Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2600,2,Mod(2001,2600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2600, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2600.2001");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2600.k (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: | \(20.7611045255\) |
| 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: |
\( x^{20} + 60x^{16} + 1134x^{12} + 6924x^{8} + 3545x^{4} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{30} \) |
| Twist minimal: | no (minimal twist has level 520) |
| 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
\( x^{20} + 60x^{16} + 1134x^{12} + 6924x^{8} + 3545x^{4} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 411\nu^{16} + 25181\nu^{12} + 501937\nu^{8} + 3176491\nu^{4} - 1179284 ) / 810196 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1102\nu^{18} - 59139\nu^{14} - 881097\nu^{10} - 1813185\nu^{6} + 22790499\nu^{2} ) / 1620392 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1233\nu^{16} + 75543\nu^{12} + 1505811\nu^{8} + 10339669\nu^{4} + 6994696 ) / 810196 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4055\nu^{16} + 246469\nu^{12} + 4699877\nu^{8} + 28189727\nu^{4} + 5860080 ) / 1620392 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5971\nu^{16} + 344145\nu^{12} + 6136961\nu^{8} + 34182331\nu^{4} + 8413200 ) / 1620392 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 788 \nu^{19} + 2369 \nu^{17} + 50743 \nu^{15} + 134301 \nu^{13} + 1075207 \nu^{11} + \cdots - 1721336 \nu ) / 3240784 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5767\nu^{18} + 346431\nu^{14} + 6564959\nu^{10} + 40432645\nu^{6} + 24430702\nu^{2} ) / 1620392 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 5521 \nu^{19} - 6013 \nu^{17} + 337273 \nu^{15} - 355589 \nu^{13} + 6616403 \nu^{11} + \cdots + 6834912 \nu ) / 3240784 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -14212\nu^{18} - 853487\nu^{14} - 16161429\nu^{10} - 99088269\nu^{6} - 51538947\nu^{2} ) / 1620392 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 18335\nu^{18} + 1099194\nu^{14} + 20718640\nu^{10} + 125194558\nu^{6} + 53140121\nu^{2} ) / 1620392 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 16267 \nu^{19} + 3644 \nu^{17} + 979392 \nu^{15} + 221288 \nu^{13} + 18671114 \nu^{11} + \cdots - 11595144 \nu ) / 3240784 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 41403 \nu^{19} - 4466 \nu^{17} + 2484918 \nu^{15} - 271650 \nu^{13} + 46978476 \nu^{11} + \cdots - 20074520 \nu ) / 6481568 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 42979 \nu^{19} - 272 \nu^{17} + 2586404 \nu^{15} + 3048 \nu^{13} + 49128890 \nu^{11} + \cdots + 23517192 \nu ) / 6481568 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 39827 \nu^{19} + 9204 \nu^{17} + 2383432 \nu^{15} + 540252 \nu^{13} + 44828062 \nu^{11} + \cdots + 16631848 \nu ) / 6481568 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 37526 \nu^{19} + 21492 \nu^{18} + 7657 \nu^{17} - 2248369 \nu^{15} + 1284238 \nu^{14} + \cdots + 15994616 \nu ) / 3240784 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 37526 \nu^{19} + 21492 \nu^{18} - 7657 \nu^{17} + 2248369 \nu^{15} + 1284238 \nu^{14} + \cdots - 15994616 \nu ) / 3240784 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 69039 \nu^{19} + 9301 \nu^{17} - 4141149 \nu^{15} + 557037 \nu^{13} - 78226127 \nu^{11} + \cdots + 32342576 \nu ) / 3240784 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 80573 \nu^{19} - 9301 \nu^{17} - 4834011 \nu^{15} - 557037 \nu^{13} - 91356045 \nu^{11} + \cdots - 38824144 \nu ) / 3240784 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 179481 \nu^{19} - 23068 \nu^{17} - 10767216 \nu^{15} - 1385724 \nu^{13} - 203430730 \nu^{11} + \cdots - 84759672 \nu ) / 6481568 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{19} - 2\beta_{18} - \beta_{17} - \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} - \beta_{11} - \beta_{6} ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{16} - \beta_{15} + \beta_{9} + 6\beta_{7} - \beta_{2} ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8 \beta_{19} - 8 \beta_{18} + 5 \beta_{17} + 5 \beta_{16} - 5 \beta_{15} + \beta_{14} + \cdots + 5 \beta_{6} ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} - 3\beta _1 - 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 34 \beta_{19} + 38 \beta_{18} + 21 \beta_{17} + 25 \beta_{16} - 25 \beta_{15} + 7 \beta_{14} + \cdots + 29 \beta_{6} ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 33\beta_{16} + 33\beta_{15} + 4\beta_{10} + 3\beta_{9} - 122\beta_{7} + 33\beta_{2} ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 152 \beta_{19} + 188 \beta_{18} - 89 \beta_{17} - 125 \beta_{16} + 125 \beta_{15} + \cdots - 161 \beta_{6} ) / 8 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{5} - 11\beta_{4} - 14\beta_{3} + 89\beta _1 + 285 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 706 \beta_{19} - 954 \beta_{18} - 385 \beta_{17} - 633 \beta_{16} + 633 \beta_{15} + \cdots - 865 \beta_{6} ) / 8 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( -925\beta_{16} - 925\beta_{15} - 344\beta_{10} - 563\beta_{9} + 2966\beta_{7} - 981\beta_{2} ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 3384 \beta_{19} - 4928 \beta_{18} + 1701 \beta_{17} + 3245 \beta_{16} - 3245 \beta_{15} + \cdots + 4565 \beta_{6} ) / 8 \)
|
| \(\nu^{12}\) | \(=\) |
\( -128\beta_{5} + 586\beta_{4} + 194\beta_{3} - 2543\beta _1 - 6831 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 16642 \beta_{19} + 25774 \beta_{18} + 7669 \beta_{17} + 16801 \beta_{16} - 16801 \beta_{15} + \cdots + 23885 \beta_{6} ) / 8 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 25173\beta_{16} + 25173\beta_{15} + 14844\beta_{10} + 23375\beta_{9} - 77850\beta_{7} + 29053\beta_{2} ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 83528 \beta_{19} + 135956 \beta_{18} - 35241 \beta_{17} - 87669 \beta_{16} + 87669 \beta_{15} + \cdots - 124577 \beta_{6} ) / 8 \)
|
| \(\nu^{16}\) | \(=\) |
\( 6621\beta_{5} - 22469\beta_{4} - 2517\beta_{3} + 72269\beta _1 + 173803 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 425922 \beta_{19} - 721346 \beta_{18} - 164849 \beta_{17} - 460273 \beta_{16} + 460273 \beta_{15} + \cdots - 649761 \beta_{6} ) / 8 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 686313 \beta_{16} - 686313 \beta_{15} - 528144 \beta_{10} - 788535 \beta_{9} + \cdots - 855641 \beta_{2} ) / 4 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 2198312 \beta_{19} - 3842488 \beta_{18} + 783973 \beta_{17} + 2428149 \beta_{16} + \cdots + 3395013 \beta_{6} ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times\).
| \(n\) | \(1301\) | \(1601\) | \(1951\) | \(1977\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2001.1 |
|
0 | −3.22659 | 0 | 0 | 0 | − | 1.65488i | 0 | 7.41088 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.2 | 0 | −3.22659 | 0 | 0 | 0 | 1.65488i | 0 | 7.41088 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.3 | 0 | −2.14926 | 0 | 0 | 0 | − | 2.12171i | 0 | 1.61930 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.4 | 0 | −2.14926 | 0 | 0 | 0 | 2.12171i | 0 | 1.61930 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.5 | 0 | −1.57201 | 0 | 0 | 0 | − | 4.19743i | 0 | −0.528799 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.6 | 0 | −1.57201 | 0 | 0 | 0 | 4.19743i | 0 | −0.528799 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.7 | 0 | −0.949078 | 0 | 0 | 0 | − | 3.85660i | 0 | −2.09925 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.8 | 0 | −0.949078 | 0 | 0 | 0 | 3.85660i | 0 | −2.09925 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.9 | 0 | −0.773218 | 0 | 0 | 0 | − | 1.12600i | 0 | −2.40213 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.10 | 0 | −0.773218 | 0 | 0 | 0 | 1.12600i | 0 | −2.40213 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.11 | 0 | 0.773218 | 0 | 0 | 0 | − | 1.12600i | 0 | −2.40213 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.12 | 0 | 0.773218 | 0 | 0 | 0 | 1.12600i | 0 | −2.40213 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.13 | 0 | 0.949078 | 0 | 0 | 0 | − | 3.85660i | 0 | −2.09925 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.14 | 0 | 0.949078 | 0 | 0 | 0 | 3.85660i | 0 | −2.09925 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.15 | 0 | 1.57201 | 0 | 0 | 0 | − | 4.19743i | 0 | −0.528799 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.16 | 0 | 1.57201 | 0 | 0 | 0 | 4.19743i | 0 | −0.528799 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.17 | 0 | 2.14926 | 0 | 0 | 0 | − | 2.12171i | 0 | 1.61930 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.18 | 0 | 2.14926 | 0 | 0 | 0 | 2.12171i | 0 | 1.61930 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.19 | 0 | 3.22659 | 0 | 0 | 0 | − | 1.65488i | 0 | 7.41088 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2001.20 | 0 | 3.22659 | 0 | 0 | 0 | 1.65488i | 0 | 7.41088 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 65.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2600.2.k.f | 20 | |
| 5.b | even | 2 | 1 | inner | 2600.2.k.f | 20 | |
| 5.c | odd | 4 | 1 | 520.2.f.a | ✓ | 10 | |
| 5.c | odd | 4 | 1 | 520.2.f.b | yes | 10 | |
| 13.b | even | 2 | 1 | inner | 2600.2.k.f | 20 | |
| 20.e | even | 4 | 1 | 1040.2.f.f | 10 | ||
| 20.e | even | 4 | 1 | 1040.2.f.g | 10 | ||
| 65.d | even | 2 | 1 | inner | 2600.2.k.f | 20 | |
| 65.h | odd | 4 | 1 | 520.2.f.a | ✓ | 10 | |
| 65.h | odd | 4 | 1 | 520.2.f.b | yes | 10 | |
| 260.p | even | 4 | 1 | 1040.2.f.f | 10 | ||
| 260.p | even | 4 | 1 | 1040.2.f.g | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 520.2.f.a | ✓ | 10 | 5.c | odd | 4 | 1 | |
| 520.2.f.a | ✓ | 10 | 65.h | odd | 4 | 1 | |
| 520.2.f.b | yes | 10 | 5.c | odd | 4 | 1 | |
| 520.2.f.b | yes | 10 | 65.h | odd | 4 | 1 | |
| 1040.2.f.f | 10 | 20.e | even | 4 | 1 | ||
| 1040.2.f.f | 10 | 260.p | even | 4 | 1 | ||
| 1040.2.f.g | 10 | 20.e | even | 4 | 1 | ||
| 1040.2.f.g | 10 | 260.p | even | 4 | 1 | ||
| 2600.2.k.f | 20 | 1.a | even | 1 | 1 | trivial | |
| 2600.2.k.f | 20 | 5.b | even | 2 | 1 | inner | |
| 2600.2.k.f | 20 | 13.b | even | 2 | 1 | inner | |
| 2600.2.k.f | 20 | 65.d | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 19T_{3}^{8} + 112T_{3}^{6} - 256T_{3}^{4} + 224T_{3}^{2} - 64 \)
acting on \(S_{2}^{\mathrm{new}}(2600, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( (T^{10} - 19 T^{8} + \cdots - 64)^{2} \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( (T^{10} + 41 T^{8} + \cdots + 4096)^{2} \)
|
| $11$ |
\( (T^{10} + 64 T^{8} + \cdots + 1024)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 137858491849 \)
|
| $17$ |
\( (T^{10} - 107 T^{8} + \cdots - 262144)^{2} \)
|
| $19$ |
\( (T^{10} + 92 T^{8} + \cdots + 256)^{2} \)
|
| $23$ |
\( (T^{10} - 44 T^{8} + \cdots - 256)^{2} \)
|
| $29$ |
\( (T^{5} + 4 T^{4} - 56 T^{3} + \cdots + 64)^{4} \)
|
| $31$ |
\( (T^{10} + 108 T^{8} + \cdots + 256)^{2} \)
|
| $37$ |
\( (T^{10} + 65 T^{8} + \cdots + 256)^{2} \)
|
| $41$ |
\( (T^{10} + 208 T^{8} + \cdots + 4194304)^{2} \)
|
| $43$ |
\( (T^{10} - 211 T^{8} + \cdots - 40246336)^{2} \)
|
| $47$ |
\( (T^{10} + 41 T^{8} + \cdots + 4096)^{2} \)
|
| $53$ |
\( (T^{10} - 364 T^{8} + \cdots - 294191104)^{2} \)
|
| $59$ |
\( (T^{10} + 316 T^{8} + \cdots + 59228416)^{2} \)
|
| $61$ |
\( (T^{5} + 8 T^{4} + \cdots + 128)^{4} \)
|
| $67$ |
\( (T^{10} + 224 T^{8} + \cdots + 110166016)^{2} \)
|
| $71$ |
\( (T^{10} + 475 T^{8} + \cdots + 583319104)^{2} \)
|
| $73$ |
\( (T^{10} + 212 T^{8} + \cdots + 45589504)^{2} \)
|
| $79$ |
\( (T^{5} + 14 T^{4} + \cdots + 83968)^{4} \)
|
| $83$ |
\( (T^{10} + 628 T^{8} + \cdots + 17618845696)^{2} \)
|
| $89$ |
\( (T^{10} + 524 T^{8} + \cdots + 16777216)^{2} \)
|
| $97$ |
\( (T^{10} + 464 T^{8} + \cdots + 22429696)^{2} \)
|
| show more | |
| show less |