Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(314,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 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("315.314");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 315.g (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: | \(2.51528766367\) |
| 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: |
\( x^{16} + 32x^{14} + 324x^{12} + 1328x^{10} + 2314x^{8} + 1920x^{6} + 780x^{4} + 144x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| 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
\( x^{16} + 32x^{14} + 324x^{12} + 1328x^{10} + 2314x^{8} + 1920x^{6} + 780x^{4} + 144x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 421 \nu^{15} + 12551 \nu^{13} + 107523 \nu^{11} + 279263 \nu^{9} - 67895 \nu^{7} - 640401 \nu^{5} + \cdots - 77157 \nu ) / 1872 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 68 \nu^{15} - 2211 \nu^{13} - 23101 \nu^{11} - 100054 \nu^{9} - 188622 \nu^{7} - 156497 \nu^{5} + \cdots - 3072 \nu ) / 312 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 103 \nu^{14} + 3280 \nu^{12} + 32851 \nu^{10} + 131324 \nu^{8} + 214559 \nu^{6} + 152528 \nu^{4} + \cdots + 3084 ) / 312 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1028 \nu^{15} + 57 \nu^{14} - 32587 \nu^{13} + 1839 \nu^{12} - 323232 \nu^{11} + 18993 \nu^{10} + \cdots + 12465 ) / 1872 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1028 \nu^{15} - 57 \nu^{14} - 32587 \nu^{13} - 1839 \nu^{12} - 323232 \nu^{11} - 18993 \nu^{10} + \cdots - 12465 ) / 1872 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -87\nu^{14} - 2720\nu^{12} - 26182\nu^{10} - 96114\nu^{8} - 129001\nu^{6} - 65828\nu^{4} - 9972\nu^{2} + 66 ) / 156 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 824 \nu^{15} + 25954 \nu^{13} + 253929 \nu^{11} + 966469 \nu^{9} + 1418954 \nu^{7} + 860592 \nu^{5} + \cdots + 9891 \nu ) / 936 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 207 \nu^{15} + 6517 \nu^{13} + 63713 \nu^{11} + 242383 \nu^{9} + 357611 \nu^{7} + 225809 \nu^{5} + \cdots + 4839 \nu ) / 208 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 2059 \nu^{15} - 64577 \nu^{13} - 625989 \nu^{11} - 2335421 \nu^{9} - 3273079 \nu^{7} + \cdots - 11529 \nu ) / 1872 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 719 \nu^{15} - 70 \nu^{14} + 22773 \nu^{13} - 2138 \nu^{12} + 225511 \nu^{11} - 19500 \nu^{10} + \cdots + 600 ) / 624 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 127 \nu^{15} + 155 \nu^{14} - 4016 \nu^{13} + 4905 \nu^{12} - 39637 \nu^{11} + 48477 \nu^{10} + \cdots + 4579 ) / 208 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 719 \nu^{15} - 70 \nu^{14} - 22773 \nu^{13} - 2138 \nu^{12} - 225511 \nu^{11} - 19500 \nu^{10} + \cdots + 600 ) / 624 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 959 \nu^{15} + 395 \nu^{14} - 30276 \nu^{13} + 12577 \nu^{12} - 297739 \nu^{11} + 125931 \nu^{10} + \cdots + 14337 ) / 624 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 214 \nu^{14} - 6749 \nu^{12} - 66222 \nu^{10} - 253811 \nu^{8} - 380182 \nu^{6} - 243405 \nu^{4} + \cdots - 5247 ) / 156 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 1847 \nu^{15} - 3339 \nu^{14} - 58600 \nu^{13} - 105243 \nu^{12} - 582465 \nu^{11} - 1031277 \nu^{10} + \cdots - 59715 ) / 1872 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{15} - 2\beta_{14} - \beta_{12} - \beta_{10} - \beta_{9} - 2\beta_{4} + 2\beta_{3} - \beta _1 - 8 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3 \beta_{13} - 2 \beta_{12} - 3 \beta_{11} - \beta_{10} - 5 \beta_{9} - 3 \beta_{8} + 11 \beta_{7} + \cdots - \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 14 \beta_{15} + 17 \beta_{14} + 4 \beta_{13} + 7 \beta_{12} + 4 \beta_{11} + 11 \beta_{10} + \cdots + 47 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 60 \beta_{13} + 29 \beta_{12} + 60 \beta_{11} + 31 \beta_{10} + 112 \beta_{9} + 80 \beta_{8} + \cdots + 38 \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 430 \beta_{15} - 558 \beta_{14} - 182 \beta_{13} - 241 \beta_{12} - 182 \beta_{11} - 423 \beta_{10} + \cdots - 1412 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1099 \beta_{13} - 446 \beta_{12} - 1099 \beta_{11} - 653 \beta_{10} - 2070 \beta_{9} - 1561 \beta_{8} + \cdots - 814 \beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 3524 \beta_{15} + 4662 \beta_{14} + 1700 \beta_{13} + 2122 \beta_{12} + 1700 \beta_{11} + \cdots + 11519 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 19488 \beta_{13} + 7254 \beta_{12} + 19488 \beta_{11} + 12234 \beta_{10} + 36520 \beta_{9} + \cdots + 15212 \beta_1 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 118974 \beta_{15} - 158222 \beta_{14} - 60316 \beta_{13} - 74227 \beta_{12} - 60316 \beta_{11} + \cdots - 388448 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 340417 \beta_{13} - 121702 \beta_{12} - 340417 \beta_{11} - 218715 \beta_{10} - 635141 \beta_{9} + \cdots - 271105 \beta_1 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 1016822 \beta_{15} + 1354137 \beta_{14} + 525872 \beta_{13} + 644851 \beta_{12} + 525872 \beta_{11} + \cdots + 3318917 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 5905588 \beta_{13} + 2073133 \beta_{12} + 5905588 \beta_{11} + 3832455 \beta_{10} + \cdots + 4741670 \beta_1 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 34945962 \beta_{15} - 46555626 \beta_{14} - 18222830 \beta_{13} - 22325383 \beta_{12} - 18222830 \beta_{11} + \cdots - 114055124 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 102133549 \beta_{13} - 35564202 \beta_{12} - 102133549 \beta_{11} - 66569347 \beta_{10} + \cdots - 82285540 \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(136\) | \(281\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 314.1 |
|
−2.33441 | 0 | 3.44949 | −1.33239 | − | 1.79576i | 0 | −2.53958 | + | 0.741964i | −3.38371 | 0 | 3.11034 | + | 4.19204i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.2 | −2.33441 | 0 | 3.44949 | −1.33239 | + | 1.79576i | 0 | −2.53958 | − | 0.741964i | −3.38371 | 0 | 3.11034 | − | 4.19204i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.3 | −2.33441 | 0 | 3.44949 | 1.33239 | − | 1.79576i | 0 | 2.53958 | − | 0.741964i | −3.38371 | 0 | −3.11034 | + | 4.19204i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.4 | −2.33441 | 0 | 3.44949 | 1.33239 | + | 1.79576i | 0 | 2.53958 | + | 0.741964i | −3.38371 | 0 | −3.11034 | − | 4.19204i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.5 | −0.741964 | 0 | −1.44949 | −2.05542 | − | 0.880486i | 0 | 1.24519 | + | 2.33441i | 2.55940 | 0 | 1.52505 | + | 0.653289i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.6 | −0.741964 | 0 | −1.44949 | −2.05542 | + | 0.880486i | 0 | 1.24519 | − | 2.33441i | 2.55940 | 0 | 1.52505 | − | 0.653289i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.7 | −0.741964 | 0 | −1.44949 | 2.05542 | − | 0.880486i | 0 | −1.24519 | − | 2.33441i | 2.55940 | 0 | −1.52505 | + | 0.653289i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.8 | −0.741964 | 0 | −1.44949 | 2.05542 | + | 0.880486i | 0 | −1.24519 | + | 2.33441i | 2.55940 | 0 | −1.52505 | − | 0.653289i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.9 | 0.741964 | 0 | −1.44949 | −2.05542 | − | 0.880486i | 0 | −1.24519 | + | 2.33441i | −2.55940 | 0 | −1.52505 | − | 0.653289i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.10 | 0.741964 | 0 | −1.44949 | −2.05542 | + | 0.880486i | 0 | −1.24519 | − | 2.33441i | −2.55940 | 0 | −1.52505 | + | 0.653289i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.11 | 0.741964 | 0 | −1.44949 | 2.05542 | − | 0.880486i | 0 | 1.24519 | − | 2.33441i | −2.55940 | 0 | 1.52505 | − | 0.653289i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.12 | 0.741964 | 0 | −1.44949 | 2.05542 | + | 0.880486i | 0 | 1.24519 | + | 2.33441i | −2.55940 | 0 | 1.52505 | + | 0.653289i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.13 | 2.33441 | 0 | 3.44949 | −1.33239 | − | 1.79576i | 0 | 2.53958 | + | 0.741964i | 3.38371 | 0 | −3.11034 | − | 4.19204i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.14 | 2.33441 | 0 | 3.44949 | −1.33239 | + | 1.79576i | 0 | 2.53958 | − | 0.741964i | 3.38371 | 0 | −3.11034 | + | 4.19204i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.15 | 2.33441 | 0 | 3.44949 | 1.33239 | − | 1.79576i | 0 | −2.53958 | − | 0.741964i | 3.38371 | 0 | 3.11034 | − | 4.19204i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 314.16 | 2.33441 | 0 | 3.44949 | 1.33239 | + | 1.79576i | 0 | −2.53958 | + | 0.741964i | 3.38371 | 0 | 3.11034 | + | 4.19204i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
| 35.c | odd | 2 | 1 | inner |
| 105.g | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 315.2.g.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 315.2.g.a | ✓ | 16 |
| 4.b | odd | 2 | 1 | 5040.2.k.g | 16 | ||
| 5.b | even | 2 | 1 | inner | 315.2.g.a | ✓ | 16 |
| 5.c | odd | 4 | 2 | 1575.2.b.h | 16 | ||
| 7.b | odd | 2 | 1 | inner | 315.2.g.a | ✓ | 16 |
| 12.b | even | 2 | 1 | 5040.2.k.g | 16 | ||
| 15.d | odd | 2 | 1 | inner | 315.2.g.a | ✓ | 16 |
| 15.e | even | 4 | 2 | 1575.2.b.h | 16 | ||
| 20.d | odd | 2 | 1 | 5040.2.k.g | 16 | ||
| 21.c | even | 2 | 1 | inner | 315.2.g.a | ✓ | 16 |
| 28.d | even | 2 | 1 | 5040.2.k.g | 16 | ||
| 35.c | odd | 2 | 1 | inner | 315.2.g.a | ✓ | 16 |
| 35.f | even | 4 | 2 | 1575.2.b.h | 16 | ||
| 60.h | even | 2 | 1 | 5040.2.k.g | 16 | ||
| 84.h | odd | 2 | 1 | 5040.2.k.g | 16 | ||
| 105.g | even | 2 | 1 | inner | 315.2.g.a | ✓ | 16 |
| 105.k | odd | 4 | 2 | 1575.2.b.h | 16 | ||
| 140.c | even | 2 | 1 | 5040.2.k.g | 16 | ||
| 420.o | odd | 2 | 1 | 5040.2.k.g | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.2.g.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 315.2.g.a | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 315.2.g.a | ✓ | 16 | 5.b | even | 2 | 1 | inner |
| 315.2.g.a | ✓ | 16 | 7.b | odd | 2 | 1 | inner |
| 315.2.g.a | ✓ | 16 | 15.d | odd | 2 | 1 | inner |
| 315.2.g.a | ✓ | 16 | 21.c | even | 2 | 1 | inner |
| 315.2.g.a | ✓ | 16 | 35.c | odd | 2 | 1 | inner |
| 315.2.g.a | ✓ | 16 | 105.g | even | 2 | 1 | inner |
| 1575.2.b.h | 16 | 5.c | odd | 4 | 2 | ||
| 1575.2.b.h | 16 | 15.e | even | 4 | 2 | ||
| 1575.2.b.h | 16 | 35.f | even | 4 | 2 | ||
| 1575.2.b.h | 16 | 105.k | odd | 4 | 2 | ||
| 5040.2.k.g | 16 | 4.b | odd | 2 | 1 | ||
| 5040.2.k.g | 16 | 12.b | even | 2 | 1 | ||
| 5040.2.k.g | 16 | 20.d | odd | 2 | 1 | ||
| 5040.2.k.g | 16 | 28.d | even | 2 | 1 | ||
| 5040.2.k.g | 16 | 60.h | even | 2 | 1 | ||
| 5040.2.k.g | 16 | 84.h | odd | 2 | 1 | ||
| 5040.2.k.g | 16 | 140.c | even | 2 | 1 | ||
| 5040.2.k.g | 16 | 420.o | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(315, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 6 T^{2} + 3)^{4} \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( (T^{8} - 4 T^{6} + \cdots + 625)^{2} \)
|
| $7$ |
\( (T^{8} - 4 T^{6} + \cdots + 2401)^{2} \)
|
| $11$ |
\( (T^{2} + 2)^{8} \)
|
| $13$ |
\( (T^{4} - 32 T^{2} + 40)^{4} \)
|
| $17$ |
\( (T^{4} + 64 T^{2} + 1000)^{4} \)
|
| $19$ |
\( (T^{4} + 72 T^{2} + 1080)^{4} \)
|
| $23$ |
\( (T^{4} - 36 T^{2} + 300)^{4} \)
|
| $29$ |
\( (T^{4} + 100 T^{2} + 2116)^{4} \)
|
| $31$ |
\( (T^{4} + 72 T^{2} + 120)^{4} \)
|
| $37$ |
\( (T^{4} + 72 T^{2} + 432)^{4} \)
|
| $41$ |
\( (T^{4} - 120 T^{2} + 3000)^{4} \)
|
| $43$ |
\( (T^{4} + 72 T^{2} + 1200)^{4} \)
|
| $47$ |
\( (T^{4} + 64 T^{2} + 160)^{4} \)
|
| $53$ |
\( (T^{4} - 108 T^{2} + 972)^{4} \)
|
| $59$ |
\( (T^{4} - 144 T^{2} + 480)^{4} \)
|
| $61$ |
\( (T^{4} + 192 T^{2} + 7680)^{4} \)
|
| $67$ |
\( (T^{4} + 144 T^{2} + 4800)^{4} \)
|
| $71$ |
\( (T^{4} + 100 T^{2} + 2116)^{4} \)
|
| $73$ |
\( (T^{4} - 128 T^{2} + 40)^{4} \)
|
| $79$ |
\( (T + 4)^{16} \)
|
| $83$ |
\( (T^{4} + 64 T^{2} + 160)^{4} \)
|
| $89$ |
\( (T^{4} - 216 T^{2} + 9720)^{4} \)
|
| $97$ |
\( (T^{4} - 32 T^{2} + 40)^{4} \)
|
| show more | |
| show less |