Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,4,Mod(23,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.23");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 77.e (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: | \(4.54314707044\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Relative dimension: | \(9\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 3 x^{17} + 53 x^{16} - 158 x^{15} + 1893 x^{14} - 5348 x^{13} + 36847 x^{12} - 90579 x^{11} + \cdots + 46656 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 3 x^{17} + 53 x^{16} - 158 x^{15} + 1893 x^{14} - 5348 x^{13} + 36847 x^{12} - 90579 x^{11} + \cdots + 46656 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 17\!\cdots\!43 \nu^{17} + \cdots + 42\!\cdots\!34 ) / 38\!\cdots\!22 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 46\!\cdots\!19 \nu^{17} + \cdots - 53\!\cdots\!08 ) / 30\!\cdots\!76 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 10\!\cdots\!12 \nu^{17} + \cdots + 33\!\cdots\!52 ) / 36\!\cdots\!12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 10\!\cdots\!62 \nu^{17} + \cdots + 19\!\cdots\!24 ) / 27\!\cdots\!84 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 10\!\cdots\!23 \nu^{17} + \cdots + 23\!\cdots\!48 ) / 12\!\cdots\!04 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 22\!\cdots\!04 \nu^{17} + \cdots - 56\!\cdots\!00 ) / 24\!\cdots\!08 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 20\!\cdots\!17 \nu^{17} + \cdots + 68\!\cdots\!28 ) / 22\!\cdots\!72 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 24\!\cdots\!64 \nu^{17} + \cdots - 26\!\cdots\!76 ) / 11\!\cdots\!36 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 20\!\cdots\!49 \nu^{17} + \cdots - 48\!\cdots\!68 ) / 82\!\cdots\!36 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 63\!\cdots\!19 \nu^{17} + \cdots - 14\!\cdots\!00 ) / 24\!\cdots\!08 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 38\!\cdots\!07 \nu^{17} + \cdots - 48\!\cdots\!20 ) / 11\!\cdots\!36 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 21\!\cdots\!03 \nu^{17} + \cdots - 89\!\cdots\!72 ) / 61\!\cdots\!52 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 20\!\cdots\!28 \nu^{17} + \cdots - 43\!\cdots\!12 ) / 55\!\cdots\!68 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 42\!\cdots\!20 \nu^{17} + \cdots - 10\!\cdots\!64 ) / 11\!\cdots\!36 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 11\!\cdots\!86 \nu^{17} + \cdots - 21\!\cdots\!88 ) / 27\!\cdots\!84 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 77\!\cdots\!93 \nu^{17} + \cdots + 23\!\cdots\!32 ) / 12\!\cdots\!04 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{16} + 11\beta_{5} + \beta_{2} - 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - 20\beta_{3} + 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( -23\beta_{16} + \beta_{15} + 3\beta_{14} - \beta_{10} + 2\beta_{9} + \beta_{8} - 3\beta_{7} - 215\beta_{5} + 5\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{17} + \beta_{15} + 35 \beta_{14} - 35 \beta_{13} + 28 \beta_{12} - 12 \beta_{11} + 26 \beta_{8} + \cdots - 306 \)
|
| \(\nu^{6}\) | \(=\) |
\( 28 \beta_{17} - 107 \beta_{13} + 61 \beta_{12} + 119 \beta_{11} + 37 \beta_{10} - 61 \beta_{9} + \cdots + 4508 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 10 \beta_{16} - 15 \beta_{15} - 964 \beta_{14} + 598 \beta_{10} - 661 \beta_{9} - 598 \beta_{8} + \cdots + 9523 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 638 \beta_{17} + 11403 \beta_{16} - 638 \beta_{15} - 2907 \beta_{14} + 2907 \beta_{13} - 1491 \beta_{12} + \cdots - 97392 \)
|
| \(\nu^{9}\) | \(=\) |
\( 128 \beta_{17} + 24251 \beta_{13} - 14851 \beta_{12} + 690 \beta_{11} - 13188 \beta_{10} + 14851 \beta_{9} + \cdots + 206053 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 253463 \beta_{16} + 13816 \beta_{15} + 71422 \beta_{14} - 24365 \beta_{10} + 34138 \beta_{9} + \cdots + 149433 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 14731 \beta_{17} + 17696 \beta_{16} - 14731 \beta_{15} + 584326 \beta_{14} - 584326 \beta_{13} + \cdots - 4949053 \)
|
| \(\nu^{12}\) | \(=\) |
\( 299013 \beta_{17} - 1670764 \beta_{13} + 765325 \beta_{12} + 2104238 \beta_{11} + 579854 \beta_{10} + \cdots + 47595766 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 516174 \beta_{16} + 582976 \beta_{15} - 13757490 \beta_{14} + 6146989 \beta_{10} + \cdots + 105512874 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 6590839 \beta_{17} + 125597045 \beta_{16} - 6590839 \beta_{15} - 38018537 \beta_{14} + \cdots - 1065193589 \)
|
| \(\nu^{15}\) | \(=\) |
\( 18152026 \beta_{17} + 319624082 \beta_{13} - 157601310 \beta_{12} - 6966770 \beta_{11} + \cdots + 2718750580 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 2800292381 \beta_{16} + 148610844 \beta_{15} + 850406736 \beta_{14} - 322629772 \beta_{10} + \cdots + 2269728736 \beta_1 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 507977352 \beta_{17} + 353036928 \beta_{16} - 507977352 \beta_{15} + 7366407065 \beta_{14} + \cdots - 63002934585 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).
| \(n\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(-\beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 23.1 |
|
−2.40648 | + | 4.16815i | 0.0813211 | + | 0.140852i | −7.58230 | − | 13.1329i | 5.94772 | − | 10.3017i | −0.782791 | 18.5130 | − | 0.518012i | 34.4830 | 13.4868 | − | 23.3598i | 28.6261 | + | 49.5819i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.2 | −2.35081 | + | 4.07172i | 4.49880 | + | 7.79214i | −7.05261 | − | 12.2155i | 1.59013 | − | 2.75419i | −42.3033 | −17.9423 | + | 4.59075i | 28.7044 | −26.9783 | + | 46.7279i | 7.47619 | + | 12.9491i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.3 | −0.510531 | + | 0.884265i | 2.68150 | + | 4.64450i | 3.47872 | + | 6.02531i | −7.38305 | + | 12.7878i | −5.47596 | −4.70121 | − | 17.9136i | −15.2725 | −0.880919 | + | 1.52580i | −7.53855 | − | 13.0572i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.4 | −0.116526 | + | 0.201829i | 4.49858 | + | 7.79177i | 3.97284 | + | 6.88117i | 9.27777 | − | 16.0696i | −2.09681 | 15.9987 | − | 9.32966i | −3.71617 | −26.9745 | + | 46.7212i | 2.16220 | + | 3.74504i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.5 | 0.247941 | − | 0.429447i | −1.50212 | − | 2.60176i | 3.87705 | + | 6.71525i | 2.44498 | − | 4.23483i | −1.48976 | 12.7987 | + | 13.3863i | 7.81219 | 8.98724 | − | 15.5664i | −1.21242 | − | 2.09998i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.6 | 0.661138 | − | 1.14512i | −4.88554 | − | 8.46200i | 3.12579 | + | 5.41403i | 7.63232 | − | 13.2196i | −12.9200 | −17.5557 | − | 5.89884i | 18.8445 | −34.2369 | + | 59.3001i | −10.0920 | − | 17.4799i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.7 | 1.64472 | − | 2.84875i | 2.34196 | + | 4.05639i | −1.41023 | − | 2.44259i | −7.91028 | + | 13.7010i | 15.4075 | 16.9213 | + | 7.52800i | 17.0378 | 2.53046 | − | 4.38288i | 26.0204 | + | 45.0687i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.8 | 1.96802 | − | 3.40871i | 1.46499 | + | 2.53744i | −3.74618 | − | 6.48858i | 8.68191 | − | 15.0375i | 11.5325 | −10.7987 | + | 15.0462i | 1.99807 | 9.20759 | − | 15.9480i | −34.1723 | − | 59.1882i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.9 | 2.36253 | − | 4.09202i | −1.67949 | − | 2.90897i | −7.16307 | − | 12.4068i | −4.28151 | + | 7.41578i | −15.8714 | −14.7338 | − | 11.2213i | −29.8914 | 7.85859 | − | 13.6115i | 20.2303 | + | 35.0400i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.1 | −2.40648 | − | 4.16815i | 0.0813211 | − | 0.140852i | −7.58230 | + | 13.1329i | 5.94772 | + | 10.3017i | −0.782791 | 18.5130 | + | 0.518012i | 34.4830 | 13.4868 | + | 23.3598i | 28.6261 | − | 49.5819i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.2 | −2.35081 | − | 4.07172i | 4.49880 | − | 7.79214i | −7.05261 | + | 12.2155i | 1.59013 | + | 2.75419i | −42.3033 | −17.9423 | − | 4.59075i | 28.7044 | −26.9783 | − | 46.7279i | 7.47619 | − | 12.9491i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.3 | −0.510531 | − | 0.884265i | 2.68150 | − | 4.64450i | 3.47872 | − | 6.02531i | −7.38305 | − | 12.7878i | −5.47596 | −4.70121 | + | 17.9136i | −15.2725 | −0.880919 | − | 1.52580i | −7.53855 | + | 13.0572i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.4 | −0.116526 | − | 0.201829i | 4.49858 | − | 7.79177i | 3.97284 | − | 6.88117i | 9.27777 | + | 16.0696i | −2.09681 | 15.9987 | + | 9.32966i | −3.71617 | −26.9745 | − | 46.7212i | 2.16220 | − | 3.74504i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.5 | 0.247941 | + | 0.429447i | −1.50212 | + | 2.60176i | 3.87705 | − | 6.71525i | 2.44498 | + | 4.23483i | −1.48976 | 12.7987 | − | 13.3863i | 7.81219 | 8.98724 | + | 15.5664i | −1.21242 | + | 2.09998i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.6 | 0.661138 | + | 1.14512i | −4.88554 | + | 8.46200i | 3.12579 | − | 5.41403i | 7.63232 | + | 13.2196i | −12.9200 | −17.5557 | + | 5.89884i | 18.8445 | −34.2369 | − | 59.3001i | −10.0920 | + | 17.4799i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.7 | 1.64472 | + | 2.84875i | 2.34196 | − | 4.05639i | −1.41023 | + | 2.44259i | −7.91028 | − | 13.7010i | 15.4075 | 16.9213 | − | 7.52800i | 17.0378 | 2.53046 | + | 4.38288i | 26.0204 | − | 45.0687i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.8 | 1.96802 | + | 3.40871i | 1.46499 | − | 2.53744i | −3.74618 | + | 6.48858i | 8.68191 | + | 15.0375i | 11.5325 | −10.7987 | − | 15.0462i | 1.99807 | 9.20759 | + | 15.9480i | −34.1723 | + | 59.1882i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.9 | 2.36253 | + | 4.09202i | −1.67949 | + | 2.90897i | −7.16307 | + | 12.4068i | −4.28151 | − | 7.41578i | −15.8714 | −14.7338 | + | 11.2213i | −29.8914 | 7.85859 | + | 13.6115i | 20.2303 | − | 35.0400i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 77.4.e.b | ✓ | 18 |
| 7.c | even | 3 | 1 | inner | 77.4.e.b | ✓ | 18 |
| 7.c | even | 3 | 1 | 539.4.a.j | 9 | ||
| 7.d | odd | 6 | 1 | 539.4.a.k | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.4.e.b | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 77.4.e.b | ✓ | 18 | 7.c | even | 3 | 1 | inner |
| 539.4.a.j | 9 | 7.c | even | 3 | 1 | ||
| 539.4.a.k | 9 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{18} - 3 T_{2}^{17} + 53 T_{2}^{16} - 158 T_{2}^{15} + 1893 T_{2}^{14} - 5348 T_{2}^{13} + \cdots + 46656 \)
acting on \(S_{4}^{\mathrm{new}}(77, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} - 3 T^{17} + \cdots + 46656 \)
|
| $3$ |
\( T^{18} + \cdots + 9129993601 \)
|
| $5$ |
\( T^{18} + \cdots + 33\!\cdots\!84 \)
|
| $7$ |
\( T^{18} + \cdots + 65\!\cdots\!43 \)
|
| $11$ |
\( (T^{2} + 11 T + 121)^{9} \)
|
| $13$ |
\( (T^{9} + \cdots + 129605201872125)^{2} \)
|
| $17$ |
\( T^{18} + \cdots + 40\!\cdots\!76 \)
|
| $19$ |
\( T^{18} + \cdots + 16\!\cdots\!16 \)
|
| $23$ |
\( T^{18} + \cdots + 50\!\cdots\!64 \)
|
| $29$ |
\( (T^{9} + \cdots + 52\!\cdots\!59)^{2} \)
|
| $31$ |
\( T^{18} + \cdots + 62\!\cdots\!24 \)
|
| $37$ |
\( T^{18} + \cdots + 14\!\cdots\!16 \)
|
| $41$ |
\( (T^{9} + \cdots - 35\!\cdots\!48)^{2} \)
|
| $43$ |
\( (T^{9} + \cdots + 91\!\cdots\!32)^{2} \)
|
| $47$ |
\( T^{18} + \cdots + 33\!\cdots\!44 \)
|
| $53$ |
\( T^{18} + \cdots + 26\!\cdots\!16 \)
|
| $59$ |
\( T^{18} + \cdots + 12\!\cdots\!49 \)
|
| $61$ |
\( T^{18} + \cdots + 37\!\cdots\!64 \)
|
| $67$ |
\( T^{18} + \cdots + 46\!\cdots\!76 \)
|
| $71$ |
\( (T^{9} + \cdots - 37\!\cdots\!56)^{2} \)
|
| $73$ |
\( T^{18} + \cdots + 31\!\cdots\!16 \)
|
| $79$ |
\( T^{18} + \cdots + 11\!\cdots\!89 \)
|
| $83$ |
\( (T^{9} + \cdots + 30\!\cdots\!84)^{2} \)
|
| $89$ |
\( T^{18} + \cdots + 87\!\cdots\!96 \)
|
| $97$ |
\( (T^{9} + \cdots - 49\!\cdots\!19)^{2} \)
|
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