Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2475,4,Mod(1,2475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2475.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2475.a (trivial) |
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: | yes |
| Analytic conductor: | \(146.029727264\) |
| 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: |
\( x^{16} - 103 x^{14} + 4248 x^{12} - 89496 x^{10} + 1015487 x^{8} - 5956953 x^{6} + 15313728 x^{4} + \cdots + 861184 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 495) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{16} - 103 x^{14} + 4248 x^{12} - 89496 x^{10} + 1015487 x^{8} - 5956953 x^{6} + 15313728 x^{4} + \cdots + 861184 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 21\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 705 \nu^{14} - 66773 \nu^{12} + 2494984 \nu^{10} - 46466048 \nu^{8} + 446257439 \nu^{6} + \cdots - 1157200224 ) / 32676512 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1274 \nu^{14} - 126327 \nu^{12} + 4826530 \nu^{10} - 88431958 \nu^{8} + 781901728 \nu^{6} + \cdots + 995416640 ) / 32676512 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5713 \nu^{14} - 534646 \nu^{12} + 19096082 \nu^{10} - 323388550 \nu^{8} + 2627255209 \nu^{6} + \cdots - 10188624512 ) / 130706048 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5713 \nu^{14} - 534646 \nu^{12} + 19096082 \nu^{10} - 323388550 \nu^{8} + 2627255209 \nu^{6} + \cdots + 659977472 ) / 130706048 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 4926 \nu^{14} + 426661 \nu^{12} - 13578378 \nu^{10} + 185973318 \nu^{8} - 834779380 \nu^{6} + \cdots - 6063288256 ) / 65353024 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 5386 \nu^{14} - 508415 \nu^{12} + 18679902 \nu^{10} - 336634802 \nu^{8} + 3081286060 \nu^{6} + \cdots - 5103373312 ) / 65353024 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 21138 \nu^{15} - 2474899 \nu^{13} + 119115486 \nu^{11} - 2986471498 \nu^{9} + \cdots - 348660175424 \nu ) / 3790475392 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 1628 \nu^{15} - 181977 \nu^{13} + 8176126 \nu^{11} - 188381418 \nu^{9} + 2349346418 \nu^{7} + \cdots - 33299387200 \nu ) / 130706048 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 57002 \nu^{15} - 5705529 \nu^{13} + 226639762 \nu^{11} - 4547664614 \nu^{9} + \cdots - 243711549056 \nu ) / 3790475392 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 44701 \nu^{15} - 4305155 \nu^{13} + 162772644 \nu^{11} - 3065070576 \nu^{9} + \cdots - 351980851136 \nu ) / 1895237696 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 124183 \nu^{15} + 11733625 \nu^{13} - 428495776 \nu^{11} + 7544310280 \nu^{9} + \cdots - 324791247616 \nu ) / 3790475392 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 262875 \nu^{15} + 27562252 \nu^{13} - 1159302874 \nu^{11} + 24910690550 \nu^{9} + \cdots + 1985419610432 \nu ) / 3790475392 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 21\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 27\beta_{2} + 268 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{15} - \beta_{14} - 2\beta_{13} + 4\beta_{12} + \beta_{11} + 2\beta_{10} + 37\beta_{3} + 492\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -5\beta_{9} + \beta_{8} + 44\beta_{7} - 36\beta_{6} - 2\beta_{5} + 10\beta_{4} + 699\beta_{2} + 6176 \)
|
| \(\nu^{7}\) | \(=\) |
\( 53 \beta_{15} - 61 \beta_{14} - 90 \beta_{13} + 154 \beta_{12} + 69 \beta_{11} + 112 \beta_{10} + \cdots + 12146 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -289\beta_{9} + 45\beta_{8} + 1475\beta_{7} - 1067\beta_{6} - 86\beta_{5} + 590\beta_{4} + 18169\beta_{2} + 150697 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2020 \beta_{15} - 2620 \beta_{14} - 3128 \beta_{13} + 4334 \beta_{12} + 3020 \beta_{11} + \cdots + 309211 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 11588 \beta_{9} + 1508 \beta_{8} + 45124 \beta_{7} - 30196 \beta_{6} - 2812 \beta_{5} + \cdots + 3804913 \)
|
| \(\nu^{11}\) | \(=\) |
\( 67988 \beta_{15} - 96180 \beta_{14} - 98904 \beta_{13} + 108280 \beta_{12} + 110116 \beta_{11} + \cdots + 8034937 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 401060 \beta_{9} + 44852 \beta_{8} + 1328017 \beta_{7} - 846257 \beta_{6} - 87176 \beta_{5} + \cdots + 98267608 \)
|
| \(\nu^{13}\) | \(=\) |
\( 2153397 \beta_{15} - 3237749 \beta_{14} - 2981450 \beta_{13} + 2545996 \beta_{12} + \cdots + 211955216 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 12858873 \beta_{9} + 1244229 \beta_{8} + 38368228 \beta_{7} - 23741948 \beta_{6} - 2707338 \beta_{5} + \cdots + 2579782132 \)
|
| \(\nu^{15}\) | \(=\) |
\( 65897505 \beta_{15} - 103403561 \beta_{14} - 87390162 \beta_{13} + 57586338 \beta_{12} + \cdots + 5657453086 \beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.31333 | 0 | 20.2314 | 0 | 0 | −15.6271 | −64.9896 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.85092 | 0 | 15.5314 | 0 | 0 | −24.5618 | −36.5345 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.59829 | 0 | 13.1443 | 0 | 0 | 8.59388 | −23.6550 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.54699 | 0 | 4.58114 | 0 | 0 | 35.4720 | 12.1266 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.37022 | 0 | 3.35835 | 0 | 0 | 5.87458 | 15.6434 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.28277 | 0 | −2.78895 | 0 | 0 | −11.2428 | 24.6287 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.919139 | 0 | −7.15518 | 0 | 0 | −26.1261 | 13.9297 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.312176 | 0 | −7.90255 | 0 | 0 | 15.1298 | 4.96439 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.312176 | 0 | −7.90255 | 0 | 0 | −15.1298 | −4.96439 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.919139 | 0 | −7.15518 | 0 | 0 | 26.1261 | −13.9297 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.28277 | 0 | −2.78895 | 0 | 0 | 11.2428 | −24.6287 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 3.37022 | 0 | 3.35835 | 0 | 0 | −5.87458 | −15.6434 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 3.54699 | 0 | 4.58114 | 0 | 0 | −35.4720 | −12.1266 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 4.59829 | 0 | 13.1443 | 0 | 0 | −8.59388 | 23.6550 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 4.85092 | 0 | 15.5314 | 0 | 0 | 24.5618 | 36.5345 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 5.31333 | 0 | 20.2314 | 0 | 0 | 15.6271 | 64.9896 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2475.4.a.ca | 16 | |
| 3.b | odd | 2 | 1 | 2475.4.a.bz | 16 | ||
| 5.b | even | 2 | 1 | inner | 2475.4.a.ca | 16 | |
| 5.c | odd | 4 | 2 | 495.4.c.f | yes | 16 | |
| 15.d | odd | 2 | 1 | 2475.4.a.bz | 16 | ||
| 15.e | even | 4 | 2 | 495.4.c.e | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 495.4.c.e | ✓ | 16 | 15.e | even | 4 | 2 | |
| 495.4.c.f | yes | 16 | 5.c | odd | 4 | 2 | |
| 2475.4.a.bz | 16 | 3.b | odd | 2 | 1 | ||
| 2475.4.a.bz | 16 | 15.d | odd | 2 | 1 | ||
| 2475.4.a.ca | 16 | 1.a | even | 1 | 1 | trivial | |
| 2475.4.a.ca | 16 | 5.b | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):
|
\( T_{2}^{16} - 103 T_{2}^{14} + 4248 T_{2}^{12} - 89496 T_{2}^{10} + 1015487 T_{2}^{8} - 5956953 T_{2}^{6} + \cdots + 861184 \)
|
|
\( T_{7}^{16} - 3252 T_{7}^{14} + 4013884 T_{7}^{12} - 2442206112 T_{7}^{10} + 793486410480 T_{7}^{8} + \cdots + 93\!\cdots\!56 \)
|
|
\( T_{29}^{8} - 326 T_{29}^{7} - 92952 T_{29}^{6} + 36589120 T_{29}^{5} - 131527344 T_{29}^{4} + \cdots - 29\!\cdots\!36 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 103 T^{14} + \cdots + 861184 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + \cdots + 93\!\cdots\!56 \)
|
| $11$ |
\( (T - 11)^{16} \)
|
| $13$ |
\( T^{16} + \cdots + 19\!\cdots\!96 \)
|
| $17$ |
\( T^{16} + \cdots + 12\!\cdots\!64 \)
|
| $19$ |
\( (T^{8} + \cdots - 271885102112768)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 18\!\cdots\!04 \)
|
| $29$ |
\( (T^{8} + \cdots - 29\!\cdots\!36)^{2} \)
|
| $31$ |
\( (T^{8} + \cdots + 40\!\cdots\!16)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 10\!\cdots\!56 \)
|
| $41$ |
\( (T^{8} + \cdots - 99\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 38\!\cdots\!56 \)
|
| $47$ |
\( T^{16} + \cdots + 37\!\cdots\!84 \)
|
| $53$ |
\( T^{16} + \cdots + 51\!\cdots\!00 \)
|
| $59$ |
\( (T^{8} + \cdots + 35\!\cdots\!52)^{2} \)
|
| $61$ |
\( (T^{8} + \cdots - 14\!\cdots\!44)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 89\!\cdots\!96 \)
|
| $71$ |
\( (T^{8} + \cdots - 77\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 66\!\cdots\!64 \)
|
| $79$ |
\( (T^{8} + \cdots - 14\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 52\!\cdots\!00 \)
|
| $89$ |
\( (T^{8} + \cdots + 63\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 83\!\cdots\!44 \)
|
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