Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2300,3,Mod(1701,2300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2300, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2300.1701");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2300.f (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: | \(62.6704608029\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} + 250 x^{18} + 26073 x^{16} + 1464600 x^{14} + 47742736 x^{12} + 908660400 x^{10} + \cdots + 85264000000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{10} \) |
| Twist minimal: | no (minimal twist has level 460) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} + 250 x^{18} + 26073 x^{16} + 1464600 x^{14} + 47742736 x^{12} + 908660400 x^{10} + \cdots + 85264000000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 143529979 \nu^{16} - 28705995800 \nu^{14} - 2491361005064 \nu^{12} + \cdots - 28\!\cdots\!52 ) / 48\!\cdots\!36 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4172665205 \nu^{16} + 834533041000 \nu^{14} + 66673526169576 \nu^{12} + \cdots + 29\!\cdots\!28 ) / 48\!\cdots\!36 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4655505199 \nu^{16} - 931101039800 \nu^{14} - 74252375553688 \nu^{12} + \cdots - 33\!\cdots\!72 ) / 48\!\cdots\!36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 9932589915 \nu^{16} - 1986517983000 \nu^{14} - 158546900816232 \nu^{12} + \cdots - 86\!\cdots\!68 ) / 48\!\cdots\!36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 17628301681 \nu^{18} - 3966367878225 \nu^{16} - 366558030159138 \nu^{14} + \cdots + 10\!\cdots\!00 ) / 34\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 258853295253 \nu^{18} + 58241991431925 \nu^{16} + \cdots + 28\!\cdots\!00 ) / 45\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 102080623967 \nu^{18} + 22968140392575 \nu^{16} + \cdots + 28\!\cdots\!00 ) / 13\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3564300983719 \nu^{18} + 801967721336775 \nu^{16} + \cdots + 67\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 4052793691541 \nu^{18} + 911878580596725 \nu^{16} + \cdots + 93\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 5430913 \nu^{19} + 1357728250 \nu^{17} + 140463584649 \nu^{15} + 7726793179800 \nu^{13} + \cdots + 65\!\cdots\!50 \nu ) / 56\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 5430913 \nu^{19} + 1357728250 \nu^{17} + 140463584649 \nu^{15} + 7726793179800 \nu^{13} + \cdots + 76\!\cdots\!50 \nu ) / 56\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 934435964478 \nu^{19} - 243686819157375 \nu^{17} + \cdots - 18\!\cdots\!50 \nu ) / 43\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 677434183331 \nu^{19} - 156184513908000 \nu^{17} + \cdots - 78\!\cdots\!50 \nu ) / 13\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 10893169294127 \nu^{19} + \cdots - 60\!\cdots\!50 \nu ) / 16\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 1659210038377 \nu^{19} + 400393106383375 \nu^{17} + \cdots + 95\!\cdots\!00 \nu ) / 22\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 335386768930241 \nu^{19} + \cdots + 11\!\cdots\!00 \nu ) / 16\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 202520274060554 \nu^{19} + \cdots - 55\!\cdots\!50 \nu ) / 83\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 54405206299631 \nu^{19} + \cdots - 15\!\cdots\!50 \nu ) / 13\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 20834608323446 \nu^{19} + \cdots + 11\!\cdots\!50 \nu ) / 43\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} - \beta_{10} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - 50 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{19} + \beta_{18} + 7 \beta_{17} + 9 \beta_{16} + 7 \beta_{15} + 8 \beta_{14} + \cdots + 40 \beta_{10} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 50\beta_{9} - 50\beta_{8} - 50\beta_{7} + 50\beta_{6} - 19\beta_{4} + 27\beta_{3} - 15\beta_{2} + 3\beta _1 + 2073 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 61 \beta_{19} - 57 \beta_{18} - 435 \beta_{17} - 533 \beta_{16} - 407 \beta_{15} + \cdots - 1737 \beta_{10} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2229 \beta_{9} + 2169 \beta_{8} + 2405 \beta_{7} - 2343 \beta_{6} - 224 \beta_{5} + 1425 \beta_{4} + \cdots - 92975 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 3000 \beta_{19} + 2642 \beta_{18} + 24316 \beta_{17} + 28568 \beta_{16} + 20780 \beta_{15} + \cdots + 77715 \beta_{10} ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 97900 \beta_{9} - 91900 \beta_{8} - 115500 \beta_{7} + 109300 \beta_{6} + 22400 \beta_{5} + \cdots + 4254221 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 139045 \beta_{19} - 114961 \beta_{18} - 1309137 \beta_{17} - 1487335 \beta_{16} + \cdots - 3512654 \beta_{10} ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 4310396 \beta_{9} + 3901080 \beta_{8} + 5549226 \beta_{7} - 5140158 \beta_{6} - 1548890 \beta_{5} + \cdots - 196465125 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 6270575 \beta_{19} + 4825511 \beta_{18} + 68880243 \beta_{17} + 76115229 \beta_{16} + \cdots + 159754501 \beta_{10} ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( 95467200 \beta_{9} - 83518500 \beta_{8} - 133379450 \beta_{7} + 122074350 \beta_{6} + 46166750 \beta_{5} + \cdots + 4565558846 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 277247566 \beta_{19} - 195738964 \beta_{18} - 3565716094 \beta_{17} - 3847193046 \beta_{16} + \cdots - 7304667967 \beta_{10} ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 8513487999 \beta_{9} + 7228236727 \beta_{8} + 12830791157 \beta_{7} - 11699892911 \beta_{6} + \cdots - 426610127350 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 12032752159 \beta_{19} + 7609572511 \beta_{18} + 182376248649 \beta_{17} + \cdots + 335701914770 \beta_{10} ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 382004599800 \beta_{9} - 316257345400 \beta_{8} - 617512731400 \beta_{7} + 564702082200 \beta_{6} + \cdots + 20021206895835 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 511889313453 \beta_{19} - 278119514849 \beta_{18} - 9242955474851 \beta_{17} + \cdots - 15503033469911 \beta_{10} ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 17240744386175 \beta_{9} + 13989379503547 \beta_{8} + 29737350681007 \beta_{7} + \cdots - 943297922562875 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 21267671213242 \beta_{19} + 9147777539432 \beta_{18} + 465123238631554 \beta_{17} + \cdots + 719261262618679 \beta_{10} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).
| \(n\) | \(277\) | \(1151\) | \(1201\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1701.1 |
|
0 | −5.07205 | 0 | 0 | 0 | − | 3.07132i | 0 | 16.7257 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.2 | 0 | −5.07205 | 0 | 0 | 0 | 3.07132i | 0 | 16.7257 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.3 | 0 | −4.41085 | 0 | 0 | 0 | − | 12.7035i | 0 | 10.4556 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.4 | 0 | −4.41085 | 0 | 0 | 0 | 12.7035i | 0 | 10.4556 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.5 | 0 | −3.08989 | 0 | 0 | 0 | − | 6.21474i | 0 | 0.547411 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.6 | 0 | −3.08989 | 0 | 0 | 0 | 6.21474i | 0 | 0.547411 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.7 | 0 | −2.55211 | 0 | 0 | 0 | − | 1.70335i | 0 | −2.48674 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.8 | 0 | −2.55211 | 0 | 0 | 0 | 1.70335i | 0 | −2.48674 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.9 | 0 | −1.32589 | 0 | 0 | 0 | − | 6.21806i | 0 | −7.24200 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.10 | 0 | −1.32589 | 0 | 0 | 0 | 6.21806i | 0 | −7.24200 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.11 | 0 | 1.32589 | 0 | 0 | 0 | − | 6.21806i | 0 | −7.24200 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.12 | 0 | 1.32589 | 0 | 0 | 0 | 6.21806i | 0 | −7.24200 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.13 | 0 | 2.55211 | 0 | 0 | 0 | − | 1.70335i | 0 | −2.48674 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.14 | 0 | 2.55211 | 0 | 0 | 0 | 1.70335i | 0 | −2.48674 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.15 | 0 | 3.08989 | 0 | 0 | 0 | − | 6.21474i | 0 | 0.547411 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.16 | 0 | 3.08989 | 0 | 0 | 0 | 6.21474i | 0 | 0.547411 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.17 | 0 | 4.41085 | 0 | 0 | 0 | − | 12.7035i | 0 | 10.4556 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.18 | 0 | 4.41085 | 0 | 0 | 0 | 12.7035i | 0 | 10.4556 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.19 | 0 | 5.07205 | 0 | 0 | 0 | − | 3.07132i | 0 | 16.7257 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1701.20 | 0 | 5.07205 | 0 | 0 | 0 | 3.07132i | 0 | 16.7257 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 23.b | odd | 2 | 1 | inner |
| 115.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2300.3.f.f | 20 | |
| 5.b | even | 2 | 1 | inner | 2300.3.f.f | 20 | |
| 5.c | odd | 4 | 2 | 460.3.d.c | ✓ | 20 | |
| 23.b | odd | 2 | 1 | inner | 2300.3.f.f | 20 | |
| 115.c | odd | 2 | 1 | inner | 2300.3.f.f | 20 | |
| 115.e | even | 4 | 2 | 460.3.d.c | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 460.3.d.c | ✓ | 20 | 5.c | odd | 4 | 2 | |
| 460.3.d.c | ✓ | 20 | 115.e | even | 4 | 2 | |
| 2300.3.f.f | 20 | 1.a | even | 1 | 1 | trivial | |
| 2300.3.f.f | 20 | 5.b | even | 2 | 1 | inner | |
| 2300.3.f.f | 20 | 23.b | odd | 2 | 1 | inner | |
| 2300.3.f.f | 20 | 115.c | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 63T_{3}^{8} + 1396T_{3}^{6} - 13113T_{3}^{4} + 50195T_{3}^{2} - 54716 \)
acting on \(S_{3}^{\mathrm{new}}(2300, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( (T^{10} - 63 T^{8} + \cdots - 54716)^{2} \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( (T^{10} + 251 T^{8} + \cdots + 6595712)^{2} \)
|
| $11$ |
\( (T^{10} + 749 T^{8} + \cdots + 2364716088)^{2} \)
|
| $13$ |
\( (T^{10} - 1191 T^{8} + \cdots - 2769997500)^{2} \)
|
| $17$ |
\( (T^{10} + 981 T^{8} + \cdots + 12903200)^{2} \)
|
| $19$ |
\( (T^{10} + \cdots + 14941760470200)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 17\!\cdots\!01 \)
|
| $29$ |
\( (T^{5} - 32 T^{4} + \cdots + 643000)^{4} \)
|
| $31$ |
\( (T^{5} + 17 T^{4} + \cdots + 3731548)^{4} \)
|
| $37$ |
\( (T^{10} + \cdots + 8760963123200)^{2} \)
|
| $41$ |
\( (T^{5} + 17 T^{4} + \cdots - 63339680)^{4} \)
|
| $43$ |
\( (T^{10} + \cdots + 668852527232)^{2} \)
|
| $47$ |
\( (T^{10} + \cdots - 10\!\cdots\!36)^{2} \)
|
| $53$ |
\( (T^{10} + \cdots + 50\!\cdots\!52)^{2} \)
|
| $59$ |
\( (T^{5} - 90 T^{4} + \cdots - 1885410080)^{4} \)
|
| $61$ |
\( (T^{10} + \cdots + 290658217164408)^{2} \)
|
| $67$ |
\( (T^{10} + \cdots + 12\!\cdots\!72)^{2} \)
|
| $71$ |
\( (T^{5} - 19 T^{4} + \cdots - 110070650)^{4} \)
|
| $73$ |
\( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{10} + \cdots + 27\!\cdots\!72)^{2} \)
|
| $83$ |
\( (T^{10} + \cdots + 487502560702592)^{2} \)
|
| $89$ |
\( (T^{10} + \cdots + 90\!\cdots\!08)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots + 31\!\cdots\!68)^{2} \)
|
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