Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [625,4,Mod(1,625)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(625, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("625.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 625 = 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 625.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: | \(36.8761937536\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 3 x^{13} - 80 x^{12} + 213 x^{11} + 2444 x^{10} - 5595 x^{9} - 35820 x^{8} + 68315 x^{7} + \cdots + 210176 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5^{3} \) |
| Twist minimal: | no (minimal twist has level 25) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 3 x^{13} - 80 x^{12} + 213 x^{11} + 2444 x^{10} - 5595 x^{9} - 35820 x^{8} + 68315 x^{7} + \cdots + 210176 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 131645 \nu^{13} + 760220 \nu^{12} + 8536340 \nu^{11} - 53655029 \nu^{10} + \cdots - 94292674816 ) / 7695851520 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 56226659 \nu^{13} - 1292815844 \nu^{12} - 2456201708 \nu^{11} + 99882380075 \nu^{10} + \cdots - 38664962812160 ) / 1623824670720 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 157525 \nu^{13} - 1759612 \nu^{12} - 5384308 \nu^{11} + 122558029 \nu^{10} + \cdots + 467359559936 ) / 3847925760 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 365285 \nu^{13} + 1995260 \nu^{12} + 25614644 \nu^{11} - 143858525 \nu^{10} + \cdots - 27668619520 ) / 7695851520 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 8589093 \nu^{13} + 75639036 \nu^{12} + 480446004 \nu^{11} - 5440293981 \nu^{10} + \cdots - 7615082170624 ) / 135318722560 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 67280813 \nu^{13} + 196483724 \nu^{12} + 5283401876 \nu^{11} - 13552553669 \nu^{10} + \cdots + 23823249315584 ) / 405956167680 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 322659499 \nu^{13} + 2536521220 \nu^{12} + 19811088652 \nu^{11} - 184455235123 \nu^{10} + \cdots - 97008546336512 ) / 1623824670720 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 196719857 \nu^{13} - 609822860 \nu^{12} - 14855878916 \nu^{11} + 41075779721 \nu^{10} + \cdots + 3431392555264 ) / 405956167680 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 230338853 \nu^{13} + 978296780 \nu^{12} + 16656310868 \nu^{11} - 68279840189 \nu^{10} + \cdots - 8358035911936 ) / 405956167680 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 315612265 \nu^{13} + 1287123564 \nu^{12} + 23786812484 \nu^{11} - 91215002817 \nu^{10} + \cdots + 74488628271872 ) / 541274890240 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 467631811 \nu^{13} - 1773981796 \nu^{12} - 35457385132 \nu^{11} + 125141357419 \nu^{10} + \cdots - 92239117483264 ) / 405956167680 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + 4\beta_{3} + 19\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{13} + \beta_{11} + \beta_{9} + 4 \beta_{8} - 4 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \cdots + 241 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{13} + 6 \beta_{12} + 33 \beta_{11} + 34 \beta_{10} - 33 \beta_{9} - 12 \beta_{8} - 12 \beta_{7} + \cdots + 197 \)
|
| \(\nu^{6}\) | \(=\) |
\( 35 \beta_{13} - 6 \beta_{12} + 44 \beta_{11} - 5 \beta_{10} + 32 \beta_{9} + 128 \beta_{8} - 180 \beta_{7} + \cdots + 5536 \)
|
| \(\nu^{7}\) | \(=\) |
\( 66 \beta_{13} + 270 \beta_{12} + 938 \beta_{11} + 942 \beta_{10} - 924 \beta_{9} - 548 \beta_{8} + \cdots + 6122 \)
|
| \(\nu^{8}\) | \(=\) |
\( 972 \beta_{13} - 288 \beta_{12} + 1552 \beta_{11} - 144 \beta_{10} + 812 \beta_{9} + 3252 \beta_{8} + \cdots + 133492 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2904 \beta_{13} + 9224 \beta_{12} + 25721 \beta_{11} + 24889 \beta_{10} - 24969 \beta_{9} + \cdots + 178797 \)
|
| \(\nu^{10}\) | \(=\) |
\( 25705 \beta_{13} - 8880 \beta_{12} + 49985 \beta_{11} - 1520 \beta_{10} + 18025 \beta_{9} + \cdots + 3294401 \)
|
| \(\nu^{11}\) | \(=\) |
\( 107145 \beta_{13} + 289430 \beta_{12} + 695185 \beta_{11} + 652050 \beta_{10} - 671305 \beta_{9} + \cdots + 5109445 \)
|
| \(\nu^{12}\) | \(=\) |
\( 680675 \beta_{13} - 210790 \beta_{12} + 1525860 \beta_{11} + 63835 \beta_{10} + 341360 \beta_{9} + \cdots + 82522152 \)
|
| \(\nu^{13}\) | \(=\) |
\( 3599250 \beta_{13} + 8773070 \beta_{12} + 18647066 \beta_{11} + 17144966 \beta_{10} - 18112356 \beta_{9} + \cdots + 144687850 \)
|
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 |
|
−5.08946 | 3.82818 | 17.9026 | 0 | −19.4834 | 12.1888 | −50.3987 | −12.3450 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.68373 | −1.46538 | 13.9373 | 0 | 6.86342 | −13.4350 | −27.8086 | −24.8527 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.48124 | −8.53177 | 4.11900 | 0 | 29.7011 | −12.5082 | 13.5107 | 45.7911 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.93263 | 6.78903 | 0.600310 | 0 | −19.9097 | 23.0864 | 21.7005 | 19.0909 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.91416 | −9.37635 | −4.33598 | 0 | 17.9479 | −14.5331 | 23.6131 | 60.9159 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.87476 | 1.60599 | −4.48528 | 0 | −3.01084 | 5.91678 | 23.4069 | −24.4208 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.333645 | 8.44193 | −7.88868 | 0 | 2.81661 | −30.0089 | −5.30118 | 44.2662 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.429681 | 0.0763482 | −7.81537 | 0 | 0.0328054 | 16.1597 | −6.79557 | −26.9942 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.02249 | 2.09955 | −3.90954 | 0 | 4.24633 | −24.6755 | −24.0869 | −22.5919 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.37453 | −6.42978 | −2.36160 | 0 | −15.2677 | −25.3460 | −24.6039 | 14.3421 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.80764 | −1.33129 | −0.117180 | 0 | −3.73779 | 35.1773 | −22.7901 | −25.2277 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 4.59185 | −8.74168 | 13.0851 | 0 | −40.1404 | 20.8866 | 23.3498 | 49.4169 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 5.10453 | 7.76437 | 18.0563 | 0 | 39.6335 | 5.45530 | 51.3326 | 33.2854 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 5.31160 | 6.27085 | 20.2131 | 0 | 33.3083 | 9.63602 | 64.8714 | 12.3235 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 625.4.a.d | 14 | |
| 5.b | even | 2 | 1 | 625.4.a.c | 14 | ||
| 25.d | even | 5 | 2 | 125.4.d.a | 28 | ||
| 25.e | even | 10 | 2 | 25.4.d.a | ✓ | 28 | |
| 25.f | odd | 20 | 4 | 125.4.e.b | 56 | ||
| 75.h | odd | 10 | 2 | 225.4.h.b | 28 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.4.d.a | ✓ | 28 | 25.e | even | 10 | 2 | |
| 125.4.d.a | 28 | 25.d | even | 5 | 2 | ||
| 125.4.e.b | 56 | 25.f | odd | 20 | 4 | ||
| 225.4.h.b | 28 | 75.h | odd | 10 | 2 | ||
| 625.4.a.c | 14 | 5.b | even | 2 | 1 | ||
| 625.4.a.d | 14 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} - 3 T_{2}^{13} - 80 T_{2}^{12} + 213 T_{2}^{11} + 2444 T_{2}^{10} - 5595 T_{2}^{9} + \cdots + 210176 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(625))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} - 3 T^{13} + \cdots + 210176 \)
|
| $3$ |
\( T^{14} - T^{13} + \cdots + 24122944 \)
|
| $5$ |
\( T^{14} \)
|
| $7$ |
\( T^{14} + \cdots + 47\!\cdots\!36 \)
|
| $11$ |
\( T^{14} + \cdots + 38\!\cdots\!64 \)
|
| $13$ |
\( T^{14} + \cdots - 13\!\cdots\!71 \)
|
| $17$ |
\( T^{14} + \cdots + 15\!\cdots\!56 \)
|
| $19$ |
\( T^{14} + \cdots - 10\!\cdots\!00 \)
|
| $23$ |
\( T^{14} + \cdots - 20\!\cdots\!56 \)
|
| $29$ |
\( T^{14} + \cdots - 59\!\cdots\!75 \)
|
| $31$ |
\( T^{14} + \cdots - 48\!\cdots\!56 \)
|
| $37$ |
\( T^{14} + \cdots + 23\!\cdots\!81 \)
|
| $41$ |
\( T^{14} + \cdots + 22\!\cdots\!84 \)
|
| $43$ |
\( T^{14} + \cdots - 31\!\cdots\!76 \)
|
| $47$ |
\( T^{14} + \cdots + 14\!\cdots\!36 \)
|
| $53$ |
\( T^{14} + \cdots + 69\!\cdots\!24 \)
|
| $59$ |
\( T^{14} + \cdots + 33\!\cdots\!00 \)
|
| $61$ |
\( T^{14} + \cdots + 11\!\cdots\!69 \)
|
| $67$ |
\( T^{14} + \cdots - 51\!\cdots\!04 \)
|
| $71$ |
\( T^{14} + \cdots + 89\!\cdots\!84 \)
|
| $73$ |
\( T^{14} + \cdots + 16\!\cdots\!24 \)
|
| $79$ |
\( T^{14} + \cdots + 10\!\cdots\!00 \)
|
| $83$ |
\( T^{14} + \cdots - 72\!\cdots\!76 \)
|
| $89$ |
\( T^{14} + \cdots - 86\!\cdots\!00 \)
|
| $97$ |
\( T^{14} + \cdots - 24\!\cdots\!09 \)
|
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