Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1050,3,Mod(601,1050)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1050, 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("1050.601");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1050.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: | \(28.6104277578\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2744 x^{12} - 10640 x^{11} + 39126 x^{10} - 108488 x^{9} + \cdots + 33124 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{20} \) |
| Twist minimal: | no (minimal twist has level 210) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2744 x^{12} - 10640 x^{11} + 39126 x^{10} - 108488 x^{9} + \cdots + 33124 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 92936 \nu^{14} - 650552 \nu^{13} + 7629434 \nu^{12} - 37319428 \nu^{11} + 225649370 \nu^{10} + \cdots - 649676357 ) / 5030871765 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 4788335 \nu^{14} + 33518345 \nu^{13} - 384267955 \nu^{12} + 1869869245 \nu^{11} + \cdots + 205883061930 ) / 119428521030 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 116229757 \nu^{14} + 813608299 \nu^{13} - 9262265428 \nu^{12} + 44996684681 \nu^{11} + \cdots + 2568347439658 ) / 2746855983690 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1093349498 \nu^{14} + 7653446486 \nu^{13} - 88689952367 \nu^{12} + 432644909884 \nu^{11} + \cdots + 12465433464302 ) / 8240567951070 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 279842740 \nu^{15} - 2098820550 \nu^{14} + 23452089504 \nu^{13} - 120606470101 \nu^{12} + \cdots + 20065240923728 ) / 21993425639458 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1064447881686 \nu^{15} - 3209377373879 \nu^{14} + 57853449494147 \nu^{13} + \cdots - 22\!\cdots\!14 ) / 68\!\cdots\!90 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1064447881686 \nu^{15} - 11495817736195 \nu^{14} + 115858532030359 \nu^{13} + \cdots + 63\!\cdots\!20 ) / 68\!\cdots\!90 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 12448 \nu^{15} - 93360 \nu^{14} + 1065318 \nu^{13} - 5508607 \nu^{12} + 32749646 \nu^{11} + \cdots - 179130952 ) / 520340730 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 3854937063308 \nu^{15} + 23962636724121 \nu^{14} - 295302862411617 \nu^{13} + \cdots + 74\!\cdots\!24 ) / 68\!\cdots\!90 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 3854937063308 \nu^{15} - 30212752287741 \nu^{14} + 339053671356957 \nu^{13} + \cdots + 53\!\cdots\!46 ) / 68\!\cdots\!90 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 3854937063308 \nu^{15} - 33861419225499 \nu^{14} + 364594339921263 \nu^{13} + \cdots + 70\!\cdots\!94 ) / 68\!\cdots\!90 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 5472163425770 \nu^{15} - 41041225693275 \nu^{14} + 464858064633033 \nu^{13} + \cdots - 62\!\cdots\!92 ) / 68\!\cdots\!90 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 7209986841170 \nu^{15} - 54074901308775 \nu^{14} + 610495540452873 \nu^{13} + \cdots - 74\!\cdots\!92 ) / 68\!\cdots\!90 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 11652755758 \nu^{15} - 87395668185 \nu^{14} + 980333632691 \nu^{13} + \cdots + 767874169626928 ) / 109967128197290 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 165988958814 \nu^{15} + 1244917191105 \nu^{14} - 14017718745515 \nu^{13} + \cdots - 92\!\cdots\!92 ) / 989704153775610 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{13} - \beta_{12} - 2\beta_{5} + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{13} - \beta_{12} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \cdots - 28 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3 \beta_{15} - 3 \beta_{14} - 9 \beta_{13} + 17 \beta_{12} - \beta_{11} + 3 \beta_{10} + 4 \beta_{9} + \cdots - 42 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 6 \beta_{15} - 6 \beta_{14} - 19 \beta_{13} + 35 \beta_{12} - 12 \beta_{11} - 12 \beta_{10} - 20 \beta_{9} + \cdots + 330 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 90 \beta_{15} + 40 \beta_{14} + 101 \beta_{13} - 185 \beta_{12} - 37 \beta_{11} - 35 \beta_{10} + \cdots + 904 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 285 \beta_{15} + 135 \beta_{14} + 351 \beta_{13} - 643 \beta_{12} + 174 \beta_{11} - 20 \beta_{10} + \cdots - 2742 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 2009 \beta_{15} - 224 \beta_{14} - 619 \beta_{13} + 1156 \beta_{12} + 1081 \beta_{11} + 56 \beta_{10} + \cdots - 13154 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 9380 \beta_{15} - 1540 \beta_{14} - 4159 \beta_{13} + 7707 \beta_{12} - 186 \beta_{11} + 1340 \beta_{10} + \cdots + 272 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 32508 \beta_{15} - 2622 \beta_{14} - 6786 \beta_{13} + 12086 \beta_{12} - 12235 \beta_{11} + \cdots + 88788 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 234915 \beta_{15} - 585 \beta_{14} - 184 \beta_{13} - 2050 \beta_{12} - 42672 \beta_{11} + \cdots + 569434 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 329714 \beta_{15} + 80366 \beta_{14} + 305101 \beta_{13} - 561877 \beta_{12} - 115963 \beta_{11} + \cdots + 2168840 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 2361744 \beta_{15} + 230076 \beta_{14} + 877983 \beta_{13} - 1603409 \beta_{12} + 435912 \beta_{11} + \cdots - 6519638 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 667368 \beta_{15} - 692783 \beta_{14} - 5309960 \beta_{13} + 9895207 \beta_{12} + 9475281 \beta_{11} + \cdots - 112895206 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 75505976 \beta_{15} - 11723894 \beta_{14} - 63474043 \beta_{13} + 117216695 \beta_{12} + \cdots + 132095548 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 124496877 \beta_{15} - 11320543 \beta_{14} + 24283611 \beta_{13} - 46833803 \beta_{12} + \cdots + 3156761778 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(451\) | \(701\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 601.1 |
|
−1.41421 | − | 1.73205i | 2.00000 | 0 | 2.44949i | −4.63050 | + | 5.24961i | −2.82843 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.2 | −1.41421 | − | 1.73205i | 2.00000 | 0 | 2.44949i | −0.853218 | + | 6.94781i | −2.82843 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.3 | −1.41421 | − | 1.73205i | 2.00000 | 0 | 2.44949i | 1.57881 | − | 6.81963i | −2.82843 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.4 | −1.41421 | − | 1.73205i | 2.00000 | 0 | 2.44949i | 6.73333 | − | 1.91369i | −2.82843 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.5 | −1.41421 | 1.73205i | 2.00000 | 0 | − | 2.44949i | −4.63050 | − | 5.24961i | −2.82843 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.6 | −1.41421 | 1.73205i | 2.00000 | 0 | − | 2.44949i | −0.853218 | − | 6.94781i | −2.82843 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.7 | −1.41421 | 1.73205i | 2.00000 | 0 | − | 2.44949i | 1.57881 | + | 6.81963i | −2.82843 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.8 | −1.41421 | 1.73205i | 2.00000 | 0 | − | 2.44949i | 6.73333 | + | 1.91369i | −2.82843 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.9 | 1.41421 | − | 1.73205i | 2.00000 | 0 | − | 2.44949i | −6.73333 | − | 1.91369i | 2.82843 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.10 | 1.41421 | − | 1.73205i | 2.00000 | 0 | − | 2.44949i | −1.57881 | − | 6.81963i | 2.82843 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.11 | 1.41421 | − | 1.73205i | 2.00000 | 0 | − | 2.44949i | 0.853218 | + | 6.94781i | 2.82843 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.12 | 1.41421 | − | 1.73205i | 2.00000 | 0 | − | 2.44949i | 4.63050 | + | 5.24961i | 2.82843 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.13 | 1.41421 | 1.73205i | 2.00000 | 0 | 2.44949i | −6.73333 | + | 1.91369i | 2.82843 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.14 | 1.41421 | 1.73205i | 2.00000 | 0 | 2.44949i | −1.57881 | + | 6.81963i | 2.82843 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.15 | 1.41421 | 1.73205i | 2.00000 | 0 | 2.44949i | 0.853218 | − | 6.94781i | 2.82843 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 601.16 | 1.41421 | 1.73205i | 2.00000 | 0 | 2.44949i | 4.63050 | − | 5.24961i | 2.82843 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1050.3.f.e | 16 | |
| 5.b | even | 2 | 1 | inner | 1050.3.f.e | 16 | |
| 5.c | odd | 4 | 2 | 210.3.h.a | ✓ | 16 | |
| 7.b | odd | 2 | 1 | inner | 1050.3.f.e | 16 | |
| 15.e | even | 4 | 2 | 630.3.h.e | 16 | ||
| 20.e | even | 4 | 2 | 1680.3.bd.a | 16 | ||
| 35.c | odd | 2 | 1 | inner | 1050.3.f.e | 16 | |
| 35.f | even | 4 | 2 | 210.3.h.a | ✓ | 16 | |
| 105.k | odd | 4 | 2 | 630.3.h.e | 16 | ||
| 140.j | odd | 4 | 2 | 1680.3.bd.a | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.3.h.a | ✓ | 16 | 5.c | odd | 4 | 2 | |
| 210.3.h.a | ✓ | 16 | 35.f | even | 4 | 2 | |
| 630.3.h.e | 16 | 15.e | even | 4 | 2 | ||
| 630.3.h.e | 16 | 105.k | odd | 4 | 2 | ||
| 1050.3.f.e | 16 | 1.a | even | 1 | 1 | trivial | |
| 1050.3.f.e | 16 | 5.b | even | 2 | 1 | inner | |
| 1050.3.f.e | 16 | 7.b | odd | 2 | 1 | inner | |
| 1050.3.f.e | 16 | 35.c | odd | 2 | 1 | inner | |
| 1680.3.bd.a | 16 | 20.e | even | 4 | 2 | ||
| 1680.3.bd.a | 16 | 140.j | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):
|
\( T_{11}^{4} - 24T_{11}^{3} + 28T_{11}^{2} + 1776T_{11} - 4844 \)
|
|
\( T_{23}^{8} - 2932T_{23}^{6} + 2745076T_{23}^{4} - 843688800T_{23}^{2} + 11186869824 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{8} \)
|
| $3$ |
\( (T^{2} + 3)^{8} \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + \cdots + 33232930569601 \)
|
| $11$ |
\( (T^{4} - 24 T^{3} + \cdots - 4844)^{4} \)
|
| $13$ |
\( (T^{8} + 1044 T^{6} + \cdots + 586802176)^{2} \)
|
| $17$ |
\( (T^{8} + 996 T^{6} + \cdots + 338560000)^{2} \)
|
| $19$ |
\( (T^{8} + 1796 T^{6} + \cdots + 2149991424)^{2} \)
|
| $23$ |
\( (T^{8} - 2932 T^{6} + \cdots + 11186869824)^{2} \)
|
| $29$ |
\( (T^{4} + 16 T^{3} + \cdots + 19600)^{4} \)
|
| $31$ |
\( (T^{8} + 3740 T^{6} + \cdots + 747256896)^{2} \)
|
| $37$ |
\( (T^{8} + \cdots + 3617786594304)^{2} \)
|
| $41$ |
\( (T^{8} + 932 T^{6} + \cdots + 1141899264)^{2} \)
|
| $43$ |
\( (T^{8} + \cdots + 9151544623104)^{2} \)
|
| $47$ |
\( (T^{8} + \cdots + 14545741676544)^{2} \)
|
| $53$ |
\( (T^{8} + \cdots + 38389325511744)^{2} \)
|
| $59$ |
\( (T^{8} + 16112 T^{6} + \cdots + 203235065856)^{2} \)
|
| $61$ |
\( (T^{8} + 13824 T^{6} + \cdots + 72666906624)^{2} \)
|
| $67$ |
\( (T^{8} - 17224 T^{6} + \cdots + 404129746944)^{2} \)
|
| $71$ |
\( (T^{4} + 96 T^{3} + \cdots - 18715436)^{4} \)
|
| $73$ |
\( (T^{8} + \cdots + 105841627140096)^{2} \)
|
| $79$ |
\( (T^{4} - 152 T^{3} + \cdots - 12612096)^{4} \)
|
| $83$ |
\( (T^{8} + \cdots + 13129665286144)^{2} \)
|
| $89$ |
\( (T^{8} + \cdots + 135150669186624)^{2} \)
|
| $97$ |
\( (T^{8} + \cdots + 5898136817664)^{2} \)
|
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