 \beta_{12} + 8240 \beta_{11} + 14589 \beta_{7} + 9802 \beta_{6} - 12032 \beta_{4} + \cdots - 12032 \beta_{2} \)
|
| \(\nu^{15}\) | \(=\) |
\( 14589 \beta_{15} + 8240 \beta_{14} + 14589 \beta_{10} + 43101 \beta_{9} - 73841 \beta_{8} + \cdots - 30740 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).
| \(n\) | \(155\) | \(199\) | \(442\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 64.1 |
|
−0.698216 | + | 2.14889i | 0 | −2.51217 | − | 1.82520i | −0.476925 | − | 1.46782i | 0 | 0.809017 | + | 0.587785i | 2.02029 | − | 1.46782i | 0 | 3.48718 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.2 | −0.221291 | + | 0.681062i | 0 | 1.20316 | + | 0.874145i | 0.476925 | + | 1.46782i | 0 | 0.809017 | + | 0.587785i | −2.02029 | + | 1.46782i | 0 | −1.10522 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.3 | 0.221291 | − | 0.681062i | 0 | 1.20316 | + | 0.874145i | −0.476925 | − | 1.46782i | 0 | 0.809017 | + | 0.587785i | 2.02029 | − | 1.46782i | 0 | −1.10522 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.4 | 0.698216 | − | 2.14889i | 0 | −2.51217 | − | 1.82520i | 0.476925 | + | 1.46782i | 0 | 0.809017 | + | 0.587785i | −2.02029 | + | 1.46782i | 0 | 3.48718 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 190.1 | −1.46183 | + | 1.06208i | 0 | 0.390899 | − | 1.20306i | −1.73855 | − | 1.26313i | 0 | −0.309017 | + | 0.951057i | −0.410415 | − | 1.26313i | 0 | 3.88301 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 190.2 | −0.276713 | + | 0.201044i | 0 | −0.581882 | + | 1.79085i | −1.73855 | − | 1.26313i | 0 | −0.309017 | + | 0.951057i | −0.410415 | − | 1.26313i | 0 | 0.735023 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 190.3 | 0.276713 | − | 0.201044i | 0 | −0.581882 | + | 1.79085i | 1.73855 | + | 1.26313i | 0 | −0.309017 | + | 0.951057i | 0.410415 | + | 1.26313i | 0 | 0.735023 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 190.4 | 1.46183 | − | 1.06208i | 0 | 0.390899 | − | 1.20306i | 1.73855 | + | 1.26313i | 0 | −0.309017 | + | 0.951057i | 0.410415 | + | 1.26313i | 0 | 3.88301 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.1 | −0.698216 | − | 2.14889i | 0 | −2.51217 | + | 1.82520i | −0.476925 | + | 1.46782i | 0 | 0.809017 | − | 0.587785i | 2.02029 | + | 1.46782i | 0 | 3.48718 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.2 | −0.221291 | − | 0.681062i | 0 | 1.20316 | − | 0.874145i | 0.476925 | − | 1.46782i | 0 | 0.809017 | − | 0.587785i | −2.02029 | − | 1.46782i | 0 | −1.10522 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.3 | 0.221291 | + | 0.681062i | 0 | 1.20316 | − | 0.874145i | −0.476925 | + | 1.46782i | 0 | 0.809017 | − | 0.587785i | 2.02029 | + | 1.46782i | 0 | −1.10522 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.4 | 0.698216 | + | 2.14889i | 0 | −2.51217 | + | 1.82520i | 0.476925 | − | 1.46782i | 0 | 0.809017 | − | 0.587785i | −2.02029 | − | 1.46782i | 0 | 3.48718 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.1 | −1.46183 | − | 1.06208i | 0 | 0.390899 | + | 1.20306i | −1.73855 | + | 1.26313i | 0 | −0.309017 | − | 0.951057i | −0.410415 | + | 1.26313i | 0 | 3.88301 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.2 | −0.276713 | − | 0.201044i | 0 | −0.581882 | − | 1.79085i | −1.73855 | + | 1.26313i | 0 | −0.309017 | − | 0.951057i | −0.410415 | + | 1.26313i | 0 | 0.735023 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.3 | 0.276713 | + | 0.201044i | 0 | −0.581882 | − | 1.79085i | 1.73855 | − | 1.26313i | 0 | −0.309017 | − | 0.951057i | 0.410415 | − | 1.26313i | 0 | 0.735023 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.4 | 1.46183 | + | 1.06208i | 0 | 0.390899 | + | 1.20306i | 1.73855 | − | 1.26313i | 0 | −0.309017 | − | 0.951057i | 0.410415 | − | 1.26313i | 0 | 3.88301 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 11.c | even | 5 | 1 | inner |
| 33.h | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 693.2.m.h | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 693.2.m.h | ✓ | 16 |
| 11.c | even | 5 | 1 | inner | 693.2.m.h | ✓ | 16 |
| 11.c | even | 5 | 1 | 7623.2.a.cu | 8 | ||
| 11.d | odd | 10 | 1 | 7623.2.a.cv | 8 | ||
| 33.f | even | 10 | 1 | 7623.2.a.cv | 8 | ||
| 33.h | odd | 10 | 1 | inner | 693.2.m.h | ✓ | 16 |
| 33.h | odd | 10 | 1 | 7623.2.a.cu | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 693.2.m.h | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 693.2.m.h | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 693.2.m.h | ✓ | 16 | 11.c | even | 5 | 1 | inner |
| 693.2.m.h | ✓ | 16 | 33.h | odd | 10 | 1 | inner |
| 7623.2.a.cu | 8 | 11.c | even | 5 | 1 | ||
| 7623.2.a.cu | 8 | 33.h | odd | 10 | 1 | ||
| 7623.2.a.cv | 8 | 11.d | odd | 10 | 1 | ||
| 7623.2.a.cv | 8 | 33.f | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 7T_{2}^{14} + 25T_{2}^{12} + 52T_{2}^{10} + 309T_{2}^{8} + 218T_{2}^{6} + 60T_{2}^{4} - 2T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 7 T^{14} + \cdots + 1 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( (T^{8} + T^{6} + 16 T^{4} + \cdots + 121)^{2} \)
|
| $7$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $11$ |
\( T^{16} + \cdots + 214358881 \)
|
| $13$ |
\( (T^{8} - 5 T^{7} + 13 T^{6} + \cdots + 16)^{2} \)
|
| $17$ |
\( T^{16} + 67 T^{14} + \cdots + 1 \)
|
| $19$ |
\( (T^{8} - 9 T^{7} + \cdots + 55696)^{2} \)
|
| $23$ |
\( (T^{8} - 61 T^{6} + \cdots + 16)^{2} \)
|
| $29$ |
\( T^{16} + \cdots + 11019960576 \)
|
| $31$ |
\( (T^{8} - 4 T^{7} + \cdots + 78961)^{2} \)
|
| $37$ |
\( (T^{8} - 12 T^{7} + \cdots + 256)^{2} \)
|
| $41$ |
\( T^{16} + 9 T^{14} + \cdots + 14641 \)
|
| $43$ |
\( (T^{4} - 8 T^{3} + \cdots - 596)^{4} \)
|
| $47$ |
\( T^{16} + \cdots + 126849222205696 \)
|
| $53$ |
\( T^{16} + \cdots + 100000000 \)
|
| $59$ |
\( T^{16} + 383 T^{14} + \cdots + 65536 \)
|
| $61$ |
\( (T^{8} + 17 T^{7} + \cdots + 1263376)^{2} \)
|
| $67$ |
\( (T^{4} + 18 T^{3} + \cdots - 3904)^{4} \)
|
| $71$ |
\( T^{16} + \cdots + 4347792138496 \)
|
| $73$ |
\( (T^{8} + 21 T^{7} + \cdots + 2062096)^{2} \)
|
| $79$ |
\( (T^{8} + 33 T^{7} + \cdots + 234256)^{2} \)
|
| $83$ |
\( T^{16} - 52 T^{14} + \cdots + 65536 \)
|
| $89$ |
\( (T^{8} - 524 T^{6} + \cdots + 71554681)^{2} \)
|
| $97$ |
\( (T^{8} - 27 T^{7} + \cdots + 59969536)^{2} \)
|
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