Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [160,5,Mod(79,160)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(160, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("160.79");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 160.e (of order \(2\), degree \(1\), 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: | \(16.5391940934\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} + 10 x^{18} - 20 x^{16} + 2640 x^{14} + 22400 x^{12} - 652288 x^{10} + 5734400 x^{8} + \cdots + 1099511627776 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{70}\cdot 5^{5} \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} + 10 x^{18} - 20 x^{16} + 2640 x^{14} + 22400 x^{12} - 652288 x^{10} + 5734400 x^{8} + \cdots + 1099511627776 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 1017 \nu^{18} - 36986 \nu^{16} - 456716 \nu^{14} - 2402512 \nu^{12} + \cdots - 43121471651840 ) / 3060164198400 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 513 \nu^{19} + 4288 \nu^{18} - 15882 \nu^{17} + 45952 \nu^{16} + 566292 \nu^{15} + \cdots + 905241667043328 ) / 97925254348800 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3453 \nu^{18} - 2274 \nu^{16} + 457156 \nu^{14} + 7409392 \nu^{12} + \cdots - 100437810216960 ) / 3060164198400 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1863 \nu^{19} - 12474 \nu^{17} + 307060 \nu^{15} + 3181104 \nu^{13} + \cdots + 101928163868672 \nu ) / 97925254348800 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1077 \nu^{18} + 22466 \nu^{16} - 212804 \nu^{14} + 1624272 \nu^{12} + 82138240 \nu^{10} + \cdots + 15418932592640 ) / 765041049600 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1387 \nu^{19} - 4288 \nu^{18} - 15918 \nu^{17} - 45952 \nu^{16} + 1801308 \nu^{15} + \cdots - 905241667043328 ) / 48962627174400 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 24 \nu^{18} - 203 \nu^{16} + 1042 \nu^{14} + 148124 \nu^{12} - 587120 \nu^{10} + \cdots + 1344526090240 ) / 9563013120 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 4579 \nu^{19} - 195234 \nu^{17} + 1282884 \nu^{15} + 21725936 \nu^{13} + \cdots + 511375986130944 \nu ) / 97925254348800 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 7657 \nu^{19} + 17152 \nu^{18} - 99942 \nu^{17} + 183808 \nu^{16} - 2114868 \nu^{15} + \cdots + 36\!\cdots\!12 ) / 97925254348800 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 171 \nu^{19} + 230336 \nu^{18} - 5294 \nu^{17} + 2294144 \nu^{16} + \cdots + 77\!\cdots\!16 ) / 32641751449600 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 5225 \nu^{19} - 87728 \nu^{18} - 60390 \nu^{17} + 496928 \nu^{16} - 2172980 \nu^{15} + \cdots - 55\!\cdots\!88 ) / 48962627174400 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 1045 \nu^{19} - 1872 \nu^{18} - 12078 \nu^{17} - 149792 \nu^{16} - 434596 \nu^{15} + \cdots + 20\!\cdots\!60 ) / 9792525434880 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 10963 \nu^{19} + 4448 \nu^{18} + 104898 \nu^{17} + 6775232 \nu^{16} + \cdots + 14\!\cdots\!68 ) / 97925254348800 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 22439 \nu^{19} - 4288 \nu^{18} - 193914 \nu^{17} - 45952 \nu^{16} - 10390796 \nu^{15} + \cdots - 905241667043328 ) / 97925254348800 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 24235 \nu^{19} - 53952 \nu^{18} + 673070 \nu^{17} + 2477184 \nu^{16} + \cdots - 58\!\cdots\!40 ) / 97925254348800 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 10317 \nu^{19} + 1072 \nu^{18} + 107298 \nu^{17} + 11488 \nu^{16} + 754492 \nu^{15} + \cdots + 226310416760832 ) / 24481313587200 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 15027 \nu^{19} - 108546 \nu^{17} - 1086460 \nu^{15} + 1004016 \nu^{13} + \cdots + 9783935500288 \nu ) / 24481313587200 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 77731 \nu^{19} + 53952 \nu^{18} - 1461538 \nu^{17} - 2477184 \nu^{16} + \cdots + 58\!\cdots\!40 ) / 97925254348800 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 89337 \nu^{19} + 924474 \nu^{17} - 2131060 \nu^{15} + 237586896 \nu^{13} + \cdots + 10\!\cdots\!28 \nu ) / 97925254348800 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{19} - 2\beta_{16} - \beta_{14} + \beta_{9} + \beta_{8} + \beta_{4} - 3\beta_{2} ) / 128 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - 2\beta_{5} + \beta_{3} + 5\beta_{2} + 3\beta _1 - 64 ) / 64 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{19} - 2 \beta_{18} + 2 \beta_{15} + 6 \beta_{14} + 6 \beta_{9} + 10 \beta_{8} + \cdots - 14 \beta_{2} ) / 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 12 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + 28 \beta_{11} + \beta_{10} + 15 \beta_{9} - 15 \beta_{8} + \cdots + 476 ) / 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7 \beta_{19} + 36 \beta_{18} - 16 \beta_{17} - 50 \beta_{16} + 12 \beta_{15} + 59 \beta_{14} + \cdots + 273 \beta_{2} ) / 32 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 10 \beta_{14} - 11 \beta_{13} + 19 \beta_{12} - 50 \beta_{11} - 2 \beta_{10} + 89 \beta_{9} + \cdots - 7554 ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 163 \beta_{19} + 168 \beta_{18} - 240 \beta_{17} + 870 \beta_{16} - 24 \beta_{15} + \cdots - 1519 \beta_{2} ) / 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 928 \beta_{14} - 136 \beta_{13} + 248 \beta_{12} + 1472 \beta_{11} - 137 \beta_{10} - 81 \beta_{9} + \cdots + 27488 ) / 8 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 177 \beta_{19} + 458 \beta_{18} + 2480 \beta_{17} - 1044 \beta_{16} + 70 \beta_{15} + \cdots - 4696 \beta_{2} ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 4356 \beta_{14} - 7002 \beta_{13} + 1146 \beta_{12} + 564 \beta_{11} + 3287 \beta_{10} + \cdots + 1024116 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 18117 \beta_{19} - 7972 \beta_{18} + 10000 \beta_{17} - 7174 \beta_{16} - 140 \beta_{15} + \cdots + 181659 \beta_{2} ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 10906 \beta_{14} + 11203 \beta_{13} + 7221 \beta_{12} + 25794 \beta_{11} + 6968 \beta_{10} + \cdots - 4428974 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 70189 \beta_{19} + 45480 \beta_{18} - 87440 \beta_{17} + 8010 \beta_{16} + 82248 \beta_{15} + \cdots - 624089 \beta_{2} ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 42928 \beta_{14} - 79744 \beta_{13} - 127600 \beta_{12} + 133712 \beta_{11} - 88035 \beta_{10} + \cdots - 45264208 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 505386 \beta_{19} + 811772 \beta_{18} - 1664032 \beta_{17} + 2390000 \beta_{16} + \cdots + 16533708 \beta_{2} \)
|
| \(\nu^{16}\) | \(=\) |
\( 2358824 \beta_{14} + 5816900 \beta_{13} + 828668 \beta_{12} + 270584 \beta_{11} - 666014 \beta_{10} + \cdots - 663272072 \)
|
| \(\nu^{17}\) | \(=\) |
\( 6097438 \beta_{19} - 10532152 \beta_{18} - 11639200 \beta_{17} + 67889660 \beta_{16} + \cdots - 127838670 \beta_{2} \)
|
| \(\nu^{18}\) | \(=\) |
\( 10785872 \beta_{14} - 64146136 \beta_{13} - 18904552 \beta_{12} - 66813328 \beta_{11} + \cdots - 22223001616 \)
|
| \(\nu^{19}\) | \(=\) |
\( 102661428 \beta_{19} - 94971104 \beta_{18} + 719474240 \beta_{17} + 931615896 \beta_{16} + \cdots + 2188601924 \beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(101\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
0 | − | 16.4147i | 0 | −15.0565 | − | 19.9575i | 0 | −3.44277 | 0 | −188.441 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.2 | 0 | − | 16.4147i | 0 | 15.0565 | + | 19.9575i | 0 | 3.44277 | 0 | −188.441 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.3 | 0 | − | 10.5360i | 0 | −24.9415 | + | 1.70938i | 0 | −26.0178 | 0 | −30.0072 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.4 | 0 | − | 10.5360i | 0 | 24.9415 | − | 1.70938i | 0 | 26.0178 | 0 | −30.0072 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.5 | 0 | − | 10.2033i | 0 | −17.9141 | + | 17.4381i | 0 | 84.8600 | 0 | −23.1073 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.6 | 0 | − | 10.2033i | 0 | 17.9141 | − | 17.4381i | 0 | −84.8600 | 0 | −23.1073 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.7 | 0 | − | 9.05347i | 0 | −4.74833 | − | 24.5449i | 0 | 65.1778 | 0 | −0.965298 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.8 | 0 | − | 9.05347i | 0 | 4.74833 | + | 24.5449i | 0 | −65.1778 | 0 | −0.965298 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.9 | 0 | − | 1.21614i | 0 | −13.8839 | + | 20.7903i | 0 | −1.36246 | 0 | 79.5210 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.10 | 0 | − | 1.21614i | 0 | 13.8839 | − | 20.7903i | 0 | 1.36246 | 0 | 79.5210 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.11 | 0 | 1.21614i | 0 | −13.8839 | − | 20.7903i | 0 | −1.36246 | 0 | 79.5210 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.12 | 0 | 1.21614i | 0 | 13.8839 | + | 20.7903i | 0 | 1.36246 | 0 | 79.5210 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.13 | 0 | 9.05347i | 0 | −4.74833 | + | 24.5449i | 0 | 65.1778 | 0 | −0.965298 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.14 | 0 | 9.05347i | 0 | 4.74833 | − | 24.5449i | 0 | −65.1778 | 0 | −0.965298 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.15 | 0 | 10.2033i | 0 | −17.9141 | − | 17.4381i | 0 | 84.8600 | 0 | −23.1073 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.16 | 0 | 10.2033i | 0 | 17.9141 | + | 17.4381i | 0 | −84.8600 | 0 | −23.1073 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.17 | 0 | 10.5360i | 0 | −24.9415 | − | 1.70938i | 0 | −26.0178 | 0 | −30.0072 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.18 | 0 | 10.5360i | 0 | 24.9415 | + | 1.70938i | 0 | 26.0178 | 0 | −30.0072 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.19 | 0 | 16.4147i | 0 | −15.0565 | + | 19.9575i | 0 | −3.44277 | 0 | −188.441 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.20 | 0 | 16.4147i | 0 | 15.0565 | − | 19.9575i | 0 | 3.44277 | 0 | −188.441 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
| 40.e | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 160.5.e.c | 20 | |
| 4.b | odd | 2 | 1 | 40.5.e.c | ✓ | 20 | |
| 5.b | even | 2 | 1 | inner | 160.5.e.c | 20 | |
| 5.c | odd | 4 | 2 | 800.5.g.i | 20 | ||
| 8.b | even | 2 | 1 | 40.5.e.c | ✓ | 20 | |
| 8.d | odd | 2 | 1 | inner | 160.5.e.c | 20 | |
| 20.d | odd | 2 | 1 | 40.5.e.c | ✓ | 20 | |
| 20.e | even | 4 | 2 | 200.5.g.i | 20 | ||
| 40.e | odd | 2 | 1 | inner | 160.5.e.c | 20 | |
| 40.f | even | 2 | 1 | 40.5.e.c | ✓ | 20 | |
| 40.i | odd | 4 | 2 | 200.5.g.i | 20 | ||
| 40.k | even | 4 | 2 | 800.5.g.i | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.5.e.c | ✓ | 20 | 4.b | odd | 2 | 1 | |
| 40.5.e.c | ✓ | 20 | 8.b | even | 2 | 1 | |
| 40.5.e.c | ✓ | 20 | 20.d | odd | 2 | 1 | |
| 40.5.e.c | ✓ | 20 | 40.f | even | 2 | 1 | |
| 160.5.e.c | 20 | 1.a | even | 1 | 1 | trivial | |
| 160.5.e.c | 20 | 5.b | even | 2 | 1 | inner | |
| 160.5.e.c | 20 | 8.d | odd | 2 | 1 | inner | |
| 160.5.e.c | 20 | 40.e | odd | 2 | 1 | inner | |
| 200.5.g.i | 20 | 20.e | even | 4 | 2 | ||
| 200.5.g.i | 20 | 40.i | odd | 4 | 2 | ||
| 800.5.g.i | 20 | 5.c | odd | 4 | 2 | ||
| 800.5.g.i | 20 | 40.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(160, [\chi])\):
|
\( T_{3}^{10} + 568T_{3}^{8} + 110072T_{3}^{6} + 8973408T_{3}^{4} + 268259472T_{3}^{2} + 377478144 \)
|
|
\( T_{7}^{10} - 12140T_{7}^{8} + 38508440T_{7}^{6} - 21234252160T_{7}^{4} + 284733514000T_{7}^{2} - 455625000000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( (T^{10} + 568 T^{8} + \cdots + 377478144)^{2} \)
|
| $5$ |
\( T^{20} + \cdots + 90\!\cdots\!25 \)
|
| $7$ |
\( (T^{10} + \cdots - 455625000000)^{2} \)
|
| $11$ |
\( (T^{5} - 42 T^{4} + \cdots + 14097087072)^{4} \)
|
| $13$ |
\( (T^{10} + \cdots - 87\!\cdots\!00)^{2} \)
|
| $17$ |
\( (T^{10} + \cdots + 17\!\cdots\!64)^{2} \)
|
| $19$ |
\( (T^{5} + 182 T^{4} + \cdots + 73451402848)^{4} \)
|
| $23$ |
\( (T^{10} + \cdots - 12\!\cdots\!00)^{2} \)
|
| $29$ |
\( (T^{10} + \cdots + 57\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{10} + \cdots + 49\!\cdots\!00)^{2} \)
|
| $37$ |
\( (T^{10} + \cdots - 38\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{5} + \cdots - 439327012557312)^{4} \)
|
| $43$ |
\( (T^{10} + \cdots + 25\!\cdots\!04)^{2} \)
|
| $47$ |
\( (T^{10} + \cdots - 23\!\cdots\!00)^{2} \)
|
| $53$ |
\( (T^{10} + \cdots - 43\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{5} + \cdots - 23\!\cdots\!52)^{4} \)
|
| $61$ |
\( (T^{10} + \cdots + 13\!\cdots\!00)^{2} \)
|
| $67$ |
\( (T^{10} + \cdots + 26\!\cdots\!64)^{2} \)
|
| $71$ |
\( (T^{10} + \cdots + 41\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots + 97\!\cdots\!64)^{2} \)
|
| $79$ |
\( (T^{10} + \cdots + 17\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{10} + \cdots + 21\!\cdots\!24)^{2} \)
|
| $89$ |
\( (T^{5} + \cdots - 40\!\cdots\!28)^{4} \)
|
| $97$ |
\( (T^{10} + \cdots + 11\!\cdots\!44)^{2} \)
|
| show more | |
| show less |