Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,18,Mod(2,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.2");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.c (of order \(3\), 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: | \(12.8255461141\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 5 x^{19} + 231900 x^{18} + 2573595 x^{17} + 36222331443 x^{16} + 522056283042 x^{15} + \cdots + 54\!\cdots\!25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{50}\cdot 3^{18}\cdot 7^{19} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 5 x^{19} + 231900 x^{18} + 2573595 x^{17} + 36222331443 x^{16} + 522056283042 x^{15} + \cdots + 54\!\cdots\!25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 28\!\cdots\!94 \nu^{19} + \cdots + 29\!\cdots\!75 ) / 11\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 28\!\cdots\!94 \nu^{19} + \cdots - 29\!\cdots\!75 ) / 11\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 66\!\cdots\!16 \nu^{19} + \cdots - 32\!\cdots\!55 ) / 11\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 48\!\cdots\!33 \nu^{19} + \cdots + 67\!\cdots\!95 ) / 13\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 75\!\cdots\!30 \nu^{19} + \cdots - 65\!\cdots\!15 ) / 34\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 50\!\cdots\!89 \nu^{19} + \cdots - 13\!\cdots\!50 ) / 19\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 75\!\cdots\!51 \nu^{19} + \cdots + 83\!\cdots\!50 ) / 17\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 72\!\cdots\!58 \nu^{19} + \cdots - 32\!\cdots\!55 ) / 19\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 10\!\cdots\!67 \nu^{19} + \cdots + 33\!\cdots\!25 ) / 22\!\cdots\!60 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 15\!\cdots\!55 \nu^{19} + \cdots - 73\!\cdots\!45 ) / 29\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 13\!\cdots\!18 \nu^{19} + \cdots - 86\!\cdots\!75 ) / 14\!\cdots\!40 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 17\!\cdots\!59 \nu^{19} + \cdots + 33\!\cdots\!60 ) / 17\!\cdots\!80 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 49\!\cdots\!03 \nu^{19} + \cdots + 11\!\cdots\!15 ) / 19\!\cdots\!20 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 18\!\cdots\!75 \nu^{19} + \cdots + 80\!\cdots\!45 ) / 44\!\cdots\!20 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 58\!\cdots\!62 \nu^{19} + \cdots - 78\!\cdots\!65 ) / 86\!\cdots\!40 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 48\!\cdots\!92 \nu^{19} + \cdots + 15\!\cdots\!85 ) / 59\!\cdots\!60 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 89\!\cdots\!54 \nu^{19} + \cdots - 41\!\cdots\!95 ) / 89\!\cdots\!40 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 26\!\cdots\!23 \nu^{19} + \cdots - 28\!\cdots\!35 ) / 23\!\cdots\!60 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 21\!\cdots\!25 \nu^{19} + \cdots + 58\!\cdots\!20 ) / 17\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{5} - \beta_{4} - 23\beta_{3} + 185510\beta_{2} - 185510 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{15} - \beta_{11} + 12 \beta_{8} - 514 \beta_{6} - 18 \beta_{5} + 514 \beta_{4} + \cdots - 4479666 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 16 \beta_{19} - 8 \beta_{17} - 168 \beta_{16} - 47 \beta_{15} - 47 \beta_{14} + 136 \beta_{13} + \cdots - 128 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 9360 \beta_{19} - 1380 \beta_{18} - 1380 \beta_{17} + 1464 \beta_{16} - 135173 \beta_{14} + \cdots + 649774163014 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 4485289 \beta_{19} - 515163 \beta_{18} + 15507280 \beta_{16} + 2038510 \beta_{15} + \cdots + 693023939838225 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 172966118 \beta_{19} + 63563632 \beta_{17} - 662251562 \beta_{16} + 3850412022 \beta_{15} + \cdots - 79789576 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 1287763873980 \beta_{19} + 173173363242 \beta_{18} + 173173363242 \beta_{17} - 2109686320152 \beta_{16} + \cdots - 13\!\cdots\!44 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 73584717689832 \beta_{19} + 37076844790440 \beta_{18} + 287758975043568 \beta_{16} + \cdots - 15\!\cdots\!06 ) / 8 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 53\!\cdots\!96 \beta_{19} + \cdots - 29\!\cdots\!08 ) / 8 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 10\!\cdots\!84 \beta_{19} + \cdots + 21\!\cdots\!82 ) / 8 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 33\!\cdots\!18 \beta_{19} + \cdots + 36\!\cdots\!90 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 62\!\cdots\!38 \beta_{19} + \cdots - 62\!\cdots\!92 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 18\!\cdots\!72 \beta_{19} + \cdots - 15\!\cdots\!42 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 45\!\cdots\!88 \beta_{19} + \cdots - 35\!\cdots\!66 ) / 8 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 81\!\cdots\!00 \beta_{19} + \cdots - 40\!\cdots\!76 ) / 8 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 36\!\cdots\!96 \beta_{19} + \cdots + 44\!\cdots\!06 ) / 8 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 85\!\cdots\!41 \beta_{19} + \cdots + 90\!\cdots\!99 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 19\!\cdots\!12 \beta_{19} + \cdots + 40\!\cdots\!32 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
−306.343 | + | 530.602i | −2643.13 | − | 4578.04i | −122157. | − | 211581.i | 112450. | − | 194769.i | 3.23883e6 | 1.21283e7 | + | 9.24856e6i | 6.93813e7 | 5.05978e7 | − | 8.76379e7i | 6.88965e7 | + | 1.19332e8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.2 | −265.112 | + | 459.187i | 9523.58 | + | 16495.3i | −75032.3 | − | 129960.i | 224503. | − | 388850.i | −1.00992e7 | −1.05114e7 | − | 1.10518e7i | 1.00703e7 | −1.16827e8 | + | 2.02350e8i | 1.19037e8 | + | 2.06177e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.3 | −175.640 | + | 304.217i | −3368.41 | − | 5834.25i | 3837.49 | + | 6646.72i | −348474. | + | 603575.i | 2.36650e6 | −1.46022e7 | − | 4.40528e6i | −4.87389e7 | 4.18777e7 | − | 7.25344e7i | −1.22412e8 | − | 2.12023e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.4 | −58.4016 | + | 101.154i | −10042.7 | − | 17394.4i | 58714.5 | + | 101697.i | 526365. | − | 911691.i | 2.34603e6 | 1.15602e7 | − | 9.94945e6i | −2.90257e7 | −1.37140e8 | + | 2.37533e8i | 6.14811e7 | + | 1.06488e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.5 | −43.5867 | + | 75.4943i | 5626.01 | + | 9744.54i | 61736.4 | + | 106931.i | −472563. | + | 818503.i | −980877. | 1.52373e7 | + | 674362.i | −2.21895e7 | 1.26603e6 | − | 2.19283e6i | −4.11949e7 | − | 7.13516e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.6 | −1.43970 | + | 2.49364i | 3646.82 | + | 6316.47i | 65531.9 | + | 113505.i | 804501. | − | 1.39344e6i | −21001.3 | −6.26914e6 | + | 1.39043e7i | −754795. | 3.79715e7 | − | 6.57686e7i | 2.31648e6 | + | 4.01227e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.7 | 158.646 | − | 274.782i | −6861.41 | − | 11884.3i | 15199.2 | + | 26325.8i | −470055. | + | 814159.i | −4.35413e6 | −5.83152e6 | + | 1.40934e7i | 5.12331e7 | −2.95879e7 | + | 5.12478e7i | 1.49144e8 | + | 2.58325e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.8 | 194.221 | − | 336.400i | 1157.82 | + | 2005.40i | −9907.51 | − | 17160.3i | 92672.9 | − | 160514.i | 899489. | −2.00510e6 | − | 1.51199e7i | 4.32169e7 | 6.18890e7 | − | 1.07195e8i | −3.59980e7 | − | 6.23504e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.9 | 282.522 | − | 489.343i | 11021.2 | + | 19089.3i | −94101.8 | − | 162989.i | −274540. | + | 475518.i | 1.24550e7 | −6.98844e6 | + | 1.35570e7i | −3.22819e7 | −1.78365e8 | + | 3.08938e8i | 1.55128e8 | + | 2.68689e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.10 | 350.134 | − | 606.449i | −4779.85 | − | 8278.95i | −179651. | − | 311165.i | 350039. | − | 606285.i | −6.69435e6 | 1.44796e7 | + | 4.79276e6i | −1.59822e8 | 1.88761e7 | − | 3.26943e7i | −2.45121e8 | − | 4.24562e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.1 | −306.343 | − | 530.602i | −2643.13 | + | 4578.04i | −122157. | + | 211581.i | 112450. | + | 194769.i | 3.23883e6 | 1.21283e7 | − | 9.24856e6i | 6.93813e7 | 5.05978e7 | + | 8.76379e7i | 6.88965e7 | − | 1.19332e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.2 | −265.112 | − | 459.187i | 9523.58 | − | 16495.3i | −75032.3 | + | 129960.i | 224503. | + | 388850.i | −1.00992e7 | −1.05114e7 | + | 1.10518e7i | 1.00703e7 | −1.16827e8 | − | 2.02350e8i | 1.19037e8 | − | 2.06177e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.3 | −175.640 | − | 304.217i | −3368.41 | + | 5834.25i | 3837.49 | − | 6646.72i | −348474. | − | 603575.i | 2.36650e6 | −1.46022e7 | + | 4.40528e6i | −4.87389e7 | 4.18777e7 | + | 7.25344e7i | −1.22412e8 | + | 2.12023e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.4 | −58.4016 | − | 101.154i | −10042.7 | + | 17394.4i | 58714.5 | − | 101697.i | 526365. | + | 911691.i | 2.34603e6 | 1.15602e7 | + | 9.94945e6i | −2.90257e7 | −1.37140e8 | − | 2.37533e8i | 6.14811e7 | − | 1.06488e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.5 | −43.5867 | − | 75.4943i | 5626.01 | − | 9744.54i | 61736.4 | − | 106931.i | −472563. | − | 818503.i | −980877. | 1.52373e7 | − | 674362.i | −2.21895e7 | 1.26603e6 | + | 2.19283e6i | −4.11949e7 | + | 7.13516e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.6 | −1.43970 | − | 2.49364i | 3646.82 | − | 6316.47i | 65531.9 | − | 113505.i | 804501. | + | 1.39344e6i | −21001.3 | −6.26914e6 | − | 1.39043e7i | −754795. | 3.79715e7 | + | 6.57686e7i | 2.31648e6 | − | 4.01227e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.7 | 158.646 | + | 274.782i | −6861.41 | + | 11884.3i | 15199.2 | − | 26325.8i | −470055. | − | 814159.i | −4.35413e6 | −5.83152e6 | − | 1.40934e7i | 5.12331e7 | −2.95879e7 | − | 5.12478e7i | 1.49144e8 | − | 2.58325e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.8 | 194.221 | + | 336.400i | 1157.82 | − | 2005.40i | −9907.51 | + | 17160.3i | 92672.9 | + | 160514.i | 899489. | −2.00510e6 | + | 1.51199e7i | 4.32169e7 | 6.18890e7 | + | 1.07195e8i | −3.59980e7 | + | 6.23504e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.9 | 282.522 | + | 489.343i | 11021.2 | − | 19089.3i | −94101.8 | + | 162989.i | −274540. | − | 475518.i | 1.24550e7 | −6.98844e6 | − | 1.35570e7i | −3.22819e7 | −1.78365e8 | − | 3.08938e8i | 1.55128e8 | − | 2.68689e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.10 | 350.134 | + | 606.449i | −4779.85 | + | 8278.95i | −179651. | + | 311165.i | 350039. | + | 606285.i | −6.69435e6 | 1.44796e7 | − | 4.79276e6i | −1.59822e8 | 1.88761e7 | + | 3.26943e7i | −2.45121e8 | + | 4.24562e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7.18.c.a | ✓ | 20 |
| 7.c | even | 3 | 1 | inner | 7.18.c.a | ✓ | 20 |
| 7.c | even | 3 | 1 | 49.18.a.f | 10 | ||
| 7.d | odd | 6 | 1 | 49.18.a.g | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.18.c.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 7.18.c.a | ✓ | 20 | 7.c | even | 3 | 1 | inner |
| 49.18.a.f | 10 | 7.c | even | 3 | 1 | ||
| 49.18.a.g | 10 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(7, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 26\!\cdots\!76 \)
|
| $3$ |
\( T^{20} + \cdots + 56\!\cdots\!21 \)
|
| $5$ |
\( T^{20} + \cdots + 56\!\cdots\!25 \)
|
| $7$ |
\( T^{20} + \cdots + 46\!\cdots\!49 \)
|
| $11$ |
\( T^{20} + \cdots + 19\!\cdots\!25 \)
|
| $13$ |
\( (T^{10} + \cdots + 35\!\cdots\!56)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 44\!\cdots\!21 \)
|
| $19$ |
\( T^{20} + \cdots + 13\!\cdots\!81 \)
|
| $23$ |
\( T^{20} + \cdots + 21\!\cdots\!21 \)
|
| $29$ |
\( (T^{10} + \cdots - 23\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 13\!\cdots\!21 \)
|
| $37$ |
\( T^{20} + \cdots + 59\!\cdots\!25 \)
|
| $41$ |
\( (T^{10} + \cdots - 56\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{10} + \cdots - 17\!\cdots\!16)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 19\!\cdots\!25 \)
|
| $53$ |
\( T^{20} + \cdots + 37\!\cdots\!81 \)
|
| $59$ |
\( T^{20} + \cdots + 11\!\cdots\!25 \)
|
| $61$ |
\( T^{20} + \cdots + 53\!\cdots\!41 \)
|
| $67$ |
\( T^{20} + \cdots + 73\!\cdots\!25 \)
|
| $71$ |
\( (T^{10} + \cdots + 17\!\cdots\!24)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 60\!\cdots\!41 \)
|
| $79$ |
\( T^{20} + \cdots + 31\!\cdots\!61 \)
|
| $83$ |
\( (T^{10} + \cdots - 23\!\cdots\!36)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 37\!\cdots\!41 \)
|
| $97$ |
\( (T^{10} + \cdots + 21\!\cdots\!44)^{2} \)
|
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