Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [845,2,Mod(146,845)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(845, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("845.146");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 845 = 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 845.e (of order \(3\), degree \(2\), not 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: | \(6.74735897080\) |
| 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: |
\( x^{18} + 17 x^{16} - 18 x^{15} + 230 x^{14} - 185 x^{13} + 996 x^{12} - 534 x^{11} + 3020 x^{10} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{18} + 17 x^{16} - 18 x^{15} + 230 x^{14} - 185 x^{13} + 996 x^{12} - 534 x^{11} + 3020 x^{10} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2598930943940 \nu^{17} + 261288719630784 \nu^{16} + 49188219965662 \nu^{15} + \cdots + 27\!\cdots\!70 ) / 67\!\cdots\!38 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 29\!\cdots\!07 \nu^{17} + \cdots + 46\!\cdots\!54 ) / 35\!\cdots\!74 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 45\!\cdots\!26 \nu^{17} + \cdots - 30\!\cdots\!93 ) / 35\!\cdots\!74 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 51\!\cdots\!88 \nu^{17} + \cdots - 10\!\cdots\!28 ) / 35\!\cdots\!74 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 63\!\cdots\!59 \nu^{17} + \cdots + 15\!\cdots\!21 ) / 35\!\cdots\!74 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 64\!\cdots\!40 \nu^{17} + \cdots + 32\!\cdots\!41 ) / 35\!\cdots\!74 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 12\!\cdots\!33 \nu^{17} + \cdots - 23\!\cdots\!49 ) / 35\!\cdots\!74 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 16\!\cdots\!89 \nu^{17} + \cdots + 11\!\cdots\!48 ) / 35\!\cdots\!74 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 22\!\cdots\!98 \nu^{17} + \cdots - 47\!\cdots\!43 ) / 43\!\cdots\!57 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 30\!\cdots\!93 \nu^{17} + \cdots + 42\!\cdots\!51 ) / 35\!\cdots\!74 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 52\!\cdots\!40 \nu^{17} + \cdots - 61\!\cdots\!81 ) / 35\!\cdots\!74 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 99\!\cdots\!12 \nu^{17} + \cdots - 87\!\cdots\!65 ) / 35\!\cdots\!74 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 15\!\cdots\!18 \nu^{17} + \cdots + 99\!\cdots\!61 ) / 35\!\cdots\!74 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 16\!\cdots\!20 \nu^{17} + \cdots - 49\!\cdots\!78 ) / 35\!\cdots\!74 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 17\!\cdots\!31 \nu^{17} + \cdots + 10\!\cdots\!26 ) / 35\!\cdots\!74 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 20\!\cdots\!56 \nu^{17} + \cdots + 25\!\cdots\!13 ) / 35\!\cdots\!74 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{15} + 2\beta_{14} + \beta_{13} + \beta_{12} - 5\beta_{10} + \beta_{5} + 2\beta_{4} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2 \beta_{17} - \beta_{16} - \beta_{14} - 2 \beta_{11} - 12 \beta_{10} + 3 \beta_{8} + 12 \beta_{7} + \cdots + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -9\beta_{12} + 3\beta_{11} + 46\beta_{10} - 27\beta_{8} + 5\beta_{7} + 6\beta_{6} + 16\beta_{2} + 6\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 27 \beta_{17} + 16 \beta_{16} - 6 \beta_{15} + 29 \beta_{14} + 12 \beta_{13} + 6 \beta_{12} + \cdots - 72 \)
|
| \(\nu^{6}\) | \(=\) |
\( 64 \beta_{17} - 12 \beta_{16} + 103 \beta_{15} - 354 \beta_{14} - 215 \beta_{13} - 64 \beta_{11} + \cdots + 630 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 154 \beta_{12} + 370 \beta_{11} + 1920 \beta_{10} + 215 \beta_{9} - 1091 \beta_{8} + \cdots + 1318 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1091 \beta_{17} + 316 \beta_{16} - 1318 \beta_{15} + 4741 \beta_{14} + 2863 \beta_{13} + 1318 \beta_{12} + \cdots - 8472 \)
|
| \(\nu^{9}\) | \(=\) |
\( 5265 \beta_{17} - 2863 \beta_{16} + 2926 \beta_{15} - 11122 \beta_{14} - 6110 \beta_{13} - 5265 \beta_{11} + \cdots + 23241 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 17918 \beta_{12} + 17415 \beta_{11} + 130797 \beta_{10} + 6110 \beta_{9} - 76701 \beta_{8} + \cdots + 49692 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 76701 \beta_{17} + 39149 \beta_{16} - 49692 \beta_{15} + 186142 \beta_{14} + 105008 \beta_{13} + \cdots - 374983 \)
|
| \(\nu^{12}\) | \(=\) |
\( 270663 \beta_{17} - 105008 \beta_{16} + 252911 \beta_{15} - 930631 \beta_{14} - 551214 \beta_{13} + \cdots + 1720394 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 800346 \beta_{12} + 1132827 \beta_{11} + 7166108 \beta_{10} + 551214 \beta_{9} - 4152038 \beta_{8} + \cdots + 3657346 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 4152038 \beta_{17} + 1704192 \beta_{16} - 3657346 \beta_{15} + 13497390 \beta_{14} + 7943412 \beta_{13} + \cdots - 25251857 \)
|
| \(\nu^{15}\) | \(=\) |
\( 16867706 \beta_{17} - 7943412 \beta_{16} + 12535678 \beta_{15} - 46573888 \beta_{14} - 26818108 \beta_{13} + \cdots + 90707172 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 53720321 \beta_{12} + 63242434 \beta_{11} + 433023846 \beta_{10} + 26818108 \beta_{9} + \cdots + 193303392 \beta_1 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 252390368 \beta_{17} + 116353757 \beta_{16} - 193303392 \beta_{15} + 717246069 \beta_{14} + \cdots - 1386576839 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) | \(677\) |
| \(\chi(n)\) | \(\beta_{10}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 146.1 |
|
−1.39471 | − | 2.41570i | 0.907309 | + | 1.57151i | −2.89042 | + | 5.00635i | 1.00000 | 2.53086 | − | 4.38358i | −0.134800 | + | 0.233481i | 10.5463 | −0.146421 | + | 0.253609i | −1.39471 | − | 2.41570i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 146.2 | −1.31799 | − | 2.28282i | −0.994722 | − | 1.72291i | −2.47417 | + | 4.28539i | 1.00000 | −2.62206 | + | 4.54154i | 1.64115 | − | 2.84256i | 7.77176 | −0.478943 | + | 0.829553i | −1.31799 | − | 2.28282i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 146.3 | −1.10278 | − | 1.91007i | 0.00652833 | + | 0.0113074i | −1.43225 | + | 2.48072i | 1.00000 | 0.0143986 | − | 0.0249391i | −2.30448 | + | 3.99148i | 1.90669 | 1.49991 | − | 2.59793i | −1.10278 | − | 1.91007i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 146.4 | −0.765438 | − | 1.32578i | −1.44363 | − | 2.50044i | −0.171790 | + | 0.297550i | 1.00000 | −2.21002 | + | 3.82786i | −1.93247 | + | 3.34713i | −2.53577 | −2.66813 | + | 4.62133i | −0.765438 | − | 1.32578i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 146.5 | −0.0120133 | − | 0.0208077i | −1.46508 | − | 2.53760i | 0.999711 | − | 1.73155i | 1.00000 | −0.0352011 | + | 0.0609700i | 0.832707 | − | 1.44229i | −0.0960927 | −2.79295 | + | 4.83752i | −0.0120133 | − | 0.0208077i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 146.6 | 0.135687 | + | 0.235017i | 0.159809 | + | 0.276797i | 0.963178 | − | 1.66827i | 1.00000 | −0.0433679 | + | 0.0751154i | 1.69076 | − | 2.92848i | 1.06551 | 1.44892 | − | 2.50961i | 0.135687 | + | 0.235017i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 146.7 | 0.523603 | + | 0.906907i | 1.37934 | + | 2.38909i | 0.451680 | − | 0.782333i | 1.00000 | −1.44445 | + | 2.50186i | −1.71183 | + | 2.96497i | 3.04042 | −2.30515 | + | 3.99264i | 0.523603 | + | 0.906907i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 146.8 | 1.14039 | + | 1.97522i | −1.60714 | − | 2.78365i | −1.60100 | + | 2.77301i | 1.00000 | 3.66554 | − | 6.34890i | −1.15185 | + | 1.99506i | −2.74149 | −3.66579 | + | 6.34933i | 1.14039 | + | 1.97522i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 146.9 | 1.29324 | + | 2.23996i | −0.442413 | − | 0.766281i | −2.34494 | + | 4.06156i | 1.00000 | 1.14429 | − | 1.98197i | −0.429191 | + | 0.743381i | −6.95735 | 1.10854 | − | 1.92005i | 1.29324 | + | 2.23996i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.1 | −1.39471 | + | 2.41570i | 0.907309 | − | 1.57151i | −2.89042 | − | 5.00635i | 1.00000 | 2.53086 | + | 4.38358i | −0.134800 | − | 0.233481i | 10.5463 | −0.146421 | − | 0.253609i | −1.39471 | + | 2.41570i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.2 | −1.31799 | + | 2.28282i | −0.994722 | + | 1.72291i | −2.47417 | − | 4.28539i | 1.00000 | −2.62206 | − | 4.54154i | 1.64115 | + | 2.84256i | 7.77176 | −0.478943 | − | 0.829553i | −1.31799 | + | 2.28282i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.3 | −1.10278 | + | 1.91007i | 0.00652833 | − | 0.0113074i | −1.43225 | − | 2.48072i | 1.00000 | 0.0143986 | + | 0.0249391i | −2.30448 | − | 3.99148i | 1.90669 | 1.49991 | + | 2.59793i | −1.10278 | + | 1.91007i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.4 | −0.765438 | + | 1.32578i | −1.44363 | + | 2.50044i | −0.171790 | − | 0.297550i | 1.00000 | −2.21002 | − | 3.82786i | −1.93247 | − | 3.34713i | −2.53577 | −2.66813 | − | 4.62133i | −0.765438 | + | 1.32578i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.5 | −0.0120133 | + | 0.0208077i | −1.46508 | + | 2.53760i | 0.999711 | + | 1.73155i | 1.00000 | −0.0352011 | − | 0.0609700i | 0.832707 | + | 1.44229i | −0.0960927 | −2.79295 | − | 4.83752i | −0.0120133 | + | 0.0208077i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.6 | 0.135687 | − | 0.235017i | 0.159809 | − | 0.276797i | 0.963178 | + | 1.66827i | 1.00000 | −0.0433679 | − | 0.0751154i | 1.69076 | + | 2.92848i | 1.06551 | 1.44892 | + | 2.50961i | 0.135687 | − | 0.235017i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.7 | 0.523603 | − | 0.906907i | 1.37934 | − | 2.38909i | 0.451680 | + | 0.782333i | 1.00000 | −1.44445 | − | 2.50186i | −1.71183 | − | 2.96497i | 3.04042 | −2.30515 | − | 3.99264i | 0.523603 | − | 0.906907i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.8 | 1.14039 | − | 1.97522i | −1.60714 | + | 2.78365i | −1.60100 | − | 2.77301i | 1.00000 | 3.66554 | + | 6.34890i | −1.15185 | − | 1.99506i | −2.74149 | −3.66579 | − | 6.34933i | 1.14039 | − | 1.97522i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.9 | 1.29324 | − | 2.23996i | −0.442413 | + | 0.766281i | −2.34494 | − | 4.06156i | 1.00000 | 1.14429 | + | 1.98197i | −0.429191 | − | 0.743381i | −6.95735 | 1.10854 | + | 1.92005i | 1.29324 | − | 2.23996i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 845.2.e.o | 18 | |
| 13.b | even | 2 | 1 | 845.2.e.p | 18 | ||
| 13.c | even | 3 | 1 | 845.2.a.o | yes | 9 | |
| 13.c | even | 3 | 1 | inner | 845.2.e.o | 18 | |
| 13.d | odd | 4 | 2 | 845.2.m.j | 36 | ||
| 13.e | even | 6 | 1 | 845.2.a.n | ✓ | 9 | |
| 13.e | even | 6 | 1 | 845.2.e.p | 18 | ||
| 13.f | odd | 12 | 2 | 845.2.c.h | 18 | ||
| 13.f | odd | 12 | 2 | 845.2.m.j | 36 | ||
| 39.h | odd | 6 | 1 | 7605.2.a.cs | 9 | ||
| 39.i | odd | 6 | 1 | 7605.2.a.cp | 9 | ||
| 65.l | even | 6 | 1 | 4225.2.a.bt | 9 | ||
| 65.n | even | 6 | 1 | 4225.2.a.bs | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 845.2.a.n | ✓ | 9 | 13.e | even | 6 | 1 | |
| 845.2.a.o | yes | 9 | 13.c | even | 3 | 1 | |
| 845.2.c.h | 18 | 13.f | odd | 12 | 2 | ||
| 845.2.e.o | 18 | 1.a | even | 1 | 1 | trivial | |
| 845.2.e.o | 18 | 13.c | even | 3 | 1 | inner | |
| 845.2.e.p | 18 | 13.b | even | 2 | 1 | ||
| 845.2.e.p | 18 | 13.e | even | 6 | 1 | ||
| 845.2.m.j | 36 | 13.d | odd | 4 | 2 | ||
| 845.2.m.j | 36 | 13.f | odd | 12 | 2 | ||
| 4225.2.a.bs | 9 | 65.n | even | 6 | 1 | ||
| 4225.2.a.bt | 9 | 65.l | even | 6 | 1 | ||
| 7605.2.a.cp | 9 | 39.i | odd | 6 | 1 | ||
| 7605.2.a.cs | 9 | 39.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):
|
\( T_{2}^{18} + 3 T_{2}^{17} + 22 T_{2}^{16} + 41 T_{2}^{15} + 240 T_{2}^{14} + 382 T_{2}^{13} + 1667 T_{2}^{12} + \cdots + 1 \)
|
|
\( T_{7}^{18} + 7 T_{7}^{17} + 64 T_{7}^{16} + 247 T_{7}^{15} + 1515 T_{7}^{14} + 4757 T_{7}^{13} + \cdots + 361201 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + 3 T^{17} + \cdots + 1 \)
|
| $3$ |
\( T^{18} + 7 T^{17} + \cdots + 1 \)
|
| $5$ |
\( (T - 1)^{18} \)
|
| $7$ |
\( T^{18} + 7 T^{17} + \cdots + 361201 \)
|
| $11$ |
\( T^{18} - 9 T^{17} + \cdots + 79138816 \)
|
| $13$ |
\( T^{18} \)
|
| $17$ |
\( T^{18} - T^{17} + \cdots + 44302336 \)
|
| $19$ |
\( T^{18} + \cdots + 116985856 \)
|
| $23$ |
\( T^{18} + \cdots + 1909078249 \)
|
| $29$ |
\( T^{18} + 12 T^{17} + \cdots + 1 \)
|
| $31$ |
\( (T^{9} + 7 T^{8} + \cdots - 10816)^{2} \)
|
| $37$ |
\( T^{18} + \cdots + 100362568044544 \)
|
| $41$ |
\( T^{18} + \cdots + 986619064369 \)
|
| $43$ |
\( T^{18} + \cdots + 5114109169 \)
|
| $47$ |
\( (T^{9} - 36 T^{8} + \cdots + 11677)^{2} \)
|
| $53$ |
\( (T^{9} + 8 T^{8} + \cdots - 4469312)^{2} \)
|
| $59$ |
\( T^{18} + \cdots + 74472226816 \)
|
| $61$ |
\( T^{18} + \cdots + 33579463009 \)
|
| $67$ |
\( T^{18} + \cdots + 13253305129 \)
|
| $71$ |
\( T^{18} + \cdots + 31208937558016 \)
|
| $73$ |
\( (T^{9} - 256 T^{7} + \cdots - 866816)^{2} \)
|
| $79$ |
\( (T^{9} - 39 T^{8} + \cdots - 10816)^{2} \)
|
| $83$ |
\( (T^{9} - 7 T^{8} + \cdots - 49784561)^{2} \)
|
| $89$ |
\( T^{18} - 19 T^{17} + \cdots + 1100401 \)
|
| $97$ |
\( T^{18} + \cdots + 11\!\cdots\!76 \)
|
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