Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [750,3,Mod(193,750)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(750, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("750.193");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 750 = 2 \cdot 3 \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 750.f (of order \(4\), 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: | \(20.4360198270\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | 16.0.6879707136000000000000.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 21x^{12} + 86x^{8} + 36x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 5^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{16} + 21x^{12} + 86x^{8} + 36x^{4} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -90\nu^{14} - 1897\nu^{10} - 7895\nu^{6} - 4115\nu^{2} ) / 671 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -35\nu^{13} - 775\nu^{9} - 3704\nu^{5} - 2905\nu ) / 671 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 228 \nu^{14} + 152 \nu^{13} + 55 \nu^{12} + 4761 \nu^{10} + 3174 \nu^{9} + 1122 \nu^{8} + 19106 \nu^{6} + \cdots + 1210 ) / 671 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 449 \nu^{15} + 138 \nu^{13} + 29 \nu^{12} - 9367 \nu^{11} + 2864 \nu^{9} + 738 \nu^{8} + \cdots + 5091 ) / 671 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 836 \nu^{14} + 173 \nu^{13} - 263 \nu^{12} - 17457 \nu^{10} + 3639 \nu^{9} - 5536 \nu^{8} + \cdots - 3712 ) / 671 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 366 \nu^{14} + 269 \nu^{13} + 110 \nu^{12} + 7625 \nu^{10} + 5573 \nu^{9} + 2244 \nu^{8} + 30317 \nu^{6} + \cdots + 1749 ) / 671 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 443\nu^{15} + 9330\nu^{11} + 38600\nu^{7} + 17310\nu^{3} ) / 671 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -760\nu^{13} - 15870\nu^{9} - 63463\nu^{5} - 16110\nu ) / 671 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 449 \nu^{15} - 748 \nu^{14} - 138 \nu^{13} - 9367 \nu^{11} - 15796 \nu^{10} - 2864 \nu^{9} + \cdots - 2060 \nu ) / 671 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 988 \nu^{15} + 594 \nu^{14} + 110 \nu^{12} - 20631 \nu^{11} + 12386 \nu^{10} + 2244 \nu^{8} + \cdots + 1749 ) / 671 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 892 \nu^{15} + 836 \nu^{14} - 263 \nu^{12} - 18697 \nu^{11} + 17457 \nu^{10} - 5536 \nu^{8} + \cdots - 3712 ) / 671 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 988 \nu^{15} + 366 \nu^{14} - 110 \nu^{12} + 20631 \nu^{11} + 7625 \nu^{10} - 2244 \nu^{8} + \cdots - 1749 ) / 671 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 1341 \nu^{15} - 311 \nu^{13} - 263 \nu^{12} + 28064 \nu^{11} - 6503 \nu^{9} - 5536 \nu^{8} + \cdots - 3712 ) / 671 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 988 \nu^{15} + 366 \nu^{14} + 165 \nu^{12} + 20631 \nu^{11} + 7625 \nu^{10} + 3366 \nu^{8} + \cdots + 2959 ) / 671 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 1169 \nu^{15} - 138 \nu^{14} + 55 \nu^{12} - 24543 \nu^{11} - 2864 \nu^{10} + 1122 \nu^{8} + \cdots + 1210 ) / 671 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{14} - \beta_{10} + \beta_{8} + 2\beta_{6} + \beta_{3} - 2\beta_{2} ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{13} - \beta_{12} - \beta_{10} + 2\beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - 18\beta_1 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{15} + 3 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} + 3 \beta_{8} + \cdots - 2 ) / 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5\beta_{14} + 4\beta_{13} - 5\beta_{12} + \beta_{11} - 3\beta_{7} + \beta_{5} + 7\beta_{4} - 3\beta_{2} - 55 ) / 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{14} + \beta_{10} - \beta_{8} - 3\beta_{6} + \beta_{3} + 3\beta_{2} + \beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 15 \beta_{13} + 23 \beta_{12} - 6 \beta_{11} + 23 \beta_{10} - 24 \beta_{9} + 15 \beta_{8} + \cdots + 194 \beta_1 ) / 10 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 51 \beta_{15} - 35 \beta_{14} + 72 \beta_{13} - 43 \beta_{12} - 36 \beta_{11} + 35 \beta_{10} + \cdots + 43 ) / 10 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 99 \beta_{14} - 57 \beta_{13} + 99 \beta_{12} - 28 \beta_{11} + 29 \beta_{7} - 28 \beta_{5} + \cdots + 722 ) / 10 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -128\beta_{14} - 128\beta_{10} + 135\beta_{8} + 433\beta_{6} - 226\beta_{3} - 585\beta_{2} - 177\beta _1 + 177 ) / 10 \)
|
| \(\nu^{10}\) | \(=\) |
\( 22 \beta_{13} - 41 \beta_{12} + 12 \beta_{11} - 41 \beta_{10} + 32 \beta_{9} - 22 \beta_{8} + \cdots - 276 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 947 \beta_{15} + 483 \beta_{14} - 1036 \beta_{13} + 715 \beta_{12} + 518 \beta_{11} - 483 \beta_{10} + \cdots - 715 ) / 10 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 1663 \beta_{14} + 858 \beta_{13} - 1663 \beta_{12} + 495 \beta_{11} - 363 \beta_{7} + 495 \beta_{5} + \cdots - 10709 ) / 10 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 1859 \beta_{14} + 1859 \beta_{10} - 2014 \beta_{8} - 6579 \beta_{6} + 3863 \beta_{3} + 9753 \beta_{2} + \cdots - 2861 ) / 10 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 3367 \beta_{13} + 6670 \beta_{12} - 2003 \beta_{11} + 6670 \beta_{10} - 4731 \beta_{9} + \cdots + 41905 \beta_1 ) / 10 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 1554 \beta_{15} - 724 \beta_{14} + 1578 \beta_{13} - 1139 \beta_{12} - 789 \beta_{11} + 724 \beta_{10} + \cdots + 1139 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(251\) |
| \(\chi(n)\) | \(\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
−1.00000 | + | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | 0 | 2.44949 | −3.25953 | + | 3.25953i | 2.00000 | + | 2.00000i | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.2 | −1.00000 | + | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | 0 | 2.44949 | 2.69401 | − | 2.69401i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.3 | −1.00000 | + | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | 0 | 2.44949 | 4.99154 | − | 4.99154i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.4 | −1.00000 | + | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | 0 | 2.44949 | 6.47296 | − | 6.47296i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.5 | −1.00000 | + | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | 0 | −2.44949 | −5.70902 | + | 5.70902i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.6 | −1.00000 | + | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | 0 | −2.44949 | 0.244525 | − | 0.244525i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.7 | −1.00000 | + | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | 0 | −2.44949 | 2.54205 | − | 2.54205i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.8 | −1.00000 | + | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | 0 | −2.44949 | 4.02347 | − | 4.02347i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.1 | −1.00000 | − | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | 0 | 2.44949 | −3.25953 | − | 3.25953i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.2 | −1.00000 | − | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | 0 | 2.44949 | 2.69401 | + | 2.69401i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.3 | −1.00000 | − | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | 0 | 2.44949 | 4.99154 | + | 4.99154i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.4 | −1.00000 | − | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | 0 | 2.44949 | 6.47296 | + | 6.47296i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.5 | −1.00000 | − | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | 0 | −2.44949 | −5.70902 | − | 5.70902i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.6 | −1.00000 | − | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | 0 | −2.44949 | 0.244525 | + | 0.244525i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.7 | −1.00000 | − | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | 0 | −2.44949 | 2.54205 | + | 2.54205i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.8 | −1.00000 | − | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | 0 | −2.44949 | 4.02347 | + | 4.02347i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 750.3.f.b | ✓ | 16 |
| 5.b | even | 2 | 1 | 750.3.f.c | yes | 16 | |
| 5.c | odd | 4 | 1 | inner | 750.3.f.b | ✓ | 16 |
| 5.c | odd | 4 | 1 | 750.3.f.c | yes | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 750.3.f.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 750.3.f.b | ✓ | 16 | 5.c | odd | 4 | 1 | inner |
| 750.3.f.c | yes | 16 | 5.b | even | 2 | 1 | |
| 750.3.f.c | yes | 16 | 5.c | odd | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} - 24 T_{7}^{15} + 288 T_{7}^{14} - 1936 T_{7}^{13} + 12808 T_{7}^{12} - 133024 T_{7}^{11} + \cdots + 4201113856 \)
acting on \(S_{3}^{\mathrm{new}}(750, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{8} \)
|
| $3$ |
\( (T^{4} + 9)^{4} \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + \cdots + 4201113856 \)
|
| $11$ |
\( (T^{8} + 12 T^{7} + \cdots + 132400)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 159765150750976 \)
|
| $17$ |
\( T^{16} + \cdots + 11\!\cdots\!41 \)
|
| $19$ |
\( T^{16} + \cdots + 13\!\cdots\!25 \)
|
| $23$ |
\( T^{16} + \cdots + 35\!\cdots\!25 \)
|
| $29$ |
\( T^{16} + \cdots + 26\!\cdots\!00 \)
|
| $31$ |
\( (T^{8} - 100 T^{7} + \cdots + 101700735361)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 20\!\cdots\!96 \)
|
| $41$ |
\( (T^{8} + 96 T^{7} + \cdots - 9683699024)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 22\!\cdots\!96 \)
|
| $47$ |
\( T^{16} + \cdots + 12\!\cdots\!25 \)
|
| $53$ |
\( T^{16} + \cdots + 16\!\cdots\!21 \)
|
| $59$ |
\( T^{16} + \cdots + 50\!\cdots\!00 \)
|
| $61$ |
\( (T^{8} + \cdots - 3717351077975)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 15\!\cdots\!96 \)
|
| $71$ |
\( (T^{8} - 88 T^{7} + \cdots - 111221483600)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 31\!\cdots\!56 \)
|
| $79$ |
\( T^{16} + \cdots + 79\!\cdots\!25 \)
|
| $83$ |
\( T^{16} + \cdots + 26\!\cdots\!21 \)
|
| $89$ |
\( T^{16} + \cdots + 16\!\cdots\!00 \)
|
| $97$ |
\( T^{16} + \cdots + 32\!\cdots\!56 \)
|
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