Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1075,6,Mod(1,1075)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1075.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1075 = 5^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1075.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: | \(172.412606299\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 43) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 18033 \nu^{9} - 1188163 \nu^{8} + 3451179 \nu^{7} + 239748231 \nu^{6} - 1005840413 \nu^{5} + \cdots - 40045363612 ) / 24633272448 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 24107 \nu^{9} - 594245 \nu^{8} + 7832247 \nu^{7} + 116173337 \nu^{6} - 397671097 \nu^{5} + \cdots - 192111505364 ) / 24633272448 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 144543 \nu^{9} - 820999 \nu^{8} + 35600547 \nu^{7} + 213763899 \nu^{6} + \cdots - 912070606876 ) / 24633272448 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 474917 \nu^{9} + 1144977 \nu^{8} - 116809809 \nu^{7} - 421392509 \nu^{6} + \cdots + 2423789997252 ) / 65688726528 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 474917 \nu^{9} - 1144977 \nu^{8} + 116809809 \nu^{7} + 421392509 \nu^{6} + \cdots - 5839603776708 ) / 65688726528 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 808641 \nu^{9} - 7898035 \nu^{8} - 155012637 \nu^{7} + 1407118071 \nu^{6} + \cdots - 7896805510540 ) / 98533089792 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 5132245 \nu^{9} + 307343 \nu^{8} + 1269871233 \nu^{7} + 1411070749 \nu^{6} + \cdots - 30463422181444 ) / 197066179584 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 2389487 \nu^{9} - 2695583 \nu^{8} + 612520467 \nu^{7} + 1209002243 \nu^{6} + \cdots - 18517909728476 ) / 49266544896 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta _1 + 52 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{9} - 5\beta_{8} - 2\beta_{7} + 3\beta_{6} - \beta_{4} + \beta_{3} + 87\beta _1 + 56 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{8} - 16\beta_{7} + 126\beta_{6} + 113\beta_{5} - 7\beta_{4} + 13\beta_{3} + 20\beta_{2} + 245\beta _1 + 4542 \)
|
| \(\nu^{5}\) | \(=\) |
\( 278 \beta_{9} - 716 \beta_{8} - 308 \beta_{7} + 608 \beta_{6} + 92 \beta_{5} - 226 \beta_{4} + \cdots + 13392 \)
|
| \(\nu^{6}\) | \(=\) |
\( 202 \beta_{9} - 1946 \beta_{8} - 2948 \beta_{7} + 16013 \beta_{6} + 11839 \beta_{5} - 2292 \beta_{4} + \cdots + 465882 \)
|
| \(\nu^{7}\) | \(=\) |
\( 34356 \beta_{9} - 90155 \beta_{8} - 42446 \beta_{7} + 97339 \beta_{6} + 22646 \beta_{5} - 41129 \beta_{4} + \cdots + 2212914 \)
|
| \(\nu^{8}\) | \(=\) |
\( 68250 \beta_{9} - 383649 \beta_{8} - 450492 \beta_{7} + 2030502 \beta_{6} + 1254459 \beta_{5} + \cdots + 51485248 \)
|
| \(\nu^{9}\) | \(=\) |
\( 4138872 \beta_{9} - 11178606 \beta_{8} - 5738016 \beta_{7} + 14588868 \beta_{6} + 3971370 \beta_{5} + \cdots + 324019770 \)
|
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 |
|
−10.7063 | −18.3440 | 82.6251 | 0 | 196.397 | 96.4803 | −542.008 | 93.5034 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.57770 | 7.84343 | 59.7324 | 0 | −75.1221 | −195.604 | −265.613 | −181.481 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −7.91219 | −12.8799 | 30.6028 | 0 | 101.908 | 172.354 | 11.0549 | −77.1083 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −5.38824 | −25.0462 | −2.96684 | 0 | 134.955 | −166.517 | 188.410 | 384.312 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.48720 | 27.5943 | −25.8138 | 0 | −68.6327 | −15.7005 | 143.795 | 518.448 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.86024 | 14.8716 | −28.5395 | 0 | 27.6647 | −202.971 | −112.618 | −21.8357 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.50018 | −16.8892 | −25.7491 | 0 | −42.2260 | −67.4603 | −144.383 | 42.2439 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.31531 | 23.8469 | −13.3781 | 0 | 102.907 | 174.859 | −195.821 | 325.673 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 8.86547 | −1.50169 | 46.5966 | 0 | −13.3132 | 124.747 | 129.406 | −240.745 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 10.5305 | −27.4953 | 78.8905 | 0 | −289.538 | 19.8137 | 493.778 | 512.989 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(43\) | \(1\) |
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 | 1075.6.a.b | 10 | |
| 5.b | even | 2 | 1 | 43.6.a.b | ✓ | 10 | |
| 15.d | odd | 2 | 1 | 387.6.a.e | 10 | ||
| 20.d | odd | 2 | 1 | 688.6.a.h | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 43.6.a.b | ✓ | 10 | 5.b | even | 2 | 1 | |
| 387.6.a.e | 10 | 15.d | odd | 2 | 1 | ||
| 688.6.a.h | 10 | 20.d | odd | 2 | 1 | ||
| 1075.6.a.b | 10 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 8 T_{2}^{9} - 229 T_{2}^{8} - 1734 T_{2}^{7} + 16722 T_{2}^{6} + 112716 T_{2}^{5} + \cdots - 20373120 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1075))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 8 T^{9} + \cdots - 20373120 \)
|
| $3$ |
\( T^{10} + \cdots + 316744901280 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + \cdots - 50\!\cdots\!40 \)
|
| $11$ |
\( T^{10} + \cdots + 46\!\cdots\!72 \)
|
| $13$ |
\( T^{10} + \cdots + 47\!\cdots\!40 \)
|
| $17$ |
\( T^{10} + \cdots - 49\!\cdots\!66 \)
|
| $19$ |
\( T^{10} + \cdots - 10\!\cdots\!16 \)
|
| $23$ |
\( T^{10} + \cdots - 24\!\cdots\!52 \)
|
| $29$ |
\( T^{10} + \cdots + 29\!\cdots\!36 \)
|
| $31$ |
\( T^{10} + \cdots - 47\!\cdots\!36 \)
|
| $37$ |
\( T^{10} + \cdots - 25\!\cdots\!00 \)
|
| $41$ |
\( T^{10} + \cdots - 22\!\cdots\!50 \)
|
| $43$ |
\( (T + 1849)^{10} \)
|
| $47$ |
\( T^{10} + \cdots + 17\!\cdots\!00 \)
|
| $53$ |
\( T^{10} + \cdots - 56\!\cdots\!00 \)
|
| $59$ |
\( T^{10} + \cdots - 13\!\cdots\!24 \)
|
| $61$ |
\( T^{10} + \cdots - 14\!\cdots\!68 \)
|
| $67$ |
\( T^{10} + \cdots + 89\!\cdots\!92 \)
|
| $71$ |
\( T^{10} + \cdots + 15\!\cdots\!32 \)
|
| $73$ |
\( T^{10} + \cdots - 95\!\cdots\!32 \)
|
| $79$ |
\( T^{10} + \cdots - 15\!\cdots\!20 \)
|
| $83$ |
\( T^{10} + \cdots + 13\!\cdots\!40 \)
|
| $89$ |
\( T^{10} + \cdots + 29\!\cdots\!88 \)
|
| $97$ |
\( T^{10} + \cdots - 18\!\cdots\!58 \)
|
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