Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [420,2,Mod(239,420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(420, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("420.239");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 420.l (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: | \(3.35371688489\) |
| 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: |
\( x^{16} - 4 x^{15} + 9 x^{14} - 16 x^{13} + 18 x^{12} - 4 x^{11} - 36 x^{10} + 102 x^{9} - 170 x^{8} + \cdots + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| 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
\( x^{16} - 4 x^{15} + 9 x^{14} - 16 x^{13} + 18 x^{12} - 4 x^{11} - 36 x^{10} + 102 x^{9} - 170 x^{8} + \cdots + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 3 \nu^{15} - 10 \nu^{14} + 17 \nu^{13} + 2 \nu^{12} - 18 \nu^{11} + 72 \nu^{10} - 100 \nu^{9} + \cdots + 2048 ) / 128 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 23 \nu^{15} - 78 \nu^{14} + 179 \nu^{13} - 266 \nu^{12} + 202 \nu^{11} + 152 \nu^{10} - 972 \nu^{9} + \cdots - 3712 ) / 256 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 25 \nu^{15} - 98 \nu^{14} + 93 \nu^{13} + 58 \nu^{12} - 330 \nu^{11} + 744 \nu^{10} - 1012 \nu^{9} + \cdots + 8576 ) / 256 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 9 \nu^{15} + 14 \nu^{14} + 15 \nu^{13} - 66 \nu^{12} + 134 \nu^{11} - 164 \nu^{10} + 92 \nu^{9} + \cdots - 1472 ) / 64 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 49 \nu^{15} + 90 \nu^{14} + 107 \nu^{13} - 498 \nu^{12} + 954 \nu^{11} - 1192 \nu^{10} + \cdots - 13952 ) / 256 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4 \nu^{15} - 13 \nu^{14} + 19 \nu^{13} - 16 \nu^{12} - 9 \nu^{11} + 63 \nu^{10} - 142 \nu^{9} + \cdots + 176 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 13 \nu^{15} + 58 \nu^{14} - 137 \nu^{13} + 202 \nu^{12} - 158 \nu^{11} - 108 \nu^{10} + 700 \nu^{9} + \cdots + 1856 ) / 64 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 3 \nu^{15} - 114 \nu^{14} + 593 \nu^{13} - 1254 \nu^{12} + 1582 \nu^{11} - 904 \nu^{10} + \cdots - 22656 ) / 256 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13 \nu^{15} - 114 \nu^{14} + 361 \nu^{13} - 630 \nu^{12} + 662 \nu^{11} - 88 \nu^{10} - 1492 \nu^{9} + \cdots - 8192 ) / 128 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 3 \nu^{15} - 198 \nu^{14} + 783 \nu^{13} - 1458 \nu^{12} + 1682 \nu^{11} - 584 \nu^{10} - 2876 \nu^{9} + \cdots - 19584 ) / 256 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 5 \nu^{15} + 52 \nu^{14} - 185 \nu^{13} + 336 \nu^{12} - 374 \nu^{11} + 100 \nu^{10} + 716 \nu^{9} + \cdots + 4608 ) / 64 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 43 \nu^{15} + 262 \nu^{14} - 743 \nu^{13} + 1218 \nu^{12} - 1170 \nu^{11} - 104 \nu^{10} + \cdots + 12928 ) / 256 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 63 \nu^{15} + 286 \nu^{14} - 699 \nu^{13} + 1098 \nu^{12} - 922 \nu^{11} - 392 \nu^{10} + \cdots + 12928 ) / 256 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 97 \nu^{15} + 410 \nu^{14} - 789 \nu^{13} + 910 \nu^{12} - 342 \nu^{11} - 1512 \nu^{10} + \cdots - 2176 ) / 256 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 79 \nu^{15} + 370 \nu^{14} - 803 \nu^{13} + 1086 \nu^{12} - 706 \nu^{11} - 992 \nu^{10} + \cdots + 5120 ) / 128 \)
|
| \(\nu\) | \(=\) |
\( ( - \beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{12} + \beta_{9} - 6 \beta_{8} - \beta_{6} + 4 \beta_{5} + \cdots + 6 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{15} - 5 \beta_{14} + 3 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} - 5 \beta_{9} - 6 \beta_{7} + \cdots - 6 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{15} + \beta_{14} + 5\beta_{12} - 6\beta_{11} - 3\beta_{9} - 2\beta_{7} + \beta_{6} - \beta_{4} - 4\beta_{2} + 2 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 7 \beta_{15} - \beta_{14} + 12 \beta_{13} + 3 \beta_{12} - 2 \beta_{11} - 4 \beta_{10} + 11 \beta_{9} + \cdots + 18 ) / 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 11 \beta_{15} - \beta_{14} - 4 \beta_{13} - 5 \beta_{12} + 6 \beta_{11} - \beta_{9} - 12 \beta_{8} + \cdots - 18 ) / 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 3 \beta_{15} - 11 \beta_{14} - 4 \beta_{13} + 9 \beta_{12} + 10 \beta_{11} + 8 \beta_{10} - 7 \beta_{9} + \cdots - 2 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 19 \beta_{15} + 7 \beta_{14} + 4 \beta_{13} + 35 \beta_{12} - 6 \beta_{11} - 24 \beta_{10} + \cdots + 30 ) / 12 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 23 \beta_{15} - 65 \beta_{14} + 48 \beta_{13} + 15 \beta_{12} + 50 \beta_{11} - 32 \beta_{10} + \cdots - 6 ) / 12 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 7 \beta_{15} - 43 \beta_{14} - 16 \beta_{13} + 37 \beta_{12} - 14 \beta_{11} - 3 \beta_{9} + 16 \beta_{8} + \cdots - 38 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 17 \beta_{15} - 145 \beta_{14} + 135 \beta_{12} + 10 \beta_{11} - 40 \beta_{10} + 11 \beta_{9} + \cdots - 78 ) / 12 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 203 \beta_{15} + 191 \beta_{14} + 128 \beta_{13} - 41 \beta_{12} - 162 \beta_{11} - 264 \beta_{10} + \cdots + 6 ) / 12 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 5 \beta_{15} - 67 \beta_{14} + 48 \beta_{13} + 37 \beta_{12} + 30 \beta_{11} + 32 \beta_{10} + \cdots - 2 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 245 \beta_{15} - 425 \beta_{14} - 128 \beta_{13} + 431 \beta_{12} + 30 \beta_{11} + 264 \beta_{10} + \cdots - 594 ) / 12 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 49 \beta_{15} - 593 \beta_{14} + 336 \beta_{13} + 135 \beta_{12} + 578 \beta_{11} - 176 \beta_{10} + \cdots - 846 ) / 12 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 73 \beta_{15} - 59 \beta_{14} + 72 \beta_{13} - 27 \beta_{12} - 102 \beta_{11} - 248 \beta_{10} + \cdots + 50 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(241\) | \(281\) | \(337\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 239.1 |
|
−1.38390 | − | 0.291259i | 0.751690 | + | 1.56044i | 1.83034 | + | 0.806145i | −1.60157 | − | 1.56044i | −0.585769 | − | 2.37842i | 1.00000 | −2.29820 | − | 1.64872i | −1.86993 | + | 2.34593i | 1.76192 | + | 2.62595i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.2 | −1.38390 | + | 0.291259i | 0.751690 | − | 1.56044i | 1.83034 | − | 0.806145i | −1.60157 | + | 1.56044i | −0.585769 | + | 2.37842i | 1.00000 | −2.29820 | + | 1.64872i | −1.86993 | − | 2.34593i | 1.76192 | − | 2.62595i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.3 | −1.25945 | − | 0.643263i | 1.48716 | − | 0.887900i | 1.17242 | + | 1.62032i | 2.05223 | + | 0.887900i | −2.44415 | + | 0.161632i | 1.00000 | −0.434319 | − | 2.79488i | 1.42327 | − | 2.64089i | −2.01352 | − | 2.43839i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.4 | −1.25945 | + | 0.643263i | 1.48716 | + | 0.887900i | 1.17242 | − | 1.62032i | 2.05223 | − | 0.887900i | −2.44415 | − | 0.161632i | 1.00000 | −0.434319 | + | 2.79488i | 1.42327 | + | 2.64089i | −2.01352 | + | 2.43839i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.5 | −0.979286 | − | 1.02029i | −1.71822 | − | 0.218455i | −0.0819979 | + | 1.99832i | −2.22537 | + | 0.218455i | 1.45974 | + | 1.96702i | 1.00000 | 2.11917 | − | 1.87326i | 2.90455 | + | 0.750707i | 2.40216 | + | 2.05660i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.6 | −0.979286 | + | 1.02029i | −1.71822 | + | 0.218455i | −0.0819979 | − | 1.99832i | −2.22537 | − | 0.218455i | 1.45974 | − | 1.96702i | 1.00000 | 2.11917 | + | 1.87326i | 2.90455 | − | 0.750707i | 2.40216 | − | 2.05660i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.7 | −0.538162 | − | 1.30782i | −0.520627 | − | 1.65195i | −1.42076 | + | 1.40763i | 1.50700 | + | 1.65195i | −1.88027 | + | 1.56990i | 1.00000 | 2.60553 | + | 1.10056i | −2.45790 | + | 1.72010i | 1.34944 | − | 2.85990i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.8 | −0.538162 | + | 1.30782i | −0.520627 | + | 1.65195i | −1.42076 | − | 1.40763i | 1.50700 | − | 1.65195i | −1.88027 | − | 1.56990i | 1.00000 | 2.60553 | − | 1.10056i | −2.45790 | − | 1.72010i | 1.34944 | + | 2.85990i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.9 | 0.538162 | − | 1.30782i | −0.520627 | − | 1.65195i | −1.42076 | − | 1.40763i | −1.50700 | + | 1.65195i | −2.44063 | − | 0.208135i | 1.00000 | −2.60553 | + | 1.10056i | −2.45790 | + | 1.72010i | 1.34944 | + | 2.85990i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.10 | 0.538162 | + | 1.30782i | −0.520627 | + | 1.65195i | −1.42076 | + | 1.40763i | −1.50700 | − | 1.65195i | −2.44063 | + | 0.208135i | 1.00000 | −2.60553 | − | 1.10056i | −2.45790 | − | 1.72010i | 1.34944 | − | 2.85990i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.11 | 0.979286 | − | 1.02029i | −1.71822 | − | 0.218455i | −0.0819979 | − | 1.99832i | 2.22537 | + | 0.218455i | −1.90552 | + | 1.53916i | 1.00000 | −2.11917 | − | 1.87326i | 2.90455 | + | 0.750707i | 2.40216 | − | 2.05660i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.12 | 0.979286 | + | 1.02029i | −1.71822 | + | 0.218455i | −0.0819979 | + | 1.99832i | 2.22537 | − | 0.218455i | −1.90552 | − | 1.53916i | 1.00000 | −2.11917 | + | 1.87326i | 2.90455 | − | 0.750707i | 2.40216 | + | 2.05660i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.13 | 1.25945 | − | 0.643263i | 1.48716 | − | 0.887900i | 1.17242 | − | 1.62032i | −2.05223 | + | 0.887900i | 1.30184 | − | 2.07490i | 1.00000 | 0.434319 | − | 2.79488i | 1.42327 | − | 2.64089i | −2.01352 | + | 2.43839i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.14 | 1.25945 | + | 0.643263i | 1.48716 | + | 0.887900i | 1.17242 | + | 1.62032i | −2.05223 | − | 0.887900i | 1.30184 | + | 2.07490i | 1.00000 | 0.434319 | + | 2.79488i | 1.42327 | + | 2.64089i | −2.01352 | − | 2.43839i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.15 | 1.38390 | − | 0.291259i | 0.751690 | + | 1.56044i | 1.83034 | − | 0.806145i | 1.60157 | − | 1.56044i | 1.49475 | + | 1.94055i | 1.00000 | 2.29820 | − | 1.64872i | −1.86993 | + | 2.34593i | 1.76192 | − | 2.62595i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.16 | 1.38390 | + | 0.291259i | 0.751690 | − | 1.56044i | 1.83034 | + | 0.806145i | 1.60157 | + | 1.56044i | 1.49475 | − | 1.94055i | 1.00000 | 2.29820 | + | 1.64872i | −1.86993 | − | 2.34593i | 1.76192 | + | 2.62595i | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 20.d | odd | 2 | 1 | inner |
| 60.h | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 420.2.l.h | yes | 16 |
| 3.b | odd | 2 | 1 | inner | 420.2.l.h | yes | 16 |
| 4.b | odd | 2 | 1 | 420.2.l.g | ✓ | 16 | |
| 5.b | even | 2 | 1 | 420.2.l.g | ✓ | 16 | |
| 12.b | even | 2 | 1 | 420.2.l.g | ✓ | 16 | |
| 15.d | odd | 2 | 1 | 420.2.l.g | ✓ | 16 | |
| 20.d | odd | 2 | 1 | inner | 420.2.l.h | yes | 16 |
| 60.h | even | 2 | 1 | inner | 420.2.l.h | yes | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 420.2.l.g | ✓ | 16 | 4.b | odd | 2 | 1 | |
| 420.2.l.g | ✓ | 16 | 5.b | even | 2 | 1 | |
| 420.2.l.g | ✓ | 16 | 12.b | even | 2 | 1 | |
| 420.2.l.g | ✓ | 16 | 15.d | odd | 2 | 1 | |
| 420.2.l.h | yes | 16 | 1.a | even | 1 | 1 | trivial |
| 420.2.l.h | yes | 16 | 3.b | odd | 2 | 1 | inner |
| 420.2.l.h | yes | 16 | 20.d | odd | 2 | 1 | inner |
| 420.2.l.h | yes | 16 | 60.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\):
|
\( T_{11}^{8} - 66T_{11}^{6} + 1400T_{11}^{4} - 11360T_{11}^{2} + 31104 \)
|
|
\( T_{17}^{8} - 76T_{17}^{6} + 1968T_{17}^{4} - 19520T_{17}^{2} + 55296 \)
|
|
\( T_{43}^{4} + 4T_{43}^{3} - 96T_{43}^{2} - 256T_{43} + 1616 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 3 T^{14} + \cdots + 256 \)
|
| $3$ |
\( (T^{8} + 4 T^{5} - 2 T^{4} + \cdots + 81)^{2} \)
|
| $5$ |
\( T^{16} - 16 T^{14} + \cdots + 390625 \)
|
| $7$ |
\( (T - 1)^{16} \)
|
| $11$ |
\( (T^{8} - 66 T^{6} + \cdots + 31104)^{2} \)
|
| $13$ |
\( (T^{8} + 38 T^{6} + \cdots + 864)^{2} \)
|
| $17$ |
\( (T^{8} - 76 T^{6} + \cdots + 55296)^{2} \)
|
| $19$ |
\( (T^{8} + 38 T^{6} + \cdots + 864)^{2} \)
|
| $23$ |
\( (T^{8} + 112 T^{6} + \cdots + 135424)^{2} \)
|
| $29$ |
\( (T^{8} + 164 T^{6} + \cdots + 1048576)^{2} \)
|
| $31$ |
\( (T^{8} + 90 T^{6} + \cdots + 13824)^{2} \)
|
| $37$ |
\( (T^{8} + 148 T^{6} + \cdots + 884736)^{2} \)
|
| $41$ |
\( (T^{8} + 244 T^{6} + \cdots + 5914624)^{2} \)
|
| $43$ |
\( (T^{4} + 4 T^{3} + \cdots + 1616)^{4} \)
|
| $47$ |
\( (T^{8} + 224 T^{6} + \cdots + 5914624)^{2} \)
|
| $53$ |
\( (T^{8} - 146 T^{6} + \cdots + 3456)^{2} \)
|
| $59$ |
\( (T^{8} - 168 T^{6} + \cdots + 13824)^{2} \)
|
| $61$ |
\( (T^{4} - 10 T^{3} + \cdots - 712)^{4} \)
|
| $67$ |
\( (T^{4} - 14 T^{3} + \cdots - 864)^{4} \)
|
| $71$ |
\( (T^{8} - 342 T^{6} + \cdots + 4478976)^{2} \)
|
| $73$ |
\( (T^{8} + 350 T^{6} + \cdots + 39567744)^{2} \)
|
| $79$ |
\( (T^{8} + 332 T^{6} + \cdots + 4990464)^{2} \)
|
| $83$ |
\( (T^{8} + 380 T^{6} + \cdots + 92416)^{2} \)
|
| $89$ |
\( (T^{8} + 528 T^{6} + \cdots + 7661824)^{2} \)
|
| $97$ |
\( (T^{8} + 510 T^{6} + \cdots + 59308416)^{2} \)
|
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