Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1098,2,Mod(217,1098)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1098, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1098.217");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1098 = 2 \cdot 3^{2} \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1098.k (of order \(5\), degree \(4\), 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: | \(8.76757414194\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{11} + 11 x^{10} + 10 x^{9} + 34 x^{8} - 107 x^{7} + 287 x^{6} - 358 x^{5} + 1201 x^{4} + \cdots + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 122) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - x^{11} + 11 x^{10} + 10 x^{9} + 34 x^{8} - 107 x^{7} + 287 x^{6} - 358 x^{5} + 1201 x^{4} + \cdots + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 21153490778 \nu^{11} - 6193527277 \nu^{10} - 155437853560 \nu^{9} + \cdots - 64805677621815 ) / 93357540306509 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 9331785129676 \nu^{11} + 1254734164555 \nu^{10} - 94082739621734 \nu^{9} + \cdots + 433528644174732 ) / 25\!\cdots\!43 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4893899876987 \nu^{11} + 4657702459307 \nu^{10} - 51959464984000 \nu^{9} + \cdots + 559864730096370 ) / 840217862758581 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17008035049102 \nu^{11} - 5604089434468 \nu^{10} + 182289088591328 \nu^{9} + \cdots - 19\!\cdots\!95 ) / 25\!\cdots\!43 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 6325450483821 \nu^{11} + 20858009306156 \nu^{10} - 73466071098077 \nu^{9} + \cdots + 64\!\cdots\!35 ) / 840217862758581 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 20735730744310 \nu^{11} - 6054031113349 \nu^{10} + 214119930809489 \nu^{9} + \cdots + 709474105123749 ) / 25\!\cdots\!43 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 20735730744310 \nu^{11} - 6054031113349 \nu^{10} + 214119930809489 \nu^{9} + \cdots + 709474105123749 ) / 25\!\cdots\!43 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 21601892540605 \nu^{11} - 22173036791611 \nu^{10} + 237453592710176 \nu^{9} + \cdots - 56\!\cdots\!79 ) / 25\!\cdots\!43 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 3842550010997 \nu^{11} - 1071610021918 \nu^{10} + 42485843012446 \nu^{9} + \cdots + 137335017680154 ) / 280072620919527 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 5221911531233 \nu^{11} + 5377144423103 \nu^{10} - 58265748642134 \nu^{9} + \cdots + 24\!\cdots\!60 ) / 280072620919527 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 76308497076955 \nu^{11} - 22344450153802 \nu^{10} + 792989615101115 \nu^{9} + \cdots + 26\!\cdots\!95 ) / 25\!\cdots\!43 \)
|
| \(\nu\) | \(=\) |
\( -\beta_{7} + \beta_{6} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} - \beta_{8} - \beta_{7} - \beta_{5} + 5\beta_{4} + \beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{11} + \beta_{10} + 2 \beta_{9} - 5 \beta_{8} + 10 \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots - 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 10 \beta_{11} - 3 \beta_{10} + 28 \beta_{8} + 49 \beta_{7} - 15 \beta_{6} + 10 \beta_{5} - 34 \beta_{4} + \cdots - 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 30 \beta_{10} - 30 \beta_{9} + 60 \beta_{8} - 24 \beta_{7} - 47 \beta_{6} + 45 \beta_{5} - 60 \beta_{4} + \cdots + 137 \)
|
| \(\nu^{6}\) | \(=\) |
\( 116\beta_{11} - 54\beta_{9} - 726\beta_{7} + 244\beta_{6} + 126\beta_{4} - 99\beta_{2} + 99\beta _1 + 225 \)
|
| \(\nu^{7}\) | \(=\) |
\( 215 \beta_{11} + 397 \beta_{10} + 215 \beta_{9} - 909 \beta_{8} - 909 \beta_{7} + 610 \beta_{6} + \cdots - 610 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 792 \beta_{11} + 792 \beta_{10} + 1454 \beta_{9} - 2795 \beta_{8} + 6360 \beta_{7} - 792 \beta_{6} + \cdots - 5568 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 5175 \beta_{11} - 2973 \beta_{10} + 9405 \beta_{8} + 27852 \beta_{7} - 10917 \beta_{6} + 5175 \beta_{5} + \cdots - 4230 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 18757 \beta_{10} - 18757 \beta_{9} + 46900 \beta_{8} - 21420 \beta_{7} - 16474 \beta_{6} + \cdots + 71252 \)
|
| \(\nu^{11}\) | \(=\) |
\( 67568 \beta_{11} - 40177 \beta_{9} - 425904 \beta_{7} + 125240 \beta_{6} + 104437 \beta_{4} + \cdots + 183674 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1098\mathbb{Z}\right)^\times\).
| \(n\) | \(245\) | \(307\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 217.1 |
|
−0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | −1.22442 | + | 3.76837i | 0 | 1.62864 | + | 1.18328i | 0.309017 | + | 0.951057i | 0 | −1.22442 | − | 3.76837i | ||||||||||||||||||||||||||||||||||||||||||
| 217.2 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | 0.557840 | − | 1.71685i | 0 | −0.785868 | − | 0.570967i | 0.309017 | + | 0.951057i | 0 | 0.557840 | + | 1.71685i | |||||||||||||||||||||||||||||||||||||||||||
| 217.3 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | 1.04854 | − | 3.22709i | 0 | 2.77526 | + | 2.01635i | 0.309017 | + | 0.951057i | 0 | 1.04854 | + | 3.22709i | |||||||||||||||||||||||||||||||||||||||||||
| 253.1 | −0.809017 | − | 0.587785i | 0 | 0.309017 | + | 0.951057i | −1.22442 | − | 3.76837i | 0 | 1.62864 | − | 1.18328i | 0.309017 | − | 0.951057i | 0 | −1.22442 | + | 3.76837i | |||||||||||||||||||||||||||||||||||||||||||
| 253.2 | −0.809017 | − | 0.587785i | 0 | 0.309017 | + | 0.951057i | 0.557840 | + | 1.71685i | 0 | −0.785868 | + | 0.570967i | 0.309017 | − | 0.951057i | 0 | 0.557840 | − | 1.71685i | |||||||||||||||||||||||||||||||||||||||||||
| 253.3 | −0.809017 | − | 0.587785i | 0 | 0.309017 | + | 0.951057i | 1.04854 | + | 3.22709i | 0 | 2.77526 | − | 2.01635i | 0.309017 | − | 0.951057i | 0 | 1.04854 | − | 3.22709i | |||||||||||||||||||||||||||||||||||||||||||
| 325.1 | 0.309017 | + | 0.951057i | 0 | −0.809017 | + | 0.587785i | −1.59188 | + | 1.15657i | 0 | 0.425021 | − | 1.30808i | −0.809017 | − | 0.587785i | 0 | −1.59188 | − | 1.15657i | |||||||||||||||||||||||||||||||||||||||||||
| 325.2 | 0.309017 | + | 0.951057i | 0 | −0.809017 | + | 0.587785i | 0.944271 | − | 0.686053i | 0 | −0.354999 | + | 1.09258i | −0.809017 | − | 0.587785i | 0 | 0.944271 | + | 0.686053i | |||||||||||||||||||||||||||||||||||||||||||
| 325.3 | 0.309017 | + | 0.951057i | 0 | −0.809017 | + | 0.587785i | 3.26564 | − | 2.37263i | 0 | 1.31194 | − | 4.03775i | −0.809017 | − | 0.587785i | 0 | 3.26564 | + | 2.37263i | |||||||||||||||||||||||||||||||||||||||||||
| 973.1 | 0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | −1.59188 | − | 1.15657i | 0 | 0.425021 | + | 1.30808i | −0.809017 | + | 0.587785i | 0 | −1.59188 | + | 1.15657i | |||||||||||||||||||||||||||||||||||||||||||
| 973.2 | 0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | 0.944271 | + | 0.686053i | 0 | −0.354999 | − | 1.09258i | −0.809017 | + | 0.587785i | 0 | 0.944271 | − | 0.686053i | |||||||||||||||||||||||||||||||||||||||||||
| 973.3 | 0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | 3.26564 | + | 2.37263i | 0 | 1.31194 | + | 4.03775i | −0.809017 | + | 0.587785i | 0 | 3.26564 | − | 2.37263i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 61.e | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1098.2.k.j | 12 | |
| 3.b | odd | 2 | 1 | 122.2.e.a | ✓ | 12 | |
| 12.b | even | 2 | 1 | 976.2.v.b | 12 | ||
| 61.e | even | 5 | 1 | inner | 1098.2.k.j | 12 | |
| 183.l | odd | 10 | 1 | 7442.2.a.o | 6 | ||
| 183.n | odd | 10 | 1 | 122.2.e.a | ✓ | 12 | |
| 183.n | odd | 10 | 1 | 7442.2.a.n | 6 | ||
| 732.bb | even | 10 | 1 | 976.2.v.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 122.2.e.a | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 122.2.e.a | ✓ | 12 | 183.n | odd | 10 | 1 | |
| 976.2.v.b | 12 | 12.b | even | 2 | 1 | ||
| 976.2.v.b | 12 | 732.bb | even | 10 | 1 | ||
| 1098.2.k.j | 12 | 1.a | even | 1 | 1 | trivial | |
| 1098.2.k.j | 12 | 61.e | even | 5 | 1 | inner | |
| 7442.2.a.n | 6 | 183.n | odd | 10 | 1 | ||
| 7442.2.a.o | 6 | 183.l | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1098, [\chi])\):
|
\( T_{5}^{12} - 6 T_{5}^{11} + 36 T_{5}^{10} - 141 T_{5}^{9} + 576 T_{5}^{8} - 1280 T_{5}^{7} + \cdots + 50625 \)
|
|
\( T_{7}^{12} - 10 T_{7}^{11} + 62 T_{7}^{10} - 235 T_{7}^{9} + 584 T_{7}^{8} - 820 T_{7}^{7} + 898 T_{7}^{6} + \cdots + 2025 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} - 6 T^{11} + \cdots + 50625 \)
|
| $7$ |
\( T^{12} - 10 T^{11} + \cdots + 2025 \)
|
| $11$ |
\( (T^{6} - 2 T^{5} - 30 T^{4} + \cdots + 81)^{2} \)
|
| $13$ |
\( (T^{6} + 4 T^{5} + \cdots + 549)^{2} \)
|
| $17$ |
\( T^{12} + 7 T^{11} + \cdots + 245025 \)
|
| $19$ |
\( T^{12} + 6 T^{11} + \cdots + 1946025 \)
|
| $23$ |
\( T^{12} + 8 T^{11} + \cdots + 2025 \)
|
| $29$ |
\( (T^{6} + 15 T^{5} + \cdots - 6561)^{2} \)
|
| $31$ |
\( T^{12} - T^{11} + \cdots + 10725625 \)
|
| $37$ |
\( T^{12} - 5 T^{10} + \cdots + 4844401 \)
|
| $41$ |
\( T^{12} + 2 T^{11} + \cdots + 64304361 \)
|
| $43$ |
\( T^{12} + \cdots + 849780801 \)
|
| $47$ |
\( (T^{6} - 5 T^{5} + \cdots - 801)^{2} \)
|
| $53$ |
\( T^{12} + 8 T^{11} + \cdots + 531441 \)
|
| $59$ |
\( T^{12} + 4 T^{11} + \cdots + 4100625 \)
|
| $61$ |
\( T^{12} + \cdots + 51520374361 \)
|
| $67$ |
\( T^{12} + \cdots + 135047641 \)
|
| $71$ |
\( T^{12} - 13 T^{11} + \cdots + 40947201 \)
|
| $73$ |
\( T^{12} + \cdots + 4909664761 \)
|
| $79$ |
\( T^{12} + \cdots + 617472801 \)
|
| $83$ |
\( T^{12} + \cdots + 122744241 \)
|
| $89$ |
\( T^{12} + \cdots + 981506241 \)
|
| $97$ |
\( T^{12} + \cdots + 1955762176 \)
|
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