Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [14,15,Mod(3,14)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14.3");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.d (of order \(6\), degree \(2\), 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: | \(17.4060555413\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 4 x^{19} + 5366534 x^{18} + 786490608 x^{17} + 19826027932115 x^{16} + \cdots + 13\!\cdots\!24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{76}\cdot 3^{12}\cdot 7^{14} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 4 x^{19} + 5366534 x^{18} + 786490608 x^{17} + 19826027932115 x^{16} + \cdots + 13\!\cdots\!24 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 39\!\cdots\!16 \nu^{19} + \cdots + 90\!\cdots\!32 ) / 10\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 28\!\cdots\!43 \nu^{19} + \cdots - 39\!\cdots\!36 ) / 18\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\!\cdots\!51 \nu^{19} + \cdots - 11\!\cdots\!24 ) / 79\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 21\!\cdots\!05 \nu^{19} + \cdots + 18\!\cdots\!00 ) / 31\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 60\!\cdots\!95 \nu^{19} + \cdots - 22\!\cdots\!24 ) / 79\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 57\!\cdots\!13 \nu^{19} + \cdots - 26\!\cdots\!12 ) / 21\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 43\!\cdots\!83 \nu^{19} + \cdots - 96\!\cdots\!28 ) / 15\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10\!\cdots\!65 \nu^{19} + \cdots + 12\!\cdots\!64 ) / 10\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 64\!\cdots\!27 \nu^{19} + \cdots - 28\!\cdots\!16 ) / 15\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 27\!\cdots\!58 \nu^{19} + \cdots + 43\!\cdots\!20 ) / 54\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 15\!\cdots\!11 \nu^{19} + \cdots - 60\!\cdots\!16 ) / 21\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 18\!\cdots\!77 \nu^{19} + \cdots - 15\!\cdots\!52 ) / 21\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 15\!\cdots\!11 \nu^{19} + \cdots + 13\!\cdots\!52 ) / 54\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 15\!\cdots\!86 \nu^{19} + \cdots - 11\!\cdots\!68 ) / 37\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 80\!\cdots\!75 \nu^{19} + \cdots + 48\!\cdots\!08 ) / 15\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 37\!\cdots\!17 \nu^{19} + \cdots + 95\!\cdots\!56 ) / 37\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 97\!\cdots\!45 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 75\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 24\!\cdots\!91 \nu^{19} + \cdots - 10\!\cdots\!48 ) / 15\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 12\!\cdots\!51 \nu^{19} + \cdots + 33\!\cdots\!88 ) / 72\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} - \beta_{10} + 2\beta_{8} + \beta_{6} + 5\beta_{5} - \beta_{4} + 5\beta_{3} + 6\beta_{2} + 150\beta_1 ) / 384 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 32 \beta_{18} - 32 \beta_{17} + 32 \beta_{12} + 106 \beta_{11} - 138 \beta_{10} - 64 \beta_{9} + \cdots - 206072110 ) / 192 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 13708 \beta_{19} - 8538 \beta_{17} + 18227 \beta_{16} - 11353 \beta_{15} - 59943 \beta_{14} + \cdots - 46527608787 ) / 384 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 14256582 \beta_{19} + 10104385 \beta_{18} + 15327269 \beta_{17} + 15110363 \beta_{16} + 10104385 \beta_{15} + \cdots - 13919196 ) / 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 34660198122 \beta_{19} - 49147590849 \beta_{18} + 110574842517 \beta_{17} + 34660198122 \beta_{16} + \cdots + 21\!\cdots\!91 ) / 384 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 199245358320764 \beta_{19} + 2198983401774 \beta_{17} - 230196384470633 \beta_{16} + \cdots + 94\!\cdots\!33 ) / 192 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 89\!\cdots\!38 \beta_{19} + \cdots - 64\!\cdots\!96 ) / 384 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 19\!\cdots\!10 \beta_{19} + \cdots - 18\!\cdots\!83 ) / 16 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 16\!\cdots\!00 \beta_{19} + \cdots - 22\!\cdots\!15 ) / 384 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 10\!\cdots\!54 \beta_{19} + \cdots - 67\!\cdots\!00 ) / 192 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 78\!\cdots\!26 \beta_{19} + \cdots + 65\!\cdots\!95 ) / 384 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 44\!\cdots\!32 \beta_{19} + \cdots + 20\!\cdots\!21 ) / 32 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 30\!\cdots\!46 \beta_{19} + \cdots - 76\!\cdots\!44 ) / 384 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 11\!\cdots\!90 \beta_{19} + \cdots - 30\!\cdots\!49 ) / 192 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 77\!\cdots\!96 \beta_{19} + \cdots - 50\!\cdots\!39 ) / 384 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 49\!\cdots\!46 \beta_{19} + \cdots - 13\!\cdots\!88 ) / 8 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 16\!\cdots\!58 \beta_{19} + \cdots + 13\!\cdots\!47 ) / 384 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 38\!\cdots\!32 \beta_{19} + \cdots + 17\!\cdots\!15 ) / 192 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 22\!\cdots\!66 \beta_{19} + \cdots - 10\!\cdots\!28 ) / 384 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−45.2548 | − | 78.3837i | −3571.50 | − | 2062.01i | −4096.00 | + | 7094.48i | 81056.3 | − | 46797.9i | 373263.i | 587107. | + | 577519.i | 741455. | 6.11227e6 | + | 1.05868e7i | −7.33638e6 | − | 4.23566e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.2 | −45.2548 | − | 78.3837i | −1320.27 | − | 762.260i | −4096.00 | + | 7094.48i | −130389. | + | 75280.2i | 137984.i | 789127. | + | 235587.i | 741455. | −1.22940e6 | − | 2.12939e6i | 1.18015e7 | + | 6.81358e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.3 | −45.2548 | − | 78.3837i | −746.556 | − | 431.025i | −4096.00 | + | 7094.48i | 45757.7 | − | 26418.2i | 78023.8i | −602075. | − | 561898.i | 741455. | −2.01992e6 | − | 3.49860e6i | −4.14152e6 | − | 2.39111e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.4 | −45.2548 | − | 78.3837i | 1034.58 | + | 597.314i | −4096.00 | + | 7094.48i | 5108.24 | − | 2949.25i | − | 108125.i | −382860. | + | 729137.i | 741455. | −1.67792e6 | − | 2.90624e6i | −462345. | − | 266935.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.5 | −45.2548 | − | 78.3837i | 2710.52 | + | 1564.92i | −4096.00 | + | 7094.48i | 10170.7 | − | 5872.03i | − | 283280.i | 552858. | − | 610386.i | 741455. | 2.50645e6 | + | 4.34129e6i | −920543. | − | 531476.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.6 | 45.2548 | + | 78.3837i | −3149.23 | − | 1818.21i | −4096.00 | + | 7094.48i | −112471. | + | 64935.4i | − | 329131.i | −629509. | + | 530982.i | −741455. | 4.22027e6 | + | 7.30973e6i | −1.01797e7 | − | 5.87728e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.7 | 45.2548 | + | 78.3837i | −1563.47 | − | 902.668i | −4096.00 | + | 7094.48i | 25206.6 | − | 14553.0i | − | 163400.i | 823165. | + | 24943.0i | −741455. | −761866. | − | 1.31959e6i | 2.28144e6 | + | 1.31719e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.8 | 45.2548 | + | 78.3837i | −262.346 | − | 151.465i | −4096.00 | + | 7094.48i | 106192. | − | 61309.7i | − | 27418.1i | −789690. | + | 233694.i | −741455. | −2.34560e6 | − | 4.06270e6i | 9.61136e6 | + | 5.54912e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.9 | 45.2548 | + | 78.3837i | 1298.66 | + | 749.784i | −4096.00 | + | 7094.48i | −67507.4 | + | 38975.4i | 135725.i | −346065. | − | 747303.i | −741455. | −1.26713e6 | − | 2.19474e6i | −6.11007e6 | − | 3.52765e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.10 | 45.2548 | + | 78.3837i | 3382.61 | + | 1952.95i | −4096.00 | + | 7094.48i | 38553.6 | − | 22259.0i | 353522.i | 725749. | + | 389245.i | −741455. | 5.23657e6 | + | 9.07000e6i | 3.48948e6 | + | 2.01465e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.1 | −45.2548 | + | 78.3837i | −3571.50 | + | 2062.01i | −4096.00 | − | 7094.48i | 81056.3 | + | 46797.9i | − | 373263.i | 587107. | − | 577519.i | 741455. | 6.11227e6 | − | 1.05868e7i | −7.33638e6 | + | 4.23566e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.2 | −45.2548 | + | 78.3837i | −1320.27 | + | 762.260i | −4096.00 | − | 7094.48i | −130389. | − | 75280.2i | − | 137984.i | 789127. | − | 235587.i | 741455. | −1.22940e6 | + | 2.12939e6i | 1.18015e7 | − | 6.81358e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.3 | −45.2548 | + | 78.3837i | −746.556 | + | 431.025i | −4096.00 | − | 7094.48i | 45757.7 | + | 26418.2i | − | 78023.8i | −602075. | + | 561898.i | 741455. | −2.01992e6 | + | 3.49860e6i | −4.14152e6 | + | 2.39111e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.4 | −45.2548 | + | 78.3837i | 1034.58 | − | 597.314i | −4096.00 | − | 7094.48i | 5108.24 | + | 2949.25i | 108125.i | −382860. | − | 729137.i | 741455. | −1.67792e6 | + | 2.90624e6i | −462345. | + | 266935.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.5 | −45.2548 | + | 78.3837i | 2710.52 | − | 1564.92i | −4096.00 | − | 7094.48i | 10170.7 | + | 5872.03i | 283280.i | 552858. | + | 610386.i | 741455. | 2.50645e6 | − | 4.34129e6i | −920543. | + | 531476.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.6 | 45.2548 | − | 78.3837i | −3149.23 | + | 1818.21i | −4096.00 | − | 7094.48i | −112471. | − | 64935.4i | 329131.i | −629509. | − | 530982.i | −741455. | 4.22027e6 | − | 7.30973e6i | −1.01797e7 | + | 5.87728e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.7 | 45.2548 | − | 78.3837i | −1563.47 | + | 902.668i | −4096.00 | − | 7094.48i | 25206.6 | + | 14553.0i | 163400.i | 823165. | − | 24943.0i | −741455. | −761866. | + | 1.31959e6i | 2.28144e6 | − | 1.31719e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.8 | 45.2548 | − | 78.3837i | −262.346 | + | 151.465i | −4096.00 | − | 7094.48i | 106192. | + | 61309.7i | 27418.1i | −789690. | − | 233694.i | −741455. | −2.34560e6 | + | 4.06270e6i | 9.61136e6 | − | 5.54912e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.9 | 45.2548 | − | 78.3837i | 1298.66 | − | 749.784i | −4096.00 | − | 7094.48i | −67507.4 | − | 38975.4i | − | 135725.i | −346065. | + | 747303.i | −741455. | −1.26713e6 | + | 2.19474e6i | −6.11007e6 | + | 3.52765e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.10 | 45.2548 | − | 78.3837i | 3382.61 | − | 1952.95i | −4096.00 | − | 7094.48i | 38553.6 | + | 22259.0i | − | 353522.i | 725749. | − | 389245.i | −741455. | 5.23657e6 | − | 9.07000e6i | 3.48948e6 | − | 2.01465e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 14.15.d.a | ✓ | 20 |
| 3.b | odd | 2 | 1 | 126.15.n.b | 20 | ||
| 7.b | odd | 2 | 1 | 98.15.d.b | 20 | ||
| 7.c | even | 3 | 1 | 98.15.b.c | 20 | ||
| 7.c | even | 3 | 1 | 98.15.d.b | 20 | ||
| 7.d | odd | 6 | 1 | inner | 14.15.d.a | ✓ | 20 |
| 7.d | odd | 6 | 1 | 98.15.b.c | 20 | ||
| 21.g | even | 6 | 1 | 126.15.n.b | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.15.d.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 14.15.d.a | ✓ | 20 | 7.d | odd | 6 | 1 | inner |
| 98.15.b.c | 20 | 7.c | even | 3 | 1 | ||
| 98.15.b.c | 20 | 7.d | odd | 6 | 1 | ||
| 98.15.d.b | 20 | 7.b | odd | 2 | 1 | ||
| 98.15.d.b | 20 | 7.c | even | 3 | 1 | ||
| 126.15.n.b | 20 | 3.b | odd | 2 | 1 | ||
| 126.15.n.b | 20 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 8192 T^{2} + 67108864)^{5} \)
|
| $3$ |
\( T^{20} + \cdots + 55\!\cdots\!49 \)
|
| $5$ |
\( T^{20} + \cdots + 68\!\cdots\!25 \)
|
| $7$ |
\( T^{20} + \cdots + 20\!\cdots\!01 \)
|
| $11$ |
\( T^{20} + \cdots + 29\!\cdots\!89 \)
|
| $13$ |
\( T^{20} + \cdots + 19\!\cdots\!16 \)
|
| $17$ |
\( T^{20} + \cdots + 57\!\cdots\!21 \)
|
| $19$ |
\( T^{20} + \cdots + 34\!\cdots\!25 \)
|
| $23$ |
\( T^{20} + \cdots + 33\!\cdots\!49 \)
|
| $29$ |
\( (T^{10} + \cdots - 12\!\cdots\!48)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 17\!\cdots\!29 \)
|
| $37$ |
\( T^{20} + \cdots + 65\!\cdots\!81 \)
|
| $41$ |
\( T^{20} + \cdots + 13\!\cdots\!24 \)
|
| $43$ |
\( (T^{10} + \cdots + 15\!\cdots\!28)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 68\!\cdots\!89 \)
|
| $53$ |
\( T^{20} + \cdots + 68\!\cdots\!69 \)
|
| $59$ |
\( T^{20} + \cdots + 22\!\cdots\!49 \)
|
| $61$ |
\( T^{20} + \cdots + 14\!\cdots\!29 \)
|
| $67$ |
\( T^{20} + \cdots + 13\!\cdots\!21 \)
|
| $71$ |
\( (T^{10} + \cdots + 14\!\cdots\!36)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 14\!\cdots\!21 \)
|
| $79$ |
\( T^{20} + \cdots + 24\!\cdots\!25 \)
|
| $83$ |
\( T^{20} + \cdots + 21\!\cdots\!36 \)
|
| $89$ |
\( T^{20} + \cdots + 70\!\cdots\!81 \)
|
| $97$ |
\( T^{20} + \cdots + 18\!\cdots\!56 \)
|
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