Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [65,6,Mod(51,65)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65.51");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 65 = 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 65.c (of order \(2\), degree \(1\), 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: | \(10.4249482878\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} + 227x^{12} + 16583x^{10} + 514217x^{8} + 6872896x^{6} + 35265600x^{4} + 63141120x^{2} + 26873856 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2}\cdot 5^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} + 227x^{12} + 16583x^{10} + 514217x^{8} + 6872896x^{6} + 35265600x^{4} + 63141120x^{2} + 26873856 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 153323 \nu^{12} - 33150733 \nu^{10} - 2224588081 \nu^{8} - 62419381495 \nu^{6} + \cdots - 16260941604528 ) / 412866726480 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 6095749 \nu^{12} + 1269875447 \nu^{10} + 77435901659 \nu^{8} + 1693739423717 \nu^{6} + \cdots - 96699000904704 ) / 3302933811840 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1587505 \nu^{12} - 354871457 \nu^{10} - 25078400129 \nu^{8} - 725931604763 \nu^{6} + \cdots - 9437062940928 ) / 550488968640 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 8179421 \nu^{12} - 1833956407 \nu^{10} - 130694362027 \nu^{8} - 3869788643461 \nu^{6} + \cdots - 118009485315072 ) / 660586762368 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13852441 \nu^{12} - 3107294447 \nu^{10} - 221973779099 \nu^{8} - 6640014720509 \nu^{6} + \cdots - 227071566103680 ) / 1100977937280 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 51973429 \nu^{12} - 11771418959 \nu^{10} - 855185469923 \nu^{8} - 26158724567405 \nu^{6} + \cdots - 15\!\cdots\!04 ) / 3302933811840 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 55653181 \nu^{13} - 12567036551 \nu^{11} - 908575583867 \nu^{9} + \cdots - 14\!\cdots\!36 \nu ) / 356716851678720 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 73690502 \nu^{13} + 16449406381 \nu^{11} + 1160165094667 \nu^{9} + \cdots + 21\!\cdots\!08 \nu ) / 44589606459840 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 409593517 \nu^{13} + 91606124039 \nu^{11} + 6485680353563 \nu^{9} + \cdots + 39\!\cdots\!20 \nu ) / 237811234452480 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 1280023163 \nu^{13} + 289041840673 \nu^{11} + 20897238428941 \nu^{9} + \cdots + 41\!\cdots\!28 \nu ) / 713433703357440 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 1325093989 \nu^{13} + 299854797119 \nu^{11} + 21753367545683 \nu^{9} + \cdots + 46\!\cdots\!64 \nu ) / 713433703357440 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 1022703043 \nu^{13} - 233041870889 \nu^{11} - 17098508673173 \nu^{9} + \cdots - 41\!\cdots\!92 \nu ) / 356716851678720 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 1144031411 \nu^{13} + 259880838241 \nu^{11} + 19015898569597 \nu^{9} + \cdots + 79\!\cdots\!16 \nu ) / 178358425839360 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{10} + 23\beta_{7} ) / 25 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{4} - 16\beta_{3} - 4\beta_{2} - 29\beta _1 - 819 ) / 25 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 25\beta_{12} + 150\beta_{11} - 269\beta_{10} + 25\beta_{9} - 25\beta_{8} - 1756\beta_{7} ) / 25 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -20\beta_{6} + 55\beta_{5} - 146\beta_{4} + 489\beta_{3} + 121\beta_{2} + 656\beta _1 + 13341 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 800 \beta_{13} - 3850 \beta_{12} - 25100 \beta_{11} + 31722 \beta_{10} - 1600 \beta_{9} + \cdots + 184803 \beta_{7} ) / 25 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 13900 \beta_{6} - 42900 \beta_{5} + 99134 \beta_{4} - 312736 \beta_{3} - 74834 \beta_{2} + \cdots - 7093099 ) / 25 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 134400 \beta_{13} + 500575 \beta_{12} + 3346000 \beta_{11} - 3822189 \beta_{10} + \cdots - 21484536 \beta_{7} ) / 25 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 342140 \beta_{6} + 1109175 \beta_{5} - 2502852 \beta_{4} + 7750853 \beta_{3} + 1819407 \beta_{2} + \cdots + 165327137 ) / 5 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 17926200 \beta_{13} - 62192600 \beta_{12} - 419927850 \beta_{11} + 464780942 \beta_{10} + \cdots + 2579020983 \beta_{7} ) / 25 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 207763700 \beta_{6} - 687307000 \beta_{5} + 1544398164 \beta_{4} - 4754946656 \beta_{3} + \cdots - 99324212379 ) / 25 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 2248957000 \beta_{13} + 7625764425 \beta_{12} + 51714169850 \beta_{11} + \cdots - 312885440216 \beta_{7} ) / 25 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 5051399940 \beta_{6} + 16844572395 \beta_{5} - 37831049478 \beta_{4} + 116257812317 \beta_{3} + \cdots + 2410992161373 ) / 5 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 276858622800 \beta_{13} - 931533803850 \beta_{12} - 6328800141400 \beta_{11} + \cdots + 38095163507263 \beta_{7} ) / 25 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(41\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 51.1 |
|
− | 9.04461i | −18.8112 | −49.8050 | − | 25.0000i | 170.140i | 194.877i | 161.040i | 110.862 | −226.115 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.2 | − | 8.60393i | 1.77599 | −42.0277 | 25.0000i | − | 15.2805i | 192.014i | 86.2773i | −239.846 | 215.098 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.3 | − | 6.25184i | 1.69265 | −7.08552 | 25.0000i | − | 10.5822i | − | 79.8024i | − | 155.761i | −240.135 | 156.296 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.4 | − | 4.31108i | 27.4571 | 13.4146 | 25.0000i | − | 118.370i | − | 31.4512i | − | 195.786i | 510.890 | 107.777 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.5 | − | 3.90915i | −14.1735 | 16.7186 | − | 25.0000i | 55.4063i | − | 16.6406i | − | 190.448i | −42.1117 | −97.7287 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.6 | − | 3.56769i | −28.5641 | 19.2716 | 25.0000i | 101.908i | 143.616i | − | 182.921i | 572.907 | 89.1924 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.7 | − | 1.21921i | −11.3769 | 30.5135 | 25.0000i | 13.8708i | − | 6.14028i | − | 76.2172i | −113.566 | 30.4803 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.8 | 1.21921i | −11.3769 | 30.5135 | − | 25.0000i | − | 13.8708i | 6.14028i | 76.2172i | −113.566 | 30.4803 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.9 | 3.56769i | −28.5641 | 19.2716 | − | 25.0000i | − | 101.908i | − | 143.616i | 182.921i | 572.907 | 89.1924 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.10 | 3.90915i | −14.1735 | 16.7186 | 25.0000i | − | 55.4063i | 16.6406i | 190.448i | −42.1117 | −97.7287 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.11 | 4.31108i | 27.4571 | 13.4146 | − | 25.0000i | 118.370i | 31.4512i | 195.786i | 510.890 | 107.777 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.12 | 6.25184i | 1.69265 | −7.08552 | − | 25.0000i | 10.5822i | 79.8024i | 155.761i | −240.135 | 156.296 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.13 | 8.60393i | 1.77599 | −42.0277 | − | 25.0000i | 15.2805i | − | 192.014i | − | 86.2773i | −239.846 | 215.098 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.14 | 9.04461i | −18.8112 | −49.8050 | 25.0000i | − | 170.140i | − | 194.877i | − | 161.040i | 110.862 | −226.115 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 65.6.c.b | ✓ | 14 |
| 13.b | even | 2 | 1 | inner | 65.6.c.b | ✓ | 14 |
| 13.d | odd | 4 | 1 | 845.6.a.i | 7 | ||
| 13.d | odd | 4 | 1 | 845.6.a.j | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.6.c.b | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 65.6.c.b | ✓ | 14 | 13.b | even | 2 | 1 | inner |
| 845.6.a.i | 7 | 13.d | odd | 4 | 1 | ||
| 845.6.a.j | 7 | 13.d | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} + 243 T_{2}^{12} + 22303 T_{2}^{10} + 978289 T_{2}^{8} + 21825120 T_{2}^{6} + \cdots + 1271920896 \)
acting on \(S_{6}^{\mathrm{new}}(65, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} + \cdots + 1271920896 \)
|
| $3$ |
\( (T^{7} + 42 T^{6} + \cdots - 7151544)^{2} \)
|
| $5$ |
\( (T^{2} + 625)^{7} \)
|
| $7$ |
\( T^{14} + \cdots + 18\!\cdots\!00 \)
|
| $11$ |
\( T^{14} + \cdots + 14\!\cdots\!00 \)
|
| $13$ |
\( T^{14} + \cdots + 97\!\cdots\!57 \)
|
| $17$ |
\( (T^{7} + \cdots + 44\!\cdots\!00)^{2} \)
|
| $19$ |
\( T^{14} + \cdots + 13\!\cdots\!16 \)
|
| $23$ |
\( (T^{7} + \cdots + 28\!\cdots\!96)^{2} \)
|
| $29$ |
\( (T^{7} + \cdots + 15\!\cdots\!60)^{2} \)
|
| $31$ |
\( T^{14} + \cdots + 15\!\cdots\!00 \)
|
| $37$ |
\( T^{14} + \cdots + 19\!\cdots\!96 \)
|
| $41$ |
\( T^{14} + \cdots + 32\!\cdots\!00 \)
|
| $43$ |
\( (T^{7} + \cdots + 19\!\cdots\!68)^{2} \)
|
| $47$ |
\( T^{14} + \cdots + 98\!\cdots\!04 \)
|
| $53$ |
\( (T^{7} + \cdots - 17\!\cdots\!16)^{2} \)
|
| $59$ |
\( T^{14} + \cdots + 87\!\cdots\!96 \)
|
| $61$ |
\( (T^{7} + \cdots - 59\!\cdots\!12)^{2} \)
|
| $67$ |
\( T^{14} + \cdots + 15\!\cdots\!84 \)
|
| $71$ |
\( T^{14} + \cdots + 23\!\cdots\!00 \)
|
| $73$ |
\( T^{14} + \cdots + 18\!\cdots\!96 \)
|
| $79$ |
\( (T^{7} + \cdots - 12\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{14} + \cdots + 70\!\cdots\!96 \)
|
| $89$ |
\( T^{14} + \cdots + 30\!\cdots\!76 \)
|
| $97$ |
\( T^{14} + \cdots + 77\!\cdots\!04 \)
|
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