Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,17,Mod(3,7)]
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([1]))
N = Newforms(chi, 17, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.3");
S:= CuspForms(chi, 17);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 17 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.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: | \(11.3627180700\) |
| 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: |
\( x^{20} - 2 x^{19} + 470221 x^{18} - 17792382 x^{17} + 149610140535 x^{16} - 5885709493452 x^{15} + \cdots + 26\!\cdots\!25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{38}\cdot 3^{16}\cdot 7^{22} \) |
| 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
\( x^{20} - 2 x^{19} + 470221 x^{18} - 17792382 x^{17} + 149610140535 x^{16} - 5885709493452 x^{15} + \cdots + 26\!\cdots\!25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 81\!\cdots\!00 \nu^{19} + \cdots - 18\!\cdots\!60 ) / 12\!\cdots\!35 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 12\!\cdots\!30 \nu^{19} + \cdots + 55\!\cdots\!00 ) / 32\!\cdots\!65 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 21\!\cdots\!78 \nu^{19} + \cdots + 21\!\cdots\!75 ) / 54\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 20\!\cdots\!61 \nu^{19} + \cdots - 17\!\cdots\!75 ) / 18\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 95\!\cdots\!79 \nu^{19} + \cdots + 15\!\cdots\!25 ) / 54\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 38\!\cdots\!71 \nu^{19} + \cdots - 35\!\cdots\!00 ) / 18\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 60\!\cdots\!39 \nu^{19} + \cdots + 16\!\cdots\!75 ) / 54\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 25\!\cdots\!27 \nu^{19} + \cdots - 39\!\cdots\!90 ) / 18\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 91\!\cdots\!86 \nu^{19} + \cdots + 37\!\cdots\!95 ) / 54\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 56\!\cdots\!08 \nu^{19} + \cdots - 21\!\cdots\!35 ) / 54\!\cdots\!60 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 57\!\cdots\!42 \nu^{19} + \cdots + 43\!\cdots\!55 ) / 54\!\cdots\!60 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 64\!\cdots\!17 \nu^{19} + \cdots + 65\!\cdots\!95 ) / 27\!\cdots\!80 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 59\!\cdots\!31 \nu^{19} + \cdots + 40\!\cdots\!90 ) / 23\!\cdots\!20 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 58\!\cdots\!77 \nu^{19} + \cdots + 10\!\cdots\!30 ) / 54\!\cdots\!60 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 39\!\cdots\!75 \nu^{19} + \cdots + 32\!\cdots\!50 ) / 18\!\cdots\!20 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 16\!\cdots\!76 \nu^{19} + \cdots + 71\!\cdots\!95 ) / 54\!\cdots\!60 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 32\!\cdots\!52 \nu^{19} + \cdots - 26\!\cdots\!15 ) / 90\!\cdots\!60 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 10\!\cdots\!29 \nu^{19} + \cdots - 15\!\cdots\!55 ) / 27\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 2\beta_{4} - 28\beta_{3} - 94050\beta_{2} - 28\beta _1 - 94050 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{16} - 2 \beta_{14} + \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + \cdots + 2561232 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 58 \beta_{19} + 8 \beta_{17} + 60 \beta_{16} + 322 \beta_{15} + 1018 \beta_{14} - 30 \beta_{13} + \cdots - 84 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5330 \beta_{19} + 11462 \beta_{18} - 7614 \beta_{17} + 353662 \beta_{15} + 343722 \beta_{14} + \cdots - 1175558557942 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 5822296 \beta_{19} - 9804750 \beta_{18} - 21680706 \beta_{17} - 26039904 \beta_{16} + \cdots + 31\!\cdots\!72 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2664712038 \beta_{19} + 5821239808 \beta_{17} + 68823969001 \beta_{16} - 42645819003 \beta_{15} + \cdots - 86287688425 \)
|
| \(\nu^{8}\) | \(=\) |
\( 5753793941412 \beta_{19} + 2559168325212 \beta_{18} + 3525587873364 \beta_{17} - 1401784311396 \beta_{15} + \cdots - 67\!\cdots\!93 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 18\!\cdots\!04 \beta_{19} + \cdots + 11\!\cdots\!36 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 75\!\cdots\!92 \beta_{19} + \cdots - 48\!\cdots\!36 \)
|
| \(\nu^{11}\) | \(=\) |
\( 35\!\cdots\!08 \beta_{19} + \cdots - 32\!\cdots\!87 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 19\!\cdots\!68 \beta_{19} + \cdots + 36\!\cdots\!43 \)
|
| \(\nu^{13}\) | \(=\) |
\( 33\!\cdots\!62 \beta_{19} + \cdots - 13\!\cdots\!28 \)
|
| \(\nu^{14}\) | \(=\) |
\( 88\!\cdots\!90 \beta_{19} + \cdots - 85\!\cdots\!72 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 34\!\cdots\!56 \beta_{19} + \cdots + 24\!\cdots\!26 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 76\!\cdots\!96 \beta_{19} + \cdots - 10\!\cdots\!52 \)
|
| \(\nu^{17}\) | \(=\) |
\( 74\!\cdots\!92 \beta_{19} + \cdots - 64\!\cdots\!24 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 36\!\cdots\!84 \beta_{19} + \cdots + 50\!\cdots\!82 \)
|
| \(\nu^{19}\) | \(=\) |
\( 25\!\cdots\!44 \beta_{19} + \cdots - 20\!\cdots\!05 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−246.634 | − | 427.182i | 6439.04 | + | 3717.58i | −88888.5 | + | 153959.i | 360858. | − | 208341.i | − | 3.66753e6i | −3.56165e6 | − | 4.53294e6i | 5.53649e7 | 6.11747e6 | + | 1.05958e7i | −1.77999e8 | − | 1.02768e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.2 | −183.065 | − | 317.078i | −5337.54 | − | 3081.63i | −34257.8 | + | 59336.2i | −149102. | + | 86084.1i | 2.25656e6i | 303703. | + | 5.75680e6i | 1.09092e6 | −2.53044e6 | − | 4.38285e6i | 5.45908e7 | + | 3.15180e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.3 | −116.323 | − | 201.477i | 2610.90 | + | 1507.40i | 5705.88 | − | 9882.88i | −301331. | + | 173973.i | − | 701383.i | 2.93221e6 | − | 4.96337e6i | −1.79016e7 | −1.69788e7 | − | 2.94082e7i | 7.01035e7 | + | 4.04742e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.4 | −59.9555 | − | 103.846i | −7943.79 | − | 4586.35i | 25578.7 | − | 44303.6i | 625327. | − | 361033.i | 1.09991e6i | −1.45978e6 | − | 5.57691e6i | −1.39928e7 | 2.05458e7 | + | 3.55864e7i | −7.49836e7 | − | 4.32918e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.5 | −54.0903 | − | 93.6871i | 8486.99 | + | 4899.97i | 26916.5 | − | 46620.7i | 214019. | − | 123564.i | − | 1.06016e6i | 516705. | + | 5.74160e6i | −1.29134e7 | 2.64960e7 | + | 4.58924e7i | −2.31527e7 | − | 1.33672e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.6 | 49.5238 | + | 85.7778i | 289.874 | + | 167.359i | 27862.8 | − | 48259.7i | −159884. | + | 92309.1i | 33153.0i | −5.76147e6 | + | 195847.i | 1.20107e7 | −2.14673e7 | − | 3.71825e7i | −1.58361e7 | − | 9.14300e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.7 | 94.8350 | + | 164.259i | −9421.11 | − | 5439.28i | 14780.6 | − | 25600.8i | −477118. | + | 275464.i | − | 2.06334e6i | 5.57858e6 | + | 1.45338e6i | 1.80371e7 | 3.76482e7 | + | 6.52085e7i | −9.04950e7 | − | 5.22473e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.8 | 141.862 | + | 245.712i | 1577.48 | + | 910.759i | −7481.54 | + | 12958.4i | 402298. | − | 232267.i | 516808.i | 5.58849e6 | + | 1.41483e6i | 1.43487e7 | −1.98644e7 | − | 3.44061e7i | 1.14141e8 | + | 6.58996e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.9 | 185.212 | + | 320.796i | 10622.1 | + | 6132.66i | −35838.6 | + | 62074.3i | −453136. | + | 261618.i | 4.54336e6i | 694120. | − | 5.72286e6i | −2.27485e6 | 5.36956e7 | + | 9.30036e7i | −1.67852e8 | − | 9.69094e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.10 | 234.636 | + | 406.401i | −4044.92 | − | 2335.34i | −77340.0 | + | 133957.i | 59013.4 | − | 34071.4i | − | 2.19181e6i | −5.75487e6 | + | 338210.i | −4.18327e7 | −1.06158e7 | − | 1.83871e7i | 2.76933e7 | + | 1.59887e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.1 | −246.634 | + | 427.182i | 6439.04 | − | 3717.58i | −88888.5 | − | 153959.i | 360858. | + | 208341.i | 3.66753e6i | −3.56165e6 | + | 4.53294e6i | 5.53649e7 | 6.11747e6 | − | 1.05958e7i | −1.77999e8 | + | 1.02768e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.2 | −183.065 | + | 317.078i | −5337.54 | + | 3081.63i | −34257.8 | − | 59336.2i | −149102. | − | 86084.1i | − | 2.25656e6i | 303703. | − | 5.75680e6i | 1.09092e6 | −2.53044e6 | + | 4.38285e6i | 5.45908e7 | − | 3.15180e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.3 | −116.323 | + | 201.477i | 2610.90 | − | 1507.40i | 5705.88 | + | 9882.88i | −301331. | − | 173973.i | 701383.i | 2.93221e6 | + | 4.96337e6i | −1.79016e7 | −1.69788e7 | + | 2.94082e7i | 7.01035e7 | − | 4.04742e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.4 | −59.9555 | + | 103.846i | −7943.79 | + | 4586.35i | 25578.7 | + | 44303.6i | 625327. | + | 361033.i | − | 1.09991e6i | −1.45978e6 | + | 5.57691e6i | −1.39928e7 | 2.05458e7 | − | 3.55864e7i | −7.49836e7 | + | 4.32918e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.5 | −54.0903 | + | 93.6871i | 8486.99 | − | 4899.97i | 26916.5 | + | 46620.7i | 214019. | + | 123564.i | 1.06016e6i | 516705. | − | 5.74160e6i | −1.29134e7 | 2.64960e7 | − | 4.58924e7i | −2.31527e7 | + | 1.33672e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.6 | 49.5238 | − | 85.7778i | 289.874 | − | 167.359i | 27862.8 | + | 48259.7i | −159884. | − | 92309.1i | − | 33153.0i | −5.76147e6 | − | 195847.i | 1.20107e7 | −2.14673e7 | + | 3.71825e7i | −1.58361e7 | + | 9.14300e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.7 | 94.8350 | − | 164.259i | −9421.11 | + | 5439.28i | 14780.6 | + | 25600.8i | −477118. | − | 275464.i | 2.06334e6i | 5.57858e6 | − | 1.45338e6i | 1.80371e7 | 3.76482e7 | − | 6.52085e7i | −9.04950e7 | + | 5.22473e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.8 | 141.862 | − | 245.712i | 1577.48 | − | 910.759i | −7481.54 | − | 12958.4i | 402298. | + | 232267.i | − | 516808.i | 5.58849e6 | − | 1.41483e6i | 1.43487e7 | −1.98644e7 | + | 3.44061e7i | 1.14141e8 | − | 6.58996e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.9 | 185.212 | − | 320.796i | 10622.1 | − | 6132.66i | −35838.6 | − | 62074.3i | −453136. | − | 261618.i | − | 4.54336e6i | 694120. | + | 5.72286e6i | −2.27485e6 | 5.36956e7 | − | 9.30036e7i | −1.67852e8 | + | 9.69094e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.10 | 234.636 | − | 406.401i | −4044.92 | + | 2335.34i | −77340.0 | − | 133957.i | 59013.4 | + | 34071.4i | 2.19181e6i | −5.75487e6 | − | 338210.i | −4.18327e7 | −1.06158e7 | + | 1.83871e7i | 2.76933e7 | − | 1.59887e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 7.17.d.a | ✓ | 20 |
| 7.c | even | 3 | 1 | 49.17.b.a | 20 | ||
| 7.d | odd | 6 | 1 | inner | 7.17.d.a | ✓ | 20 |
| 7.d | odd | 6 | 1 | 49.17.b.a | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.17.d.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 7.17.d.a | ✓ | 20 | 7.d | odd | 6 | 1 | inner |
| 49.17.b.a | 20 | 7.c | even | 3 | 1 | ||
| 49.17.b.a | 20 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{17}^{\mathrm{new}}(7, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 25\!\cdots\!56 \)
|
| $3$ |
\( T^{20} + \cdots + 22\!\cdots\!21 \)
|
| $5$ |
\( T^{20} + \cdots + 56\!\cdots\!25 \)
|
| $7$ |
\( T^{20} + \cdots + 16\!\cdots\!01 \)
|
| $11$ |
\( T^{20} + \cdots + 36\!\cdots\!69 \)
|
| $13$ |
\( T^{20} + \cdots + 11\!\cdots\!00 \)
|
| $17$ |
\( T^{20} + \cdots + 40\!\cdots\!01 \)
|
| $19$ |
\( T^{20} + \cdots + 29\!\cdots\!09 \)
|
| $23$ |
\( T^{20} + \cdots + 51\!\cdots\!21 \)
|
| $29$ |
\( (T^{10} + \cdots + 22\!\cdots\!76)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 22\!\cdots\!61 \)
|
| $37$ |
\( T^{20} + \cdots + 11\!\cdots\!81 \)
|
| $41$ |
\( T^{20} + \cdots + 74\!\cdots\!00 \)
|
| $43$ |
\( (T^{10} + \cdots - 36\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 45\!\cdots\!61 \)
|
| $53$ |
\( T^{20} + \cdots + 14\!\cdots\!25 \)
|
| $59$ |
\( T^{20} + \cdots + 10\!\cdots\!89 \)
|
| $61$ |
\( T^{20} + \cdots + 15\!\cdots\!89 \)
|
| $67$ |
\( T^{20} + \cdots + 13\!\cdots\!09 \)
|
| $71$ |
\( (T^{10} + \cdots - 15\!\cdots\!64)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 49\!\cdots\!89 \)
|
| $79$ |
\( T^{20} + \cdots + 29\!\cdots\!29 \)
|
| $83$ |
\( T^{20} + \cdots + 42\!\cdots\!00 \)
|
| $89$ |
\( T^{20} + \cdots + 33\!\cdots\!41 \)
|
| $97$ |
\( T^{20} + \cdots + 85\!\cdots\!00 \)
|
| show more | |
| show less |