Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,6,Mod(1,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, 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("950.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 950.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: | \(152.364628822\) |
| Analytic rank: | \(1\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 4 x^{10} - 1757 x^{9} + 3226 x^{8} + 1058248 x^{7} - 656072 x^{6} - 254660592 x^{5} + \cdots + 2758188563136 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3\cdot 5^{5} \) |
| Twist minimal: | no (minimal twist has level 190) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 4 x^{10} - 1757 x^{9} + 3226 x^{8} + 1058248 x^{7} - 656072 x^{6} - 254660592 x^{5} + \cdots + 2758188563136 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 80\!\cdots\!23 \nu^{10} + \cdots + 77\!\cdots\!96 ) / 50\!\cdots\!60 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 78\!\cdots\!15 \nu^{10} + \cdots + 29\!\cdots\!52 ) / 50\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 31\!\cdots\!13 \nu^{10} + \cdots + 31\!\cdots\!28 ) / 12\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 58\!\cdots\!67 \nu^{10} + \cdots - 23\!\cdots\!92 ) / 16\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 20\!\cdots\!67 \nu^{10} + \cdots - 44\!\cdots\!64 ) / 50\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 70\!\cdots\!15 \nu^{10} + \cdots - 22\!\cdots\!52 ) / 12\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 10\!\cdots\!35 \nu^{10} + \cdots - 37\!\cdots\!52 ) / 16\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 51\!\cdots\!57 \nu^{10} + \cdots - 16\!\cdots\!40 ) / 50\!\cdots\!60 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 29\!\cdots\!83 \nu^{10} + \cdots - 98\!\cdots\!64 ) / 26\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{4} + \beta_{3} + 2\beta _1 + 320 \)
|
| \(\nu^{3}\) | \(=\) |
\( 16 \beta_{10} - 12 \beta_{9} - 4 \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{4} + 10 \beta_{3} + \cdots + 838 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7 \beta_{10} + 76 \beta_{9} - 71 \beta_{8} - 44 \beta_{7} + 774 \beta_{6} - 84 \beta_{5} + 785 \beta_{4} + \cdots + 180564 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13456 \beta_{10} - 9835 \beta_{9} - 1926 \beta_{8} - 1464 \beta_{7} + 662 \beta_{6} - 1098 \beta_{5} + \cdots + 1301464 \)
|
| \(\nu^{6}\) | \(=\) |
\( 19937 \beta_{10} + 63930 \beta_{9} - 76871 \beta_{8} - 49862 \beta_{7} + 566918 \beta_{6} + \cdots + 120242488 \)
|
| \(\nu^{7}\) | \(=\) |
\( 9895860 \beta_{10} - 7122043 \beta_{9} - 666020 \beta_{8} - 1439954 \beta_{7} + 1917336 \beta_{6} + \cdots + 1337189040 \)
|
| \(\nu^{8}\) | \(=\) |
\( 26826565 \beta_{10} + 43558898 \beta_{9} - 60598215 \beta_{8} - 46337076 \beta_{7} + 416997288 \beta_{6} + \cdots + 85009753410 \)
|
| \(\nu^{9}\) | \(=\) |
\( 7091293710 \beta_{10} - 4988825667 \beta_{9} - 99639810 \beta_{8} - 1303258038 \beta_{7} + \cdots + 1203267317112 \)
|
| \(\nu^{10}\) | \(=\) |
\( 28973546793 \beta_{10} + 27371880396 \beta_{9} - 41712204915 \beta_{8} - 40940873136 \beta_{7} + \cdots + 61935850399682 \)
|
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 |
|
4.00000 | −28.2702 | 16.0000 | 0 | −113.081 | 97.4878 | 64.0000 | 556.206 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −27.2428 | 16.0000 | 0 | −108.971 | −252.422 | 64.0000 | 499.170 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −12.8957 | 16.0000 | 0 | −51.5827 | 118.602 | 64.0000 | −76.7013 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | −8.79040 | 16.0000 | 0 | −35.1616 | −22.9562 | 64.0000 | −165.729 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | −8.70979 | 16.0000 | 0 | −34.8392 | 80.6272 | 64.0000 | −167.140 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.00000 | −3.59939 | 16.0000 | 0 | −14.3975 | −110.999 | 64.0000 | −230.044 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.00000 | 9.69248 | 16.0000 | 0 | 38.7699 | 36.6998 | 64.0000 | −149.056 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.00000 | 10.0929 | 16.0000 | 0 | 40.3716 | 181.357 | 64.0000 | −141.133 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.00000 | 20.0050 | 16.0000 | 0 | 80.0200 | −200.402 | 64.0000 | 157.200 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 4.00000 | 20.1054 | 16.0000 | 0 | 80.4215 | −189.825 | 64.0000 | 161.227 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 4.00000 | 25.6125 | 16.0000 | 0 | 102.450 | 21.8301 | 64.0000 | 413.001 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(19\) | \(1\) |
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 | 950.6.a.w | 11 | |
| 5.b | even | 2 | 1 | 950.6.a.v | 11 | ||
| 5.c | odd | 4 | 2 | 190.6.b.a | ✓ | 22 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 190.6.b.a | ✓ | 22 | 5.c | odd | 4 | 2 | |
| 950.6.a.v | 11 | 5.b | even | 2 | 1 | ||
| 950.6.a.w | 11 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{11} + 4 T_{3}^{10} - 1757 T_{3}^{9} - 3226 T_{3}^{8} + 1058248 T_{3}^{7} + 656072 T_{3}^{6} + \cdots - 2758188563136 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(950))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{11} \)
|
| $3$ |
\( T^{11} + \cdots - 2758188563136 \)
|
| $5$ |
\( T^{11} \)
|
| $7$ |
\( T^{11} + \cdots + 33\!\cdots\!52 \)
|
| $11$ |
\( T^{11} + \cdots - 21\!\cdots\!00 \)
|
| $13$ |
\( T^{11} + \cdots + 87\!\cdots\!16 \)
|
| $17$ |
\( T^{11} + \cdots + 37\!\cdots\!68 \)
|
| $19$ |
\( (T + 361)^{11} \)
|
| $23$ |
\( T^{11} + \cdots - 47\!\cdots\!40 \)
|
| $29$ |
\( T^{11} + \cdots + 50\!\cdots\!00 \)
|
| $31$ |
\( T^{11} + \cdots + 65\!\cdots\!40 \)
|
| $37$ |
\( T^{11} + \cdots + 52\!\cdots\!44 \)
|
| $41$ |
\( T^{11} + \cdots + 45\!\cdots\!48 \)
|
| $43$ |
\( T^{11} + \cdots - 19\!\cdots\!04 \)
|
| $47$ |
\( T^{11} + \cdots - 29\!\cdots\!00 \)
|
| $53$ |
\( T^{11} + \cdots - 18\!\cdots\!64 \)
|
| $59$ |
\( T^{11} + \cdots - 33\!\cdots\!00 \)
|
| $61$ |
\( T^{11} + \cdots - 51\!\cdots\!64 \)
|
| $67$ |
\( T^{11} + \cdots + 33\!\cdots\!84 \)
|
| $71$ |
\( T^{11} + \cdots + 41\!\cdots\!00 \)
|
| $73$ |
\( T^{11} + \cdots - 30\!\cdots\!36 \)
|
| $79$ |
\( T^{11} + \cdots - 17\!\cdots\!00 \)
|
| $83$ |
\( T^{11} + \cdots - 22\!\cdots\!64 \)
|
| $89$ |
\( T^{11} + \cdots - 73\!\cdots\!00 \)
|
| $97$ |
\( T^{11} + \cdots + 13\!\cdots\!08 \)
|
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