Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [560,3,Mod(241,560)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(560, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("560.241");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 560 = 2^{4} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 560.bx (of order \(6\), degree \(2\), not 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: | \(15.2588948042\) |
| 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: |
\( x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} + \cdots + 441 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| 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
\( x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} + \cdots + 441 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 115287976 \nu^{11} + 155566808 \nu^{10} + 1678212789 \nu^{9} + 3282242448 \nu^{8} + \cdots - 423574271316 ) / 1182297257421 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 960485876 \nu^{11} + 1805683776 \nu^{10} - 18404798452 \nu^{9} + 23294419987 \nu^{8} + \cdots + 1461932499888 ) / 1182297257421 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 6740525471 \nu^{11} + 28434054878 \nu^{10} - 139462837212 \nu^{9} + 440457052261 \nu^{8} + \cdots + 26654089921641 ) / 7093783544526 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 12593073845 \nu^{11} + 52875316152 \nu^{10} - 270584479006 \nu^{9} + \cdots + 41846762790189 ) / 7093783544526 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 20099706409 \nu^{11} - 38300982339 \nu^{10} + 346926560957 \nu^{9} - 453614466560 \nu^{8} + \cdots + 20528318534472 ) / 7093783544526 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 22979355133 \nu^{11} - 77959111894 \nu^{10} + 482194495722 \nu^{9} - 1167796047734 \nu^{8} + \cdots - 71882680944312 ) / 7093783544526 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 9058064565 \nu^{11} - 1839528853 \nu^{10} - 143550656091 \nu^{9} - 113877990722 \nu^{8} + \cdots - 3111541612074 ) / 2364594514842 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 27689168462 \nu^{11} + 31316075951 \nu^{10} - 479491314879 \nu^{9} + 276064172287 \nu^{8} + \cdots - 5553545565645 ) / 7093783544526 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 42084124100 \nu^{11} - 50430282123 \nu^{10} + 738914826223 \nu^{9} - 485302351243 \nu^{8} + \cdots + 9745511328693 ) / 7093783544526 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 65452485346 \nu^{11} + 87736926802 \nu^{10} - 1160252919867 \nu^{9} + \cdots - 14009629724496 ) / 7093783544526 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{10} + \beta_{6} + \beta_{5} - 6\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{8} + \beta_{4} + 9\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{11} + 28\beta_{10} + 12\beta_{8} - 2\beta_{7} - 12\beta_{6} - 2\beta_{5} + 57\beta_{3} - 2\beta _1 - 59 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{11} + 2 \beta_{10} - 16 \beta_{9} - 4 \beta_{7} - 14 \beta_{6} + 10 \beta_{5} - 32 \beta_{4} + \cdots + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 34 \beta_{11} - 177 \beta_{10} + 4 \beta_{9} - 143 \beta_{8} - 34 \beta_{7} - 177 \beta_{5} + \cdots + 692 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 84 \beta_{11} - 116 \beta_{10} + 422 \beta_{9} - 189 \beta_{8} + 42 \beta_{7} + 189 \beta_{6} + \cdots + 180 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 464 \beta_{11} - 2176 \beta_{10} + 100 \beta_{9} + 928 \beta_{7} + 1732 \beta_{6} + 2620 \beta_{5} + \cdots - 464 \)
|
| \(\nu^{9}\) | \(=\) |
\( 664 \beta_{11} + 1116 \beta_{10} - 2640 \beta_{9} + 2548 \beta_{8} + 664 \beta_{7} + 1116 \beta_{5} + \cdots - 3932 \)
|
| \(\nu^{10}\) | \(=\) |
\( 11888 \beta_{11} + 53074 \beta_{10} - 3560 \beta_{9} + 21165 \beta_{8} - 5944 \beta_{7} - 21165 \beta_{6} + \cdots - 91894 \)
|
| \(\nu^{11}\) | \(=\) |
\( 9504 \beta_{11} + 18380 \beta_{10} - 32481 \beta_{9} - 19008 \beta_{7} - 34205 \beta_{6} - 2555 \beta_{5} + \cdots + 9504 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(337\) | \(351\) | \(421\) |
| \(\chi(n)\) | \(\beta_{3}\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 241.1 |
|
0 | −4.45439 | − | 2.57174i | 0 | −1.93649 | + | 1.11803i | 0 | 1.42520 | + | 6.85338i | 0 | 8.72772 | + | 15.1168i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 241.2 | 0 | −0.731043 | − | 0.422068i | 0 | −1.93649 | + | 1.11803i | 0 | 6.88972 | − | 1.23763i | 0 | −4.14372 | − | 7.17713i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 241.3 | 0 | −0.507487 | − | 0.292998i | 0 | 1.93649 | − | 1.11803i | 0 | −1.91172 | − | 6.73389i | 0 | −4.32830 | − | 7.49684i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 241.4 | 0 | 1.83681 | + | 1.06048i | 0 | 1.93649 | − | 1.11803i | 0 | −4.91879 | + | 4.98051i | 0 | −2.25076 | − | 3.89842i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 241.5 | 0 | 2.10717 | + | 1.21658i | 0 | 1.93649 | − | 1.11803i | 0 | 5.39402 | + | 4.46146i | 0 | −1.53989 | − | 2.66717i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 241.6 | 0 | 4.74894 | + | 2.74180i | 0 | −1.93649 | + | 1.11803i | 0 | −5.87843 | + | 3.80053i | 0 | 10.5350 | + | 18.2471i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 481.1 | 0 | −4.45439 | + | 2.57174i | 0 | −1.93649 | − | 1.11803i | 0 | 1.42520 | − | 6.85338i | 0 | 8.72772 | − | 15.1168i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 481.2 | 0 | −0.731043 | + | 0.422068i | 0 | −1.93649 | − | 1.11803i | 0 | 6.88972 | + | 1.23763i | 0 | −4.14372 | + | 7.17713i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 481.3 | 0 | −0.507487 | + | 0.292998i | 0 | 1.93649 | + | 1.11803i | 0 | −1.91172 | + | 6.73389i | 0 | −4.32830 | + | 7.49684i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 481.4 | 0 | 1.83681 | − | 1.06048i | 0 | 1.93649 | + | 1.11803i | 0 | −4.91879 | − | 4.98051i | 0 | −2.25076 | + | 3.89842i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 481.5 | 0 | 2.10717 | − | 1.21658i | 0 | 1.93649 | + | 1.11803i | 0 | 5.39402 | − | 4.46146i | 0 | −1.53989 | + | 2.66717i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 481.6 | 0 | 4.74894 | − | 2.74180i | 0 | −1.93649 | − | 1.11803i | 0 | −5.87843 | − | 3.80053i | 0 | 10.5350 | − | 18.2471i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 560.3.bx.c | 12 | |
| 4.b | odd | 2 | 1 | 35.3.h.a | ✓ | 12 | |
| 7.d | odd | 6 | 1 | inner | 560.3.bx.c | 12 | |
| 12.b | even | 2 | 1 | 315.3.w.c | 12 | ||
| 20.d | odd | 2 | 1 | 175.3.i.d | 12 | ||
| 20.e | even | 4 | 2 | 175.3.j.b | 24 | ||
| 28.d | even | 2 | 1 | 245.3.h.c | 12 | ||
| 28.f | even | 6 | 1 | 35.3.h.a | ✓ | 12 | |
| 28.f | even | 6 | 1 | 245.3.d.a | 12 | ||
| 28.g | odd | 6 | 1 | 245.3.d.a | 12 | ||
| 28.g | odd | 6 | 1 | 245.3.h.c | 12 | ||
| 84.j | odd | 6 | 1 | 315.3.w.c | 12 | ||
| 140.s | even | 6 | 1 | 175.3.i.d | 12 | ||
| 140.x | odd | 12 | 2 | 175.3.j.b | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.3.h.a | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 35.3.h.a | ✓ | 12 | 28.f | even | 6 | 1 | |
| 175.3.i.d | 12 | 20.d | odd | 2 | 1 | ||
| 175.3.i.d | 12 | 140.s | even | 6 | 1 | ||
| 175.3.j.b | 24 | 20.e | even | 4 | 2 | ||
| 175.3.j.b | 24 | 140.x | odd | 12 | 2 | ||
| 245.3.d.a | 12 | 28.f | even | 6 | 1 | ||
| 245.3.d.a | 12 | 28.g | odd | 6 | 1 | ||
| 245.3.h.c | 12 | 28.d | even | 2 | 1 | ||
| 245.3.h.c | 12 | 28.g | odd | 6 | 1 | ||
| 315.3.w.c | 12 | 12.b | even | 2 | 1 | ||
| 315.3.w.c | 12 | 84.j | odd | 6 | 1 | ||
| 560.3.bx.c | 12 | 1.a | even | 1 | 1 | trivial | |
| 560.3.bx.c | 12 | 7.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 6 T_{3}^{11} - 16 T_{3}^{10} + 168 T_{3}^{9} + 435 T_{3}^{8} - 4284 T_{3}^{7} + 7688 T_{3}^{6} + \cdots + 5184 \)
acting on \(S_{3}^{\mathrm{new}}(560, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} - 6 T^{11} + \cdots + 5184 \)
|
| $5$ |
\( (T^{4} - 5 T^{2} + 25)^{3} \)
|
| $7$ |
\( T^{12} + \cdots + 13841287201 \)
|
| $11$ |
\( T^{12} + \cdots + 4134617957376 \)
|
| $13$ |
\( T^{12} + 546 T^{10} + \cdots + 77792400 \)
|
| $17$ |
\( T^{12} + \cdots + 8707129344 \)
|
| $19$ |
\( T^{12} + \cdots + 1590595171344 \)
|
| $23$ |
\( T^{12} + \cdots + 44219321357121 \)
|
| $29$ |
\( (T^{6} - 32 T^{5} + \cdots + 486804)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 76\!\cdots\!04 \)
|
| $37$ |
\( T^{12} + \cdots + 110961448062976 \)
|
| $41$ |
\( T^{12} + \cdots + 77\!\cdots\!25 \)
|
| $43$ |
\( (T^{6} - 2 T^{5} + \cdots - 9258464)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 11\!\cdots\!04 \)
|
| $53$ |
\( T^{12} + \cdots + 18\!\cdots\!00 \)
|
| $59$ |
\( T^{12} + \cdots + 92\!\cdots\!84 \)
|
| $61$ |
\( T^{12} + \cdots + 23\!\cdots\!24 \)
|
| $67$ |
\( T^{12} + \cdots + 28\!\cdots\!56 \)
|
| $71$ |
\( (T^{6} - 4 T^{5} + \cdots + 46762703904)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 15\!\cdots\!64 \)
|
| $79$ |
\( T^{12} + \cdots + 560965048576 \)
|
| $83$ |
\( T^{12} + \cdots + 26\!\cdots\!00 \)
|
| $89$ |
\( T^{12} + \cdots + 81\!\cdots\!44 \)
|
| $97$ |
\( T^{12} + \cdots + 11\!\cdots\!00 \)
|
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