Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1680,3,Mod(769,1680)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1680.769");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1680.bd (of order \(2\), degree \(1\), 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: | \(45.7766844125\) |
| 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: |
\( x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + \cdots + 33750 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{2} \) |
| 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
\( x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + \cdots + 33750 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 1304 \nu^{14} - 9128 \nu^{13} + 115446 \nu^{12} - 574012 \nu^{11} + 3591842 \nu^{10} + \cdots + 99070875 ) / 560625 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 19306 \nu^{14} + 135142 \nu^{13} - 1709079 \nu^{12} + 8497628 \nu^{11} - 53165468 \nu^{10} + \cdots - 1406219625 ) / 3027375 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 181 \nu^{14} + 1267 \nu^{13} - 16034 \nu^{12} + 79733 \nu^{11} - 499338 \nu^{10} + \cdots - 14271525 ) / 26325 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1984 \nu^{15} - 14880 \nu^{14} + 171958 \nu^{13} - 892047 \nu^{12} + 4997626 \nu^{11} + \cdots + 575687250 ) / 85899825 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 447494447 \nu^{15} - 3678104460 \nu^{14} + 43437004320 \nu^{13} - 245298593960 \nu^{12} + \cdots - 41918527948500 ) / 83752329375 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 447494447 \nu^{15} + 3229118109 \nu^{14} - 40294099863 \nu^{13} + 205531523111 \nu^{12} + \cdots + 5890964889375 ) / 83752329375 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 472182246 \nu^{15} - 3541366845 \nu^{14} + 43501742065 \nu^{13} - 229050592940 \nu^{12} + \cdots - 20360648194875 ) / 83752329375 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 755997166 \nu^{15} + 5904909925 \nu^{14} - 71246353625 \nu^{13} + 387215467260 \nu^{12} + \cdots + 47103825674250 ) / 83752329375 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 755997166 \nu^{15} - 5240241701 \nu^{14} + 66593676057 \nu^{13} - 328372443834 \nu^{12} + \cdots + 2190542081625 ) / 83752329375 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 5163308 \nu^{15} - 38724810 \nu^{14} + 475369020 \nu^{13} - 2502572345 \nu^{12} + \cdots - 200352316500 ) / 404600625 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 49652491 \nu^{15} - 376911377 \nu^{14} + 4602946389 \nu^{13} - 24466123483 \nu^{12} + \cdots - 2269018608750 ) / 3641405625 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 49652491 \nu^{15} - 367875988 \nu^{14} + 4539698666 \nu^{13} - 23665127672 \nu^{12} + \cdots - 1551811091625 ) / 3641405625 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 1309724492 \nu^{15} + 9822933690 \nu^{14} - 120588431330 \nu^{13} + 634843642680 \nu^{12} + \cdots + 51303336994125 ) / 83752329375 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 65666397 \nu^{15} - 477141136 \nu^{14} + 5938288402 \nu^{13} - 30467772959 \nu^{12} + \cdots - 1356819889500 ) / 3641405625 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 2046364168 \nu^{15} + 15347731260 \nu^{14} - 188413764520 \nu^{13} + 991915545270 \nu^{12} + \cdots + 80293733362125 ) / 83752329375 \)
|
| \(\nu\) | \(=\) |
\( ( - \beta_{15} - \beta_{13} - 2 \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \cdots + 6 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 7 \beta_{15} - 12 \beta_{14} + 5 \beta_{13} + 6 \beta_{12} + 6 \beta_{11} - 2 \beta_{10} + 7 \beta_{9} + \cdots - 90 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 11 \beta_{15} - 18 \beta_{14} + 11 \beta_{13} - 9 \beta_{12} - 9 \beta_{11} + 61 \beta_{10} + \cdots - 156 ) / 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 143 \beta_{15} + 204 \beta_{14} - 97 \beta_{13} - 108 \beta_{12} - 168 \beta_{11} + 124 \beta_{10} + \cdots + 1386 ) / 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 88 \beta_{15} + 540 \beta_{14} - 328 \beta_{13} + 255 \beta_{12} + 105 \beta_{11} - 1493 \beta_{10} + \cdots + 4236 ) / 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 1496 \beta_{15} - 1815 \beta_{14} + 814 \beta_{13} + 1095 \beta_{12} + 2160 \beta_{11} - 2395 \beta_{10} + \cdots - 11916 ) / 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 1450 \beta_{15} - 14616 \beta_{14} + 9932 \beta_{13} - 6321 \beta_{12} + 1659 \beta_{11} + \cdots - 111504 ) / 12 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 61883 \beta_{15} + 65724 \beta_{14} - 21481 \beta_{13} - 50412 \beta_{12} - 106032 \beta_{11} + \cdots + 413190 ) / 12 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 103439 \beta_{15} + 385758 \beta_{14} - 270907 \beta_{13} + 143883 \beta_{12} - 154917 \beta_{11} + \cdots + 2878206 ) / 12 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 1245547 \beta_{15} - 1144488 \beta_{14} + 95411 \beta_{13} + 1245744 \beta_{12} + 2473284 \beta_{11} + \cdots - 6713352 ) / 12 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 3766714 \beta_{15} - 9997614 \beta_{14} + 6779426 \beta_{13} - 3036957 \beta_{12} + 6542943 \beta_{11} + \cdots - 73078230 ) / 12 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 12045634 \beta_{15} + 8786955 \beta_{14} + 3666955 \beta_{13} - 15943449 \beta_{12} - 27259074 \beta_{11} + \cdots + 44325837 ) / 6 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 114248852 \beta_{15} + 254014020 \beta_{14} - 158151874 \beta_{13} + 57387759 \beta_{12} + \cdots + 1821064524 ) / 12 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 436459717 \beta_{15} - 180750738 \beta_{14} - 388881655 \beta_{13} + 824613468 \beta_{12} + \cdots - 387024930 ) / 12 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 3176368585 \beta_{15} - 6304996872 \beta_{14} + 3451153529 \beta_{13} - 864267417 \beta_{12} + \cdots - 44352101334 ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(337\) | \(421\) | \(1121\) | \(1471\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 769.1 |
|
0 | −1.73205 | 0 | −4.59769 | − | 1.96501i | 0 | 6.81963 | + | 1.57881i | 0 | 3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.2 | 0 | −1.73205 | 0 | −4.59769 | + | 1.96501i | 0 | 6.81963 | − | 1.57881i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.3 | 0 | −1.73205 | 0 | −2.40341 | − | 4.38447i | 0 | −6.94781 | − | 0.853218i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.4 | 0 | −1.73205 | 0 | −2.40341 | + | 4.38447i | 0 | −6.94781 | + | 0.853218i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.5 | 0 | −1.73205 | 0 | −1.38028 | − | 4.80571i | 0 | −5.24961 | + | 4.63050i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.6 | 0 | −1.73205 | 0 | −1.38028 | + | 4.80571i | 0 | −5.24961 | − | 4.63050i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.7 | 0 | −1.73205 | 0 | 4.91728 | − | 0.905717i | 0 | 1.91369 | + | 6.73333i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.8 | 0 | −1.73205 | 0 | 4.91728 | + | 0.905717i | 0 | 1.91369 | − | 6.73333i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.9 | 0 | 1.73205 | 0 | −4.91728 | − | 0.905717i | 0 | −1.91369 | − | 6.73333i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.10 | 0 | 1.73205 | 0 | −4.91728 | + | 0.905717i | 0 | −1.91369 | + | 6.73333i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.11 | 0 | 1.73205 | 0 | 1.38028 | − | 4.80571i | 0 | 5.24961 | − | 4.63050i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.12 | 0 | 1.73205 | 0 | 1.38028 | + | 4.80571i | 0 | 5.24961 | + | 4.63050i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.13 | 0 | 1.73205 | 0 | 2.40341 | − | 4.38447i | 0 | 6.94781 | + | 0.853218i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.14 | 0 | 1.73205 | 0 | 2.40341 | + | 4.38447i | 0 | 6.94781 | − | 0.853218i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.15 | 0 | 1.73205 | 0 | 4.59769 | − | 1.96501i | 0 | −6.81963 | − | 1.57881i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 769.16 | 0 | 1.73205 | 0 | 4.59769 | + | 1.96501i | 0 | −6.81963 | + | 1.57881i | 0 | 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 | 1680.3.bd.a | 16 | |
| 4.b | odd | 2 | 1 | 210.3.h.a | ✓ | 16 | |
| 5.b | even | 2 | 1 | inner | 1680.3.bd.a | 16 | |
| 7.b | odd | 2 | 1 | inner | 1680.3.bd.a | 16 | |
| 12.b | even | 2 | 1 | 630.3.h.e | 16 | ||
| 20.d | odd | 2 | 1 | 210.3.h.a | ✓ | 16 | |
| 20.e | even | 4 | 2 | 1050.3.f.e | 16 | ||
| 28.d | even | 2 | 1 | 210.3.h.a | ✓ | 16 | |
| 35.c | odd | 2 | 1 | inner | 1680.3.bd.a | 16 | |
| 60.h | even | 2 | 1 | 630.3.h.e | 16 | ||
| 84.h | odd | 2 | 1 | 630.3.h.e | 16 | ||
| 140.c | even | 2 | 1 | 210.3.h.a | ✓ | 16 | |
| 140.j | odd | 4 | 2 | 1050.3.f.e | 16 | ||
| 420.o | odd | 2 | 1 | 630.3.h.e | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.3.h.a | ✓ | 16 | 4.b | odd | 2 | 1 | |
| 210.3.h.a | ✓ | 16 | 20.d | odd | 2 | 1 | |
| 210.3.h.a | ✓ | 16 | 28.d | even | 2 | 1 | |
| 210.3.h.a | ✓ | 16 | 140.c | even | 2 | 1 | |
| 630.3.h.e | 16 | 12.b | even | 2 | 1 | ||
| 630.3.h.e | 16 | 60.h | even | 2 | 1 | ||
| 630.3.h.e | 16 | 84.h | odd | 2 | 1 | ||
| 630.3.h.e | 16 | 420.o | odd | 2 | 1 | ||
| 1050.3.f.e | 16 | 20.e | even | 4 | 2 | ||
| 1050.3.f.e | 16 | 140.j | odd | 4 | 2 | ||
| 1680.3.bd.a | 16 | 1.a | even | 1 | 1 | trivial | |
| 1680.3.bd.a | 16 | 5.b | even | 2 | 1 | inner | |
| 1680.3.bd.a | 16 | 7.b | odd | 2 | 1 | inner | |
| 1680.3.bd.a | 16 | 35.c | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{4} + 24T_{11}^{3} + 28T_{11}^{2} - 1776T_{11} - 4844 \)
acting on \(S_{3}^{\mathrm{new}}(1680, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( (T^{2} - 3)^{8} \)
|
| $5$ |
\( T^{16} + \cdots + 152587890625 \)
|
| $7$ |
\( T^{16} + \cdots + 33232930569601 \)
|
| $11$ |
\( (T^{4} + 24 T^{3} + \cdots - 4844)^{4} \)
|
| $13$ |
\( (T^{8} - 1044 T^{6} + \cdots + 586802176)^{2} \)
|
| $17$ |
\( (T^{8} - 996 T^{6} + \cdots + 338560000)^{2} \)
|
| $19$ |
\( (T^{8} + 1796 T^{6} + \cdots + 2149991424)^{2} \)
|
| $23$ |
\( (T^{8} + 2932 T^{6} + \cdots + 11186869824)^{2} \)
|
| $29$ |
\( (T^{4} - 16 T^{3} + \cdots + 19600)^{4} \)
|
| $31$ |
\( (T^{8} + 3740 T^{6} + \cdots + 747256896)^{2} \)
|
| $37$ |
\( (T^{8} + \cdots + 3617786594304)^{2} \)
|
| $41$ |
\( (T^{8} + 932 T^{6} + \cdots + 1141899264)^{2} \)
|
| $43$ |
\( (T^{8} + \cdots + 9151544623104)^{2} \)
|
| $47$ |
\( (T^{8} + \cdots + 14545741676544)^{2} \)
|
| $53$ |
\( (T^{8} + \cdots + 38389325511744)^{2} \)
|
| $59$ |
\( (T^{8} + 16112 T^{6} + \cdots + 203235065856)^{2} \)
|
| $61$ |
\( (T^{8} + 13824 T^{6} + \cdots + 72666906624)^{2} \)
|
| $67$ |
\( (T^{8} + 17224 T^{6} + \cdots + 404129746944)^{2} \)
|
| $71$ |
\( (T^{4} - 96 T^{3} + \cdots - 18715436)^{4} \)
|
| $73$ |
\( (T^{8} + \cdots + 105841627140096)^{2} \)
|
| $79$ |
\( (T^{4} - 152 T^{3} + \cdots - 12612096)^{4} \)
|
| $83$ |
\( (T^{8} + \cdots + 13129665286144)^{2} \)
|
| $89$ |
\( (T^{8} + \cdots + 135150669186624)^{2} \)
|
| $97$ |
\( (T^{8} + \cdots + 5898136817664)^{2} \)
|
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