Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,10,Mod(18,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.18");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.c (of order \(3\), degree \(2\), not 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: | \(25.2367559720\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} - 5242 x^{14} + 24024 x^{13} + 10505991 x^{12} - 53910056 x^{11} + \cdots + 89\!\cdots\!16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 7^{12} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{16} - 4 x^{15} - 5242 x^{14} + 24024 x^{13} + 10505991 x^{12} - 53910056 x^{11} + \cdots + 89\!\cdots\!16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 34\!\cdots\!68 \nu^{15} + \cdots + 36\!\cdots\!48 ) / 47\!\cdots\!28 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 27\!\cdots\!47 \nu^{15} + \cdots - 16\!\cdots\!56 ) / 19\!\cdots\!32 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 73\!\cdots\!39 \nu^{15} + \cdots + 11\!\cdots\!84 ) / 19\!\cdots\!32 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15\!\cdots\!07 \nu^{15} + \cdots - 22\!\cdots\!84 ) / 38\!\cdots\!64 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 20\!\cdots\!50 \nu^{15} + \cdots - 45\!\cdots\!80 ) / 96\!\cdots\!16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 49\!\cdots\!11 \nu^{15} + \cdots - 16\!\cdots\!92 ) / 19\!\cdots\!32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 10\!\cdots\!79 \nu^{15} + \cdots - 49\!\cdots\!08 ) / 27\!\cdots\!76 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 27\!\cdots\!47 \nu^{15} + \cdots - 16\!\cdots\!56 ) / 39\!\cdots\!68 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 47\!\cdots\!72 \nu^{15} + \cdots - 10\!\cdots\!92 ) / 38\!\cdots\!64 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 53\!\cdots\!83 \nu^{15} + \cdots + 82\!\cdots\!68 ) / 12\!\cdots\!88 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 11\!\cdots\!99 \nu^{15} + \cdots - 24\!\cdots\!20 ) / 27\!\cdots\!76 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 28\!\cdots\!11 \nu^{15} + \cdots + 28\!\cdots\!64 ) / 39\!\cdots\!68 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 31\!\cdots\!67 \nu^{15} + \cdots - 11\!\cdots\!04 ) / 38\!\cdots\!64 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 46\!\cdots\!26 \nu^{15} + \cdots - 19\!\cdots\!80 ) / 64\!\cdots\!44 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 30\!\cdots\!65 \nu^{15} + \cdots + 49\!\cdots\!48 ) / 38\!\cdots\!64 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{8} + 49\beta_{2} ) / 49 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{8} + 14\beta_{7} - 49\beta_{3} - 49\beta_{2} + 98\beta _1 + 32242 ) / 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 147 \beta_{13} + 2 \beta_{12} + 196 \beta_{10} - 1915 \beta_{8} - 63 \beta_{7} + 294 \beta_{6} + \cdots - 42826 ) / 49 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 196 \beta_{13} - 24 \beta_{12} + 56 \beta_{11} + 196 \beta_{10} + 15204 \beta_{8} + 33516 \beta_{7} + \cdots + 39710678 ) / 49 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 980 \beta_{15} - 3920 \beta_{14} + 322175 \beta_{13} + 12776 \beta_{12} - 420 \beta_{11} + \cdots - 83367914 ) / 49 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 280 \beta_{15} - 840 \beta_{14} + 23394 \beta_{13} - 21744 \beta_{12} + 47936 \beta_{11} + \cdots + 7209233766 ) / 7 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 4506334 \beta_{15} - 10797640 \beta_{14} + 573863353 \beta_{13} + 55158334 \beta_{12} + \cdots - 139070664082 ) / 49 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 3045840 \beta_{15} - 6058752 \beta_{14} - 146231288 \beta_{13} - 601452656 \beta_{12} + \cdots + 64218557344746 ) / 49 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 13740344016 \beta_{15} - 23480646336 \beta_{14} + 937885751019 \beta_{13} + 168417661792 \beta_{12} + \cdots - 218597296881978 ) / 49 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 4414120704 \beta_{15} + 6671618520 \beta_{14} - 911261132334 \beta_{13} - 1792850694624 \beta_{12} + \cdots + 81\!\cdots\!06 ) / 49 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 34238833476694 \beta_{15} - 45420919744200 \beta_{14} + \cdots - 33\!\cdots\!30 ) / 49 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 5719482568184 \beta_{15} + 6852625430112 \beta_{14} - 346584403163292 \beta_{13} + \cdots + 14\!\cdots\!10 ) / 7 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 75\!\cdots\!48 \beta_{15} + \cdots - 48\!\cdots\!54 ) / 49 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 14\!\cdots\!88 \beta_{15} + \cdots + 13\!\cdots\!02 ) / 49 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 15\!\cdots\!54 \beta_{15} + \cdots - 70\!\cdots\!94 ) / 49 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.1 |
|
−13.9891 | − | 24.2298i | −13.0785 | + | 22.6526i | −135.388 | + | 234.499i | 184.056 | + | 318.794i | 731.823 | 0 | −6748.99 | 9499.41 | + | 16453.5i | 5149.53 | − | 8919.25i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 18.2 | −13.9891 | − | 24.2298i | 13.0785 | − | 22.6526i | −135.388 | + | 234.499i | −184.056 | − | 318.794i | −731.823 | 0 | −6748.99 | 9499.41 | + | 16453.5i | −5149.53 | + | 8919.25i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 18.3 | 1.14844 | + | 1.98916i | −129.061 | + | 223.541i | 253.362 | − | 438.836i | 486.452 | + | 842.560i | −592.877 | 0 | 2339.89 | −23472.1 | − | 40654.8i | −1117.32 | + | 1935.26i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 18.4 | 1.14844 | + | 1.98916i | 129.061 | − | 223.541i | 253.362 | − | 438.836i | −486.452 | − | 842.560i | 592.877 | 0 | 2339.89 | −23472.1 | − | 40654.8i | 1117.32 | − | 1935.26i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 18.5 | 7.67743 | + | 13.2977i | −62.0333 | + | 107.445i | 138.114 | − | 239.221i | −123.139 | − | 213.283i | −1905.03 | 0 | 12103.1 | 2145.23 | + | 3715.65i | 1890.78 | − | 3274.93i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 18.6 | 7.67743 | + | 13.2977i | 62.0333 | − | 107.445i | 138.114 | − | 239.221i | 123.139 | + | 213.283i | 1905.03 | 0 | 12103.1 | 2145.23 | + | 3715.65i | −1890.78 | + | 3274.93i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 18.7 | 21.6632 | + | 37.5218i | −49.9953 | + | 86.5945i | −682.588 | + | 1182.28i | −945.259 | − | 1637.24i | −4332.24 | 0 | −36965.0 | 4842.43 | + | 8387.34i | 40954.7 | − | 70935.6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 18.8 | 21.6632 | + | 37.5218i | 49.9953 | − | 86.5945i | −682.588 | + | 1182.28i | 945.259 | + | 1637.24i | 4332.24 | 0 | −36965.0 | 4842.43 | + | 8387.34i | −40954.7 | + | 70935.6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 30.1 | −13.9891 | + | 24.2298i | −13.0785 | − | 22.6526i | −135.388 | − | 234.499i | 184.056 | − | 318.794i | 731.823 | 0 | −6748.99 | 9499.41 | − | 16453.5i | 5149.53 | + | 8919.25i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 30.2 | −13.9891 | + | 24.2298i | 13.0785 | + | 22.6526i | −135.388 | − | 234.499i | −184.056 | + | 318.794i | −731.823 | 0 | −6748.99 | 9499.41 | − | 16453.5i | −5149.53 | − | 8919.25i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 30.3 | 1.14844 | − | 1.98916i | −129.061 | − | 223.541i | 253.362 | + | 438.836i | 486.452 | − | 842.560i | −592.877 | 0 | 2339.89 | −23472.1 | + | 40654.8i | −1117.32 | − | 1935.26i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 30.4 | 1.14844 | − | 1.98916i | 129.061 | + | 223.541i | 253.362 | + | 438.836i | −486.452 | + | 842.560i | 592.877 | 0 | 2339.89 | −23472.1 | + | 40654.8i | 1117.32 | + | 1935.26i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 30.5 | 7.67743 | − | 13.2977i | −62.0333 | − | 107.445i | 138.114 | + | 239.221i | −123.139 | + | 213.283i | −1905.03 | 0 | 12103.1 | 2145.23 | − | 3715.65i | 1890.78 | + | 3274.93i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 30.6 | 7.67743 | − | 13.2977i | 62.0333 | + | 107.445i | 138.114 | + | 239.221i | 123.139 | − | 213.283i | 1905.03 | 0 | 12103.1 | 2145.23 | − | 3715.65i | −1890.78 | − | 3274.93i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 30.7 | 21.6632 | − | 37.5218i | −49.9953 | − | 86.5945i | −682.588 | − | 1182.28i | −945.259 | + | 1637.24i | −4332.24 | 0 | −36965.0 | 4842.43 | − | 8387.34i | 40954.7 | + | 70935.6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 30.8 | 21.6632 | − | 37.5218i | 49.9953 | + | 86.5945i | −682.588 | − | 1182.28i | 945.259 | − | 1637.24i | 4332.24 | 0 | −36965.0 | 4842.43 | − | 8387.34i | −40954.7 | − | 70935.6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 49.10.c.h | 16 | |
| 7.b | odd | 2 | 1 | inner | 49.10.c.h | 16 | |
| 7.c | even | 3 | 1 | 49.10.a.g | ✓ | 8 | |
| 7.c | even | 3 | 1 | inner | 49.10.c.h | 16 | |
| 7.d | odd | 6 | 1 | 49.10.a.g | ✓ | 8 | |
| 7.d | odd | 6 | 1 | inner | 49.10.c.h | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 49.10.a.g | ✓ | 8 | 7.c | even | 3 | 1 | |
| 49.10.a.g | ✓ | 8 | 7.d | odd | 6 | 1 | |
| 49.10.c.h | 16 | 1.a | even | 1 | 1 | trivial | |
| 49.10.c.h | 16 | 7.b | odd | 2 | 1 | inner | |
| 49.10.c.h | 16 | 7.c | even | 3 | 1 | inner | |
| 49.10.c.h | 16 | 7.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(49, [\chi])\):
|
\( T_{2}^{8} - 33 T_{2}^{7} + 1995 T_{2}^{6} - 11814 T_{2}^{5} + 1551836 T_{2}^{4} - 21717168 T_{2}^{3} + \cdots + 1827733504 \)
|
|
\( T_{3}^{16} + 92702 T_{3}^{14} + 6685098888 T_{3}^{12} + 153894640052792 T_{3}^{10} + \cdots + 49\!\cdots\!16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} - 33 T^{7} + \cdots + 1827733504)^{2} \)
|
| $3$ |
\( T^{16} + \cdots + 49\!\cdots\!16 \)
|
| $5$ |
\( T^{16} + \cdots + 77\!\cdots\!00 \)
|
| $7$ |
\( T^{16} \)
|
| $11$ |
\( (T^{8} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $13$ |
\( (T^{8} + \cdots + 26\!\cdots\!56)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 12\!\cdots\!36 \)
|
| $19$ |
\( T^{16} + \cdots + 20\!\cdots\!56 \)
|
| $23$ |
\( (T^{8} + \cdots + 56\!\cdots\!56)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots - 12\!\cdots\!60)^{4} \)
|
| $31$ |
\( T^{16} + \cdots + 25\!\cdots\!36 \)
|
| $37$ |
\( (T^{8} + \cdots + 85\!\cdots\!96)^{2} \)
|
| $41$ |
\( (T^{8} + \cdots + 60\!\cdots\!64)^{2} \)
|
| $43$ |
\( (T^{4} + \cdots + 18\!\cdots\!00)^{4} \)
|
| $47$ |
\( T^{16} + \cdots + 14\!\cdots\!00 \)
|
| $53$ |
\( (T^{8} + \cdots + 16\!\cdots\!64)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 62\!\cdots\!16 \)
|
| $61$ |
\( T^{16} + \cdots + 47\!\cdots\!36 \)
|
| $67$ |
\( (T^{8} + \cdots + 13\!\cdots\!44)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots - 11\!\cdots\!68)^{4} \)
|
| $73$ |
\( T^{16} + \cdots + 48\!\cdots\!36 \)
|
| $79$ |
\( (T^{8} + \cdots + 27\!\cdots\!64)^{2} \)
|
| $83$ |
\( (T^{8} + \cdots + 59\!\cdots\!00)^{2} \)
|
| $89$ |
\( T^{16} + \cdots + 47\!\cdots\!96 \)
|
| $97$ |
\( (T^{8} + \cdots + 57\!\cdots\!96)^{2} \)
|
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