Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [612,2,Mod(145,612)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(612, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("612.145");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 612 = 2^{2} \cdot 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 612.w (of order \(8\), degree \(4\), 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: | \(4.88684460370\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{8})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 60x^{14} + 1434x^{12} + 17508x^{10} + 116445x^{8} + 414440x^{6} + 715372x^{4} + 462752x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 204) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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
\( x^{16} + 60x^{14} + 1434x^{12} + 17508x^{10} + 116445x^{8} + 414440x^{6} + 715372x^{4} + 462752x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 10183231 \nu^{14} - 730091944 \nu^{12} - 20013985161 \nu^{10} - 264194956239 \nu^{8} + \cdots + 1177839845590 ) / 450248976368 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 86362753 \nu^{14} - 4947624218 \nu^{12} - 110704440654 \nu^{10} - 1225172767250 \nu^{8} + \cdots + 834139859368 ) / 450248976368 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 28803044 \nu^{15} - 766785 \nu^{14} - 1725343305 \nu^{13} - 41712081 \nu^{12} + \cdots - 37457113132 ) / 52970467808 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 28803044 \nu^{15} + 766785 \nu^{14} - 1725343305 \nu^{13} + 41712081 \nu^{12} + \cdots + 37457113132 ) / 52970467808 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 470397100 \nu^{15} + 94265158 \nu^{14} + 28257173258 \nu^{13} + 5215141977 \nu^{12} + \cdots + 691188026456 ) / 900497952736 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 470397100 \nu^{15} - 94265158 \nu^{14} + 28257173258 \nu^{13} - 5215141977 \nu^{12} + \cdots - 691188026456 ) / 900497952736 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1789 \nu^{15} + 107628 \nu^{13} + 2581674 \nu^{11} + 31676582 \nu^{9} + 212104547 \nu^{7} + \cdots + 874905524 \nu ) / 2572256 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 630241980 \nu^{15} + 30183059 \nu^{14} + 37723759683 \nu^{13} + 1769619153 \nu^{12} + \cdots - 345529433242 ) / 900497952736 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 630241980 \nu^{15} - 30183059 \nu^{14} + 37723759683 \nu^{13} - 1769619153 \nu^{12} + \cdots + 345529433242 ) / 900497952736 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 384330901 \nu^{15} - 5768030 \nu^{14} - 23059716203 \nu^{13} - 333947021 \nu^{12} + \cdots + 487357836308 ) / 450248976368 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 384330901 \nu^{15} - 5768030 \nu^{14} + 23059716203 \nu^{13} - 333947021 \nu^{12} + \cdots + 487357836308 ) / 450248976368 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 848959971 \nu^{15} + 88634985 \nu^{14} + 50982942440 \nu^{13} + 5266939953 \nu^{12} + \cdots + 1180083186368 ) / 900497952736 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 841692656 \nu^{15} + 5768030 \nu^{14} - 50607784084 \nu^{13} + 333947021 \nu^{12} + \cdots - 487357836308 ) / 450248976368 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 1997867746 \nu^{15} - 153784274 \nu^{14} - 119731104237 \nu^{13} - 8466451041 \nu^{12} + \cdots + 2719728082012 ) / 900497952736 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 1997867746 \nu^{15} - 153784274 \nu^{14} + 119731104237 \nu^{13} - 8466451041 \nu^{12} + \cdots + 2719728082012 ) / 900497952736 \)
|
| \(\nu\) | \(=\) |
\( ( - 2 \beta_{14} + 2 \beta_{12} - \beta_{11} - \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \cdots + 2 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{15} + \beta_{14} + \beta_{11} + \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - \beta_{4} + \beta_{3} + \cdots - 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6 \beta_{15} + 24 \beta_{14} - 30 \beta_{12} + 19 \beta_{11} + 11 \beta_{10} + 24 \beta_{9} + 24 \beta_{8} + \cdots - 30 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 14 \beta_{15} - 14 \beta_{14} - 13 \beta_{11} - 13 \beta_{10} - 34 \beta_{9} + 34 \beta_{8} + \cdots + 100 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 112 \beta_{15} - 334 \beta_{14} - 24 \beta_{13} + 446 \beta_{12} - 339 \beta_{11} - 131 \beta_{10} + \cdots + 446 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 208 \beta_{15} + 208 \beta_{14} + 172 \beta_{11} + 172 \beta_{10} + 546 \beta_{9} - 546 \beta_{8} + \cdots - 1438 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1916 \beta_{15} + 5074 \beta_{14} + 724 \beta_{13} - 6990 \beta_{12} + 5949 \beta_{11} + 1765 \beta_{10} + \cdots - 6990 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 3222 \beta_{15} - 3222 \beta_{14} - 2217 \beta_{11} - 2217 \beta_{10} - 8606 \beta_{9} + 8606 \beta_{8} + \cdots + 22070 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 32568 \beta_{15} - 80942 \beta_{14} - 16624 \beta_{13} + 113510 \beta_{12} - 103867 \beta_{11} + \cdots + 113510 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( 51210 \beta_{15} + 51210 \beta_{14} + 27368 \beta_{11} + 27368 \beta_{10} + 134274 \beta_{9} + \cdots - 350586 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 554016 \beta_{15} + 1326254 \beta_{14} + 347652 \beta_{13} - 1880270 \beta_{12} + 1810189 \beta_{11} + \cdots - 1880270 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 826054 \beta_{15} - 826054 \beta_{14} - 312587 \beta_{11} - 312587 \beta_{10} - 2082902 \beta_{9} + \cdots + 5678658 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 9438728 \beta_{15} - 22060974 \beta_{14} - 6942264 \beta_{13} + 31499702 \beta_{12} - 31526063 \beta_{11} + \cdots + 31499702 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( 13442558 \beta_{15} + 13442558 \beta_{14} + 3020394 \beta_{11} + 3020394 \beta_{10} + 32213074 \beta_{9} + \cdots - 93075922 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 161081800 \beta_{15} + 370280198 \beta_{14} + 134696124 \beta_{13} - 531361998 \beta_{12} + \cdots - 531361998 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/612\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(137\) | \(307\) |
| \(\chi(n)\) | \(\beta_{3}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 145.1 |
|
0 | 0 | 0 | −0.919353 | + | 2.21951i | 0 | −1.55193 | − | 3.74669i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.2 | 0 | 0 | 0 | −0.586256 | + | 1.41535i | 0 | −1.15081 | − | 2.77830i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.3 | 0 | 0 | 0 | 0.496466 | − | 1.19857i | 0 | 1.20907 | + | 2.91895i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.4 | 0 | 0 | 0 | 1.59493 | − | 3.85050i | 0 | 0.0794551 | + | 0.191822i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 253.1 | 0 | 0 | 0 | −2.83156 | + | 1.17287i | 0 | −0.119959 | − | 0.0496888i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 253.2 | 0 | 0 | 0 | −0.00271628 | + | 0.00112512i | 0 | 3.72964 | + | 1.54487i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 253.3 | 0 | 0 | 0 | 2.63370 | − | 1.09092i | 0 | −4.87029 | − | 2.01734i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 253.4 | 0 | 0 | 0 | 3.61479 | − | 1.49729i | 0 | 2.67483 | + | 1.10795i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 325.1 | 0 | 0 | 0 | −0.919353 | − | 2.21951i | 0 | −1.55193 | + | 3.74669i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 325.2 | 0 | 0 | 0 | −0.586256 | − | 1.41535i | 0 | −1.15081 | + | 2.77830i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 325.3 | 0 | 0 | 0 | 0.496466 | + | 1.19857i | 0 | 1.20907 | − | 2.91895i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 325.4 | 0 | 0 | 0 | 1.59493 | + | 3.85050i | 0 | 0.0794551 | − | 0.191822i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 433.1 | 0 | 0 | 0 | −2.83156 | − | 1.17287i | 0 | −0.119959 | + | 0.0496888i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 433.2 | 0 | 0 | 0 | −0.00271628 | − | 0.00112512i | 0 | 3.72964 | − | 1.54487i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 433.3 | 0 | 0 | 0 | 2.63370 | + | 1.09092i | 0 | −4.87029 | + | 2.01734i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 433.4 | 0 | 0 | 0 | 3.61479 | + | 1.49729i | 0 | 2.67483 | − | 1.10795i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.d | even | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 612.2.w.c | 16 | |
| 3.b | odd | 2 | 1 | 204.2.o.a | ✓ | 16 | |
| 12.b | even | 2 | 1 | 816.2.bq.d | 16 | ||
| 17.d | even | 8 | 1 | inner | 612.2.w.c | 16 | |
| 51.g | odd | 8 | 1 | 204.2.o.a | ✓ | 16 | |
| 51.i | even | 16 | 1 | 3468.2.a.o | 8 | ||
| 51.i | even | 16 | 1 | 3468.2.a.p | 8 | ||
| 51.i | even | 16 | 2 | 3468.2.b.h | 16 | ||
| 51.i | even | 16 | 2 | 3468.2.j.h | 16 | ||
| 51.i | even | 16 | 2 | 3468.2.j.i | 16 | ||
| 204.p | even | 8 | 1 | 816.2.bq.d | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 204.2.o.a | ✓ | 16 | 3.b | odd | 2 | 1 | |
| 204.2.o.a | ✓ | 16 | 51.g | odd | 8 | 1 | |
| 612.2.w.c | 16 | 1.a | even | 1 | 1 | trivial | |
| 612.2.w.c | 16 | 17.d | even | 8 | 1 | inner | |
| 816.2.bq.d | 16 | 12.b | even | 2 | 1 | ||
| 816.2.bq.d | 16 | 204.p | even | 8 | 1 | ||
| 3468.2.a.o | 8 | 51.i | even | 16 | 1 | ||
| 3468.2.a.p | 8 | 51.i | even | 16 | 1 | ||
| 3468.2.b.h | 16 | 51.i | even | 16 | 2 | ||
| 3468.2.j.h | 16 | 51.i | even | 16 | 2 | ||
| 3468.2.j.i | 16 | 51.i | even | 16 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} - 8 T_{5}^{15} + 28 T_{5}^{14} - 32 T_{5}^{13} - 120 T_{5}^{12} + 296 T_{5}^{11} + 1344 T_{5}^{10} + \cdots + 4 \)
acting on \(S_{2}^{\mathrm{new}}(612, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} - 8 T^{15} + \cdots + 4 \)
|
| $7$ |
\( T^{16} - 12 T^{14} + \cdots + 4096 \)
|
| $11$ |
\( T^{16} + 8 T^{15} + \cdots + 4734976 \)
|
| $13$ |
\( T^{16} + \cdots + 121264144 \)
|
| $17$ |
\( T^{16} + \cdots + 6975757441 \)
|
| $19$ |
\( T^{16} + \cdots + 4421718016 \)
|
| $23$ |
\( T^{16} + \cdots + 2264237056 \)
|
| $29$ |
\( T^{16} + \cdots + 14168617024 \)
|
| $31$ |
\( T^{16} + 24 T^{15} + \cdots + 18939904 \)
|
| $37$ |
\( T^{16} + 8 T^{15} + \cdots + 2347024 \)
|
| $41$ |
\( T^{16} + \cdots + 48146014084 \)
|
| $43$ |
\( T^{16} + \cdots + 18339659776 \)
|
| $47$ |
\( T^{16} + \cdots + 303038464 \)
|
| $53$ |
\( T^{16} + 24 T^{15} + \cdots + 1183744 \)
|
| $59$ |
\( T^{16} - 32 T^{15} + \cdots + 4734976 \)
|
| $61$ |
\( T^{16} + \cdots + 2539777694224 \)
|
| $67$ |
\( (T^{8} + 8 T^{7} + \cdots - 1482496)^{2} \)
|
| $71$ |
\( T^{16} + 144 T^{14} + \cdots + 39337984 \)
|
| $73$ |
\( T^{16} + \cdots + 502805791744 \)
|
| $79$ |
\( T^{16} + \cdots + 58866405376 \)
|
| $83$ |
\( T^{16} + \cdots + 34\!\cdots\!04 \)
|
| $89$ |
\( T^{16} + \cdots + 641585799247936 \)
|
| $97$ |
\( T^{16} + \cdots + 1671373209856 \)
|
| show more | |
| show less |