Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [667,2,Mod(1,667)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(667, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("667.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 667 = 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 667.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: | \(5.32602181482\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + \cdots + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + \cdots + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 124169 \nu^{15} + 194810 \nu^{14} + 2801696 \nu^{13} - 4124788 \nu^{12} - 24534615 \nu^{11} + \cdots - 5285248 ) / 1247936 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 320937 \nu^{15} + 55514 \nu^{14} + 7810176 \nu^{13} - 279828 \nu^{12} - 74718487 \nu^{11} + \cdots + 26486912 ) / 1247936 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 399265 \nu^{15} + 390970 \nu^{14} + 9549392 \nu^{13} - 8005092 \nu^{12} - 89853839 \nu^{11} + \cdots + 9101888 ) / 1247936 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 412871 \nu^{15} - 286326 \nu^{14} - 9954704 \nu^{13} + 5268764 \nu^{12} + 94577321 \nu^{11} + \cdots - 22861184 ) / 1247936 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 524711 \nu^{15} - 423750 \nu^{14} - 12681760 \nu^{13} + 8084172 \nu^{12} + 121020601 \nu^{11} + \cdots - 29967232 ) / 1247936 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 17613 \nu^{15} - 15970 \nu^{14} - 416256 \nu^{13} + 315428 \nu^{12} + 3856563 \nu^{11} + \cdots - 482240 ) / 40256 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 767889 \nu^{15} - 741754 \nu^{14} - 18434160 \nu^{13} + 14936836 \nu^{12} + 174451711 \nu^{11} + \cdots - 25501376 ) / 1247936 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 781759 \nu^{15} - 787334 \nu^{14} - 18711408 \nu^{13} + 15967900 \nu^{12} + 176274769 \nu^{11} + \cdots - 9694400 ) / 1247936 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 785211 \nu^{15} + 785182 \nu^{14} + 18568432 \nu^{13} - 15860300 \nu^{12} - 172213973 \nu^{11} + \cdots + 12619200 ) / 1247936 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 25785 \nu^{15} + 22922 \nu^{14} + 619504 \nu^{13} - 451684 \nu^{12} - 5863511 \nu^{11} + \cdots + 1090240 ) / 40256 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 203897 \nu^{15} + 168786 \nu^{14} + 4840280 \nu^{13} - 3291564 \nu^{12} - 45092591 \nu^{11} + \cdots + 8409984 ) / 311984 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 268687 \nu^{15} + 248146 \nu^{14} + 6403548 \nu^{13} - 4997680 \nu^{12} - 59968933 \nu^{11} + \cdots + 5982064 ) / 311984 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 743697 \nu^{15} - 561258 \nu^{14} - 17803568 \nu^{13} + 10903780 \nu^{12} + 167505279 \nu^{11} + \cdots - 23649344 ) / 623968 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} - \beta_{11} - \beta_{10} + \beta_{6} - \beta_{5} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 8\beta_{2} - \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{14} + 10 \beta_{13} - \beta_{12} - 10 \beta_{11} - 10 \beta_{10} + \beta_{9} - \beta_{7} + \cdots + 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{14} - \beta_{13} + 11 \beta_{12} + 12 \beta_{11} + 11 \beta_{10} + 12 \beta_{9} + 12 \beta_{8} + \cdots + 76 \)
|
| \(\nu^{7}\) | \(=\) |
\( - \beta_{15} + 13 \beta_{14} + 83 \beta_{13} - 13 \beta_{12} - 80 \beta_{11} - 81 \beta_{10} + \cdots + 42 \)
|
| \(\nu^{8}\) | \(=\) |
\( 16 \beta_{14} - 19 \beta_{13} + 96 \beta_{12} + 112 \beta_{11} + 100 \beta_{10} + 108 \beta_{9} + \cdots + 452 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 17 \beta_{15} + 125 \beta_{14} + 651 \beta_{13} - 131 \beta_{12} - 597 \beta_{11} - 618 \beta_{10} + \cdots + 250 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 3 \beta_{15} + 171 \beta_{14} - 237 \beta_{13} + 773 \beta_{12} + 963 \beta_{11} + 854 \beta_{10} + \cdots + 2853 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 196 \beta_{15} + 1071 \beta_{14} + 4994 \beta_{13} - 1199 \beta_{12} - 4347 \beta_{11} - 4624 \beta_{10} + \cdots + 1529 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 63 \beta_{15} + 1545 \beta_{14} - 2457 \beta_{13} + 6004 \beta_{12} + 7979 \beta_{11} + 7061 \beta_{10} + \cdots + 18746 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1927 \beta_{15} + 8665 \beta_{14} + 37940 \beta_{13} - 10419 \beta_{12} - 31431 \beta_{11} + \cdots + 9651 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 838 \beta_{15} + 12813 \beta_{14} - 23034 \beta_{13} + 45821 \beta_{12} + 64739 \beta_{11} + \cdots + 126701 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 17420 \beta_{15} + 67905 \beta_{14} + 286953 \beta_{13} - 87569 \beta_{12} - 227366 \beta_{11} + \cdots + 62508 \)
|
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 |
|
−2.75586 | 2.22549 | 5.59478 | 3.97063 | −6.13314 | 1.71912 | −9.90674 | 1.95279 | −10.9425 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.42429 | −1.02036 | 3.87720 | −0.275174 | 2.47365 | −5.16310 | −4.55089 | −1.95887 | 0.667103 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.80038 | −3.34524 | 1.24138 | 2.67126 | 6.02272 | 1.66993 | 1.36580 | 8.19063 | −4.80929 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.63671 | −0.492566 | 0.678809 | −1.90242 | 0.806186 | 0.625725 | 2.16240 | −2.75738 | 3.11371 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.28888 | 3.25447 | −0.338793 | 4.18846 | −4.19461 | −3.67888 | 3.01442 | 7.59155 | −5.39841 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.03000 | −0.920112 | −0.939097 | 3.95819 | 0.947716 | 4.48231 | 3.02727 | −2.15339 | −4.07695 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.319955 | 2.34279 | −1.89763 | 1.26502 | −0.749588 | 1.17463 | 1.24707 | 2.48866 | −0.404751 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.208883 | −2.00378 | −1.95637 | −2.17278 | 0.418556 | −2.92851 | 0.826420 | 1.01513 | 0.453858 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.445942 | −0.0589774 | −1.80114 | 2.46971 | −0.0263005 | −2.81741 | −1.69509 | −2.99652 | 1.10135 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.745705 | 2.46941 | −1.44392 | −2.81935 | 1.84145 | 3.27962 | −2.56815 | 3.09797 | −2.10240 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.84222 | 1.43653 | 1.39376 | 2.07719 | 2.64640 | 4.15058 | −1.11683 | −0.936385 | 3.82664 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.88851 | 0.985425 | 1.56646 | 3.37829 | 1.86098 | 1.01841 | −0.818750 | −2.02894 | 6.37991 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.13862 | 3.19578 | 2.57368 | −0.343330 | 6.83454 | −2.49131 | 1.22687 | 7.21299 | −0.734251 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.14844 | −1.74064 | 2.61580 | −1.42439 | −3.73966 | 3.61731 | 1.32302 | 0.0298167 | −3.06021 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.51658 | −2.82858 | 4.33319 | 2.87619 | −7.11834 | −3.18638 | 5.87166 | 5.00084 | 7.23817 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.73896 | 1.50037 | 5.50188 | −1.91750 | 4.10945 | −0.472052 | 9.59151 | −0.748891 | −5.25195 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(23\) | \(-1\) |
| \(29\) | \(1\) |
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 | 667.2.a.d | ✓ | 16 |
| 3.b | odd | 2 | 1 | 6003.2.a.q | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 667.2.a.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 6003.2.a.q | 16 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 3 T_{2}^{15} - 22 T_{2}^{14} + 68 T_{2}^{13} + 187 T_{2}^{12} - 597 T_{2}^{11} - 795 T_{2}^{10} + \cdots + 64 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(667))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 3 T^{15} + \cdots + 64 \)
|
| $3$ |
\( T^{16} - 5 T^{15} + \cdots + 256 \)
|
| $5$ |
\( T^{16} - 16 T^{15} + \cdots - 33344 \)
|
| $7$ |
\( T^{16} - T^{15} + \cdots - 278528 \)
|
| $11$ |
\( T^{16} - 4 T^{15} + \cdots + 192704 \)
|
| $13$ |
\( T^{16} - 15 T^{15} + \cdots + 2778588 \)
|
| $17$ |
\( T^{16} - 20 T^{15} + \cdots + 157952 \)
|
| $19$ |
\( T^{16} + 4 T^{15} + \cdots - 10881280 \)
|
| $23$ |
\( (T - 1)^{16} \)
|
| $29$ |
\( (T + 1)^{16} \)
|
| $31$ |
\( T^{16} + \cdots + 148582656 \)
|
| $37$ |
\( T^{16} - 5 T^{15} + \cdots + 78877952 \)
|
| $41$ |
\( T^{16} - 7 T^{15} + \cdots + 4709632 \)
|
| $43$ |
\( T^{16} + \cdots + 489266892 \)
|
| $47$ |
\( T^{16} + \cdots - 3749718976 \)
|
| $53$ |
\( T^{16} + \cdots + 114419215472 \)
|
| $59$ |
\( T^{16} + \cdots + 1514229698560 \)
|
| $61$ |
\( T^{16} + \cdots + 1260900834304 \)
|
| $67$ |
\( T^{16} + \cdots + 10321323188224 \)
|
| $71$ |
\( T^{16} + \cdots + 13531730176 \)
|
| $73$ |
\( T^{16} + \cdots - 194107921412096 \)
|
| $79$ |
\( T^{16} + \cdots - 238723830651340 \)
|
| $83$ |
\( T^{16} + \cdots - 4205662739456 \)
|
| $89$ |
\( T^{16} + \cdots - 16474890357760 \)
|
| $97$ |
\( T^{16} + \cdots - 73\!\cdots\!08 \)
|
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