Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [84,3,Mod(67,84)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(84, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("84.67");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 84.l (of order \(6\), degree \(2\), 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: | \(2.28883422063\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 3 x^{13} + x^{12} + 10 x^{11} - 10 x^{10} - 24 x^{9} + 36 x^{8} + 48 x^{7} + 144 x^{6} + \cdots + 16384 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 3 x^{13} + x^{12} + 10 x^{11} - 10 x^{10} - 24 x^{9} + 36 x^{8} + 48 x^{7} + 144 x^{6} + \cdots + 16384 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5 \nu^{13} - 5 \nu^{12} + 35 \nu^{11} - 48 \nu^{10} - 506 \nu^{9} - 364 \nu^{8} + 12 \nu^{7} + \cdots - 122880 ) / 188416 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 5 \nu^{13} - 15 \nu^{12} + 49 \nu^{11} + 228 \nu^{10} + 122 \nu^{9} + 84 \nu^{8} - 604 \nu^{7} + \cdots + 40960 ) / 47104 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15 \nu^{13} - 35 \nu^{12} + 5 \nu^{11} + 220 \nu^{10} - 246 \nu^{9} - 1372 \nu^{8} - 188 \nu^{7} + \cdots - 356352 ) / 94208 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 25 \nu^{13} + 239 \nu^{12} + 23 \nu^{11} - 1512 \nu^{10} + 110 \nu^{9} + 3348 \nu^{8} + \cdots + 466944 ) / 188416 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 9 \nu^{13} - 87 \nu^{12} - 51 \nu^{11} + 174 \nu^{10} + 94 \nu^{9} - 864 \nu^{8} - 1404 \nu^{7} + \cdots - 163840 ) / 94208 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 27 \nu^{13} - 501 \nu^{12} - 621 \nu^{11} + 1184 \nu^{10} - 634 \nu^{9} - 3660 \nu^{8} + \cdots - 933888 ) / 188416 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 20 \nu^{13} + 5 \nu^{12} + 15 \nu^{11} - 133 \nu^{10} - 196 \nu^{9} + 142 \nu^{8} + 796 \nu^{7} + \cdots + 53248 ) / 47104 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 19 \nu^{13} + 52 \nu^{12} - 14 \nu^{11} - 225 \nu^{10} + 238 \nu^{9} + 962 \nu^{8} + \cdots + 319488 ) / 47104 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 3 \nu^{13} - 5 \nu^{12} - 9 \nu^{11} + 34 \nu^{10} + 10 \nu^{9} - 112 \nu^{8} + 12 \nu^{7} + \cdots - 32768 ) / 4096 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 87 \nu^{13} + 201 \nu^{12} + 53 \nu^{11} - 890 \nu^{10} - 10 \nu^{9} + 3072 \nu^{8} + \cdots + 815104 ) / 94208 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 97 \nu^{13} - 66 \nu^{12} - 224 \nu^{11} + 593 \nu^{10} + 678 \nu^{9} - 1034 \nu^{8} - 2456 \nu^{7} + \cdots - 419840 ) / 47104 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 387 \nu^{13} - 603 \nu^{12} - 947 \nu^{11} + 2792 \nu^{10} + 4234 \nu^{9} - 9412 \nu^{8} + \cdots - 2105344 ) / 188416 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 2 \beta_{13} + \beta_{12} - 4 \beta_{11} + \beta_{10} + 4 \beta_{9} + 3 \beta_{8} - \beta_{7} + \cdots - 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2 \beta_{13} + 4 \beta_{12} + 7 \beta_{11} - 6 \beta_{10} - 7 \beta_{9} + 13 \beta_{8} + \beta_{6} + \cdots + 9 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 4 \beta_{13} - 7 \beta_{12} - 7 \beta_{11} + 19 \beta_{10} + 23 \beta_{9} + 10 \beta_{8} - \beta_{7} + \cdots - 22 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 2 \beta_{13} - 9 \beta_{12} + 14 \beta_{11} + 27 \beta_{10} - 62 \beta_{9} - \beta_{8} - 3 \beta_{7} + \cdots + 99 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 2 \beta_{13} + 22 \beta_{12} + 39 \beta_{11} - 20 \beta_{10} - 103 \beta_{9} - 13 \beta_{8} + \cdots - 281 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 56 \beta_{13} - 83 \beta_{12} - \beta_{11} + 43 \beta_{10} + 17 \beta_{9} - 88 \beta_{8} - 37 \beta_{7} + \cdots - 376 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 162 \beta_{13} + 145 \beta_{12} - 132 \beta_{11} + 65 \beta_{10} + 116 \beta_{9} + 111 \beta_{8} + \cdots + 195 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 90 \beta_{13} - 296 \beta_{12} - 633 \beta_{11} - 146 \beta_{10} + 89 \beta_{9} + 9 \beta_{8} + \cdots - 491 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 148 \beta_{13} + 101 \beta_{12} + 485 \beta_{11} + 639 \beta_{10} + 1515 \beta_{9} - 646 \beta_{8} + \cdots - 14 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 422 \beta_{13} - 45 \beta_{12} - 298 \beta_{11} + 87 \beta_{10} - 1606 \beta_{9} - 1573 \beta_{8} + \cdots + 4543 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(43\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−1.94212 | − | 0.477665i | 1.50000 | + | 0.866025i | 3.54367 | + | 1.85537i | −0.560105 | − | 0.970131i | −2.49951 | − | 2.39842i | 2.94481 | + | 6.35044i | −5.99600 | − | 5.29604i | 1.50000 | + | 2.59808i | 0.624395 | + | 2.15165i | ||||||||||||||||||||||||||||||||||||||||||
| 67.2 | −1.61263 | + | 1.18297i | 1.50000 | + | 0.866025i | 1.20115 | − | 3.81539i | −4.68839 | − | 8.12053i | −3.44343 | + | 0.377879i | −2.03077 | − | 6.69895i | 2.57649 | + | 7.57375i | 1.50000 | + | 2.59808i | 17.1670 | + | 7.54918i | |||||||||||||||||||||||||||||||||||||||||||
| 67.3 | −0.674031 | − | 1.88300i | 1.50000 | + | 0.866025i | −3.09136 | + | 2.53840i | 3.85384 | + | 6.67504i | 0.619677 | − | 3.40823i | 6.80322 | − | 1.64808i | 6.86348 | + | 4.11007i | 1.50000 | + | 2.59808i | 9.97149 | − | 11.7560i | |||||||||||||||||||||||||||||||||||||||||||
| 67.4 | 0.930442 | + | 1.77039i | 1.50000 | + | 0.866025i | −2.26855 | + | 3.29449i | 2.05390 | + | 3.55745i | −0.137538 | + | 3.46137i | −1.76582 | − | 6.77362i | −7.94329 | − | 0.950890i | 1.50000 | + | 2.59808i | −4.38705 | + | 6.94620i | |||||||||||||||||||||||||||||||||||||||||||
| 67.5 | 1.09671 | − | 1.67249i | 1.50000 | + | 0.866025i | −1.59444 | − | 3.66848i | −1.27628 | − | 2.21059i | 3.09349 | − | 1.55895i | 4.08667 | − | 5.68323i | −7.88414 | − | 1.35660i | 1.50000 | + | 2.59808i | −5.09690 | − | 0.289812i | |||||||||||||||||||||||||||||||||||||||||||
| 67.6 | 1.80755 | + | 0.856011i | 1.50000 | + | 0.866025i | 2.53449 | + | 3.09457i | −2.93936 | − | 5.09112i | 1.97000 | + | 2.84940i | 6.23888 | + | 3.17433i | 1.93223 | + | 7.76315i | 1.50000 | + | 2.59808i | −0.954989 | − | 11.7186i | |||||||||||||||||||||||||||||||||||||||||||
| 67.7 | 1.89408 | − | 0.642246i | 1.50000 | + | 0.866025i | 3.17504 | − | 2.43292i | 1.05640 | + | 1.82975i | 3.39731 | + | 0.676948i | −6.77699 | + | 1.75282i | 4.45123 | − | 6.64730i | 1.50000 | + | 2.59808i | 3.17606 | + | 2.78720i | |||||||||||||||||||||||||||||||||||||||||||
| 79.1 | −1.94212 | + | 0.477665i | 1.50000 | − | 0.866025i | 3.54367 | − | 1.85537i | −0.560105 | + | 0.970131i | −2.49951 | + | 2.39842i | 2.94481 | − | 6.35044i | −5.99600 | + | 5.29604i | 1.50000 | − | 2.59808i | 0.624395 | − | 2.15165i | |||||||||||||||||||||||||||||||||||||||||||
| 79.2 | −1.61263 | − | 1.18297i | 1.50000 | − | 0.866025i | 1.20115 | + | 3.81539i | −4.68839 | + | 8.12053i | −3.44343 | − | 0.377879i | −2.03077 | + | 6.69895i | 2.57649 | − | 7.57375i | 1.50000 | − | 2.59808i | 17.1670 | − | 7.54918i | |||||||||||||||||||||||||||||||||||||||||||
| 79.3 | −0.674031 | + | 1.88300i | 1.50000 | − | 0.866025i | −3.09136 | − | 2.53840i | 3.85384 | − | 6.67504i | 0.619677 | + | 3.40823i | 6.80322 | + | 1.64808i | 6.86348 | − | 4.11007i | 1.50000 | − | 2.59808i | 9.97149 | + | 11.7560i | |||||||||||||||||||||||||||||||||||||||||||
| 79.4 | 0.930442 | − | 1.77039i | 1.50000 | − | 0.866025i | −2.26855 | − | 3.29449i | 2.05390 | − | 3.55745i | −0.137538 | − | 3.46137i | −1.76582 | + | 6.77362i | −7.94329 | + | 0.950890i | 1.50000 | − | 2.59808i | −4.38705 | − | 6.94620i | |||||||||||||||||||||||||||||||||||||||||||
| 79.5 | 1.09671 | + | 1.67249i | 1.50000 | − | 0.866025i | −1.59444 | + | 3.66848i | −1.27628 | + | 2.21059i | 3.09349 | + | 1.55895i | 4.08667 | + | 5.68323i | −7.88414 | + | 1.35660i | 1.50000 | − | 2.59808i | −5.09690 | + | 0.289812i | |||||||||||||||||||||||||||||||||||||||||||
| 79.6 | 1.80755 | − | 0.856011i | 1.50000 | − | 0.866025i | 2.53449 | − | 3.09457i | −2.93936 | + | 5.09112i | 1.97000 | − | 2.84940i | 6.23888 | − | 3.17433i | 1.93223 | − | 7.76315i | 1.50000 | − | 2.59808i | −0.954989 | + | 11.7186i | |||||||||||||||||||||||||||||||||||||||||||
| 79.7 | 1.89408 | + | 0.642246i | 1.50000 | − | 0.866025i | 3.17504 | + | 2.43292i | 1.05640 | − | 1.82975i | 3.39731 | − | 0.676948i | −6.77699 | − | 1.75282i | 4.45123 | + | 6.64730i | 1.50000 | − | 2.59808i | 3.17606 | − | 2.78720i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 28.g | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 84.3.l.d | yes | 14 |
| 3.b | odd | 2 | 1 | 252.3.y.d | 14 | ||
| 4.b | odd | 2 | 1 | 84.3.l.c | ✓ | 14 | |
| 7.c | even | 3 | 1 | 84.3.l.c | ✓ | 14 | |
| 7.c | even | 3 | 1 | 588.3.g.g | 14 | ||
| 7.d | odd | 6 | 1 | 588.3.g.f | 14 | ||
| 12.b | even | 2 | 1 | 252.3.y.e | 14 | ||
| 21.h | odd | 6 | 1 | 252.3.y.e | 14 | ||
| 28.f | even | 6 | 1 | 588.3.g.f | 14 | ||
| 28.g | odd | 6 | 1 | inner | 84.3.l.d | yes | 14 |
| 28.g | odd | 6 | 1 | 588.3.g.g | 14 | ||
| 84.n | even | 6 | 1 | 252.3.y.d | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.3.l.c | ✓ | 14 | 4.b | odd | 2 | 1 | |
| 84.3.l.c | ✓ | 14 | 7.c | even | 3 | 1 | |
| 84.3.l.d | yes | 14 | 1.a | even | 1 | 1 | trivial |
| 84.3.l.d | yes | 14 | 28.g | odd | 6 | 1 | inner |
| 252.3.y.d | 14 | 3.b | odd | 2 | 1 | ||
| 252.3.y.d | 14 | 84.n | even | 6 | 1 | ||
| 252.3.y.e | 14 | 12.b | even | 2 | 1 | ||
| 252.3.y.e | 14 | 21.h | odd | 6 | 1 | ||
| 588.3.g.f | 14 | 7.d | odd | 6 | 1 | ||
| 588.3.g.f | 14 | 28.f | even | 6 | 1 | ||
| 588.3.g.g | 14 | 7.c | even | 3 | 1 | ||
| 588.3.g.g | 14 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(84, [\chi])\):
|
\( T_{5}^{14} + 5 T_{5}^{13} + 118 T_{5}^{12} + 209 T_{5}^{11} + 8414 T_{5}^{10} + 16253 T_{5}^{9} + \cdots + 111175936 \)
|
|
\( T_{11}^{14} + 21 T_{11}^{13} - 428 T_{11}^{12} - 12075 T_{11}^{11} + 171652 T_{11}^{10} + \cdots + 540293064966912 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} - 3 T^{13} + \cdots + 16384 \)
|
| $3$ |
\( (T^{2} - 3 T + 3)^{7} \)
|
| $5$ |
\( T^{14} + \cdots + 111175936 \)
|
| $7$ |
\( T^{14} + \cdots + 678223072849 \)
|
| $11$ |
\( T^{14} + \cdots + 540293064966912 \)
|
| $13$ |
\( (T^{7} - 20 T^{6} + \cdots - 38439872)^{2} \)
|
| $17$ |
\( T^{14} + \cdots + 432618701062144 \)
|
| $19$ |
\( T^{14} + \cdots + 2501189087232 \)
|
| $23$ |
\( T^{14} + \cdots + 10\!\cdots\!88 \)
|
| $29$ |
\( (T^{7} + 53 T^{6} + \cdots - 125760256)^{2} \)
|
| $31$ |
\( T^{14} + \cdots + 36\!\cdots\!23 \)
|
| $37$ |
\( T^{14} + \cdots + 11\!\cdots\!76 \)
|
| $41$ |
\( (T^{7} + 48 T^{6} + \cdots - 15211042816)^{2} \)
|
| $43$ |
\( T^{14} + \cdots + 95\!\cdots\!92 \)
|
| $47$ |
\( T^{14} + \cdots + 32\!\cdots\!32 \)
|
| $53$ |
\( T^{14} + \cdots + 11\!\cdots\!44 \)
|
| $59$ |
\( T^{14} + \cdots + 27\!\cdots\!48 \)
|
| $61$ |
\( T^{14} + \cdots + 17\!\cdots\!24 \)
|
| $67$ |
\( T^{14} + \cdots + 17\!\cdots\!88 \)
|
| $71$ |
\( T^{14} + \cdots + 79\!\cdots\!48 \)
|
| $73$ |
\( T^{14} + \cdots + 23\!\cdots\!96 \)
|
| $79$ |
\( T^{14} + \cdots + 23\!\cdots\!23 \)
|
| $83$ |
\( T^{14} + \cdots + 14\!\cdots\!92 \)
|
| $89$ |
\( T^{14} + \cdots + 25\!\cdots\!24 \)
|
| $97$ |
\( (T^{7} + \cdots + 5504261016976)^{2} \)
|
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