Properties

Label 1680.3.l.a
Level $1680$
Weight $3$
Character orbit 1680.l
Analytic conductor $45.777$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,3,Mod(1121,1680)]
 
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, 1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.1121");
 
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.l (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: \( x^{16} + 46x^{14} + 823x^{12} + 7252x^{10} + 32831x^{8} + 71486x^{6} + 60809x^{4} + 15680x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{3} + \beta_{8} q^{5} - \beta_{7} q^{7} + (\beta_{15} - \beta_{13} + \beta_{12} - \beta_{10} + \beta_{9} + \beta_{6} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{3} + \beta_{8} q^{5} - \beta_{7} q^{7} + (\beta_{15} - \beta_{13} + \beta_{12} - \beta_{10} + \beta_{9} + \beta_{6} - \beta_{2} + 1) q^{9} + (2 \beta_{12} - 2 \beta_{11} - \beta_{9} - \beta_{5} + 1) q^{11} + ( - \beta_{15} - \beta_{14} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{6} + \beta_{3} + \beta_{2} + 1) q^{13} + ( - \beta_{15} + \beta_{13} - \beta_{8} - \beta_{4} - \beta_1) q^{15} + ( - \beta_{15} - \beta_{14} + 3 \beta_{13} + \beta_{12} - 2 \beta_{11} + \beta_{10} - 3 \beta_{8} + \cdots + 2) q^{17}+ \cdots + ( - \beta_{15} - 2 \beta_{14} - 5 \beta_{13} + 9 \beta_{12} - 6 \beta_{11} + \beta_{10} + \cdots + 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} + 22 q^{9} - 10 q^{15} + 16 q^{19} - 14 q^{21} - 80 q^{25} + 148 q^{27} + 72 q^{31} - 4 q^{33} - 40 q^{37} - 90 q^{39} - 280 q^{43} + 40 q^{45} + 112 q^{49} - 38 q^{51} - 80 q^{55} - 36 q^{57} - 56 q^{61} + 56 q^{63} + 120 q^{67} + 60 q^{69} - 208 q^{73} + 40 q^{75} + 204 q^{79} + 458 q^{81} + 100 q^{85} + 324 q^{87} + 28 q^{91} + 108 q^{93} + 728 q^{97} + 166 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 46x^{14} + 823x^{12} + 7252x^{10} + 32831x^{8} + 71486x^{6} + 60809x^{4} + 15680x^{2} + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19 \nu^{15} + 346 \nu^{14} + 314 \nu^{13} + 11114 \nu^{12} - 6979 \nu^{11} + 96196 \nu^{10} - 199188 \nu^{9} - 172404 \nu^{8} - 1667083 \nu^{7} - 5356966 \nu^{6} + \cdots - 1400928 ) / 468672 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 355 \nu^{15} + 362 \nu^{14} + 14860 \nu^{13} + 13434 \nu^{12} + 235239 \nu^{11} + 173056 \nu^{10} + 1782656 \nu^{9} + 883956 \nu^{8} + 6908345 \nu^{7} + 1385434 \nu^{6} + \cdots + 480768 ) / 468672 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 355 \nu^{15} - 454 \nu^{14} - 14860 \nu^{13} - 17010 \nu^{12} - 235239 \nu^{11} - 222000 \nu^{10} - 1782656 \nu^{9} - 1143564 \nu^{8} - 6908345 \nu^{7} + \cdots + 357696 ) / 468672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9 \nu^{15} - 536 \nu^{14} - 1136 \nu^{13} - 19136 \nu^{12} - 51355 \nu^{11} - 226568 \nu^{10} - 720276 \nu^{9} - 882084 \nu^{8} - 4021041 \nu^{7} + 703220 \nu^{6} + \cdots + 1054080 ) / 234336 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 187 \nu^{14} + 7587 \nu^{12} + 114130 \nu^{10} + 791734 \nu^{8} + 2620631 \nu^{6} + 3968375 \nu^{4} + 2637324 \nu^{2} + 780032 ) / 78112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 187 \nu^{14} - 7587 \nu^{12} - 114130 \nu^{10} - 791734 \nu^{8} - 2620631 \nu^{6} - 3968375 \nu^{4} - 2559212 \nu^{2} - 311360 ) / 78112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 321 \nu^{15} + 14812 \nu^{13} + 265971 \nu^{11} + 2352364 \nu^{9} + 10668555 \nu^{7} + 23042624 \nu^{5} + 18482577 \nu^{3} + 3269960 \nu ) / 234336 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 365 \nu^{15} + 174 \nu^{14} - 16310 \nu^{13} + 5702 \nu^{12} - 279615 \nu^{11} + 53512 \nu^{10} - 2303744 \nu^{9} - 1872 \nu^{8} - 9262303 \nu^{7} - 2088654 \nu^{6} + \cdots - 1769184 ) / 234336 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 48 \nu^{15} - 1040 \nu^{14} + 4519 \nu^{13} - 43396 \nu^{12} + 130499 \nu^{11} - 680212 \nu^{10} + 1650778 \nu^{9} - 5011884 \nu^{8} + 9843186 \nu^{7} - 17945092 \nu^{6} + \cdots - 455520 ) / 234336 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1141 \nu^{15} - 346 \nu^{14} - 45836 \nu^{13} - 11114 \nu^{12} - 677801 \nu^{11} - 96196 \nu^{10} - 4551216 \nu^{9} + 172404 \nu^{8} - 14056703 \nu^{7} + \cdots + 1400928 ) / 468672 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 695 \nu^{15} - 354 \nu^{14} + 32427 \nu^{13} - 12274 \nu^{12} + 589430 \nu^{11} - 134626 \nu^{10} + 5278382 \nu^{9} - 355776 \nu^{8} + 24214099 \nu^{7} + 1985766 \nu^{6} + \cdots + 225744 ) / 234336 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 695 \nu^{15} - 354 \nu^{14} + 29986 \nu^{13} - 12274 \nu^{12} + 494231 \nu^{11} - 134626 \nu^{10} + 3935832 \nu^{9} - 355776 \nu^{8} + 15909817 \nu^{7} + 1985766 \nu^{6} + \cdots + 225744 ) / 234336 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1823 \nu^{15} - 956 \nu^{14} - 86142 \nu^{13} - 40980 \nu^{12} - 1587409 \nu^{11} - 664816 \nu^{10} - 14396428 \nu^{9} - 5104704 \nu^{8} - 66456433 \nu^{7} + \cdots + 1631712 ) / 468672 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 511 \nu^{15} + 87 \nu^{14} + 22834 \nu^{13} + 2851 \nu^{12} + 393902 \nu^{11} + 26756 \nu^{10} + 3318976 \nu^{9} - 936 \nu^{8} + 14253143 \nu^{7} - 1044327 \nu^{6} + \cdots - 884592 ) / 117168 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{13} + 2\beta_{11} - 2\beta_{8} + 2\beta_{3} + 4\beta_{2} - 9\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{13} - \beta_{12} - \beta_{11} + 2 \beta_{10} - 2 \beta_{8} - 18 \beta_{7} - 12 \beta_{6} + \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta _1 + 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4 \beta_{15} + 4 \beta_{14} - 44 \beta_{13} + 4 \beta_{12} - 32 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + 44 \beta_{8} - 4 \beta_{6} + 4 \beta_{5} - 32 \beta_{3} - 68 \beta_{2} + 101 \beta _1 - 36 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 22 \beta_{15} - 16 \beta_{13} + 18 \beta_{12} + 20 \beta_{11} - 36 \beta_{10} - 2 \beta_{9} + 38 \beta_{8} + 261 \beta_{7} + 149 \beta_{6} - 20 \beta_{5} - 88 \beta_{4} + 34 \beta_{3} - 18 \beta_{2} - 44 \beta _1 - 692 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 68 \beta_{15} - 96 \beta_{14} + 706 \beta_{13} - 92 \beta_{12} + 450 \beta_{11} + 96 \beta_{10} + 84 \beta_{9} - 782 \beta_{8} + 96 \beta_{6} - 80 \beta_{5} + 454 \beta_{3} + 968 \beta_{2} - 1249 \beta _1 + 534 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 343 \beta_{15} - 12 \beta_{14} + 217 \beta_{13} - 271 \beta_{12} - 313 \beta_{11} + 530 \beta_{10} + 42 \beta_{9} - 572 \beta_{8} - 3598 \beta_{7} - 1932 \beta_{6} + 313 \beta_{5} + 1468 \beta_{4} - 416 \beta_{3} + 271 \beta_{2} + \cdots + 8626 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 760 \beta_{15} + 1708 \beta_{14} - 10316 \beta_{13} + 1720 \beta_{12} - 6232 \beta_{11} - 1708 \beta_{10} - 1288 \beta_{9} + 12560 \beta_{8} - 1708 \beta_{6} + 1180 \beta_{5} - 6236 \beta_{3} - 13120 \beta_{2} + \cdots - 7416 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4724 \beta_{15} + 480 \beta_{14} - 2808 \beta_{13} + 3924 \beta_{12} + 4560 \beta_{11} - 7368 \beta_{10} - 636 \beta_{9} + 8012 \beta_{8} + 48769 \beta_{7} + 25729 \beta_{6} - 4568 \beta_{5} - 22320 \beta_{4} + \cdots - 112226 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5656 \beta_{15} - 27344 \beta_{14} + 145458 \beta_{13} - 29912 \beta_{12} + 86626 \beta_{11} + 27344 \beta_{10} + 17624 \beta_{9} - 191530 \beta_{8} + 27344 \beta_{6} - 15376 \beta_{5} + \cdots + 100170 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 61357 \beta_{15} - 12128 \beta_{14} + 35657 \beta_{13} - 56229 \beta_{12} - 64673 \beta_{11} + 100330 \beta_{10} + 8444 \beta_{9} - 109142 \beta_{8} - 657138 \beta_{7} - 347836 \beta_{6} + \cdots + 1494128 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2244 \beta_{15} + 417444 \beta_{14} - 2020764 \beta_{13} + 497580 \beta_{12} - 1209904 \beta_{11} - 417444 \beta_{10} - 228540 \beta_{9} + 2841732 \beta_{8} - 417444 \beta_{6} + \cdots - 1336524 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 771362 \beta_{15} + 249912 \beta_{14} - 448648 \beta_{13} + 802782 \beta_{12} + 907004 \beta_{11} - 1355652 \beta_{10} - 104222 \beta_{9} + 1469922 \beta_{8} + 8840469 \beta_{7} + \cdots - 20142648 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1193348 \beta_{15} - 6218992 \beta_{14} + 27899826 \beta_{13} - 8009844 \beta_{12} + 16948834 \beta_{11} + 6218992 \beta_{10} + 2877004 \beta_{9} - 41525334 \beta_{8} + \cdots + 17741342 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1
2.60953i
2.60953i
2.02253i
2.02253i
0.209282i
0.209282i
3.73696i
3.73696i
0.601965i
0.601965i
2.62785i
2.62785i
1.02879i
1.02879i
3.57278i
3.57278i
0 −2.93192 0.635503i 0 2.23607i 0 2.64575 0 8.19227 + 3.72648i 0
1121.2 0 −2.93192 + 0.635503i 0 2.23607i 0 2.64575 0 8.19227 3.72648i 0
1121.3 0 −2.91626 0.703870i 0 2.23607i 0 −2.64575 0 8.00914 + 4.10533i 0
1121.4 0 −2.91626 + 0.703870i 0 2.23607i 0 −2.64575 0 8.00914 4.10533i 0
1121.5 0 −1.91543 2.30892i 0 2.23607i 0 2.64575 0 −1.66223 + 8.84517i 0
1121.6 0 −1.91543 + 2.30892i 0 2.23607i 0 2.64575 0 −1.66223 8.84517i 0
1121.7 0 −1.47033 2.61498i 0 2.23607i 0 2.64575 0 −4.67625 + 7.68978i 0
1121.8 0 −1.47033 + 2.61498i 0 2.23607i 0 2.64575 0 −4.67625 7.68978i 0
1121.9 0 −0.926467 2.85336i 0 2.23607i 0 −2.64575 0 −7.28332 + 5.28709i 0
1121.10 0 −0.926467 + 2.85336i 0 2.23607i 0 −2.64575 0 −7.28332 5.28709i 0
1121.11 0 0.217001 2.99214i 0 2.23607i 0 −2.64575 0 −8.90582 1.29859i 0
1121.12 0 0.217001 + 2.99214i 0 2.23607i 0 −2.64575 0 −8.90582 + 1.29859i 0
1121.13 0 2.94860 0.552947i 0 2.23607i 0 −2.64575 0 8.38850 3.26084i 0
1121.14 0 2.94860 + 0.552947i 0 2.23607i 0 −2.64575 0 8.38850 + 3.26084i 0
1121.15 0 2.99481 0.176471i 0 2.23607i 0 2.64575 0 8.93772 1.05699i 0
1121.16 0 2.99481 + 0.176471i 0 2.23607i 0 2.64575 0 8.93772 + 1.05699i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1121.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b 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.l.a 16
3.b odd 2 1 inner 1680.3.l.a 16
4.b odd 2 1 105.3.c.a 16
12.b even 2 1 105.3.c.a 16
20.d odd 2 1 525.3.c.b 16
20.e even 4 2 525.3.f.b 32
60.h even 2 1 525.3.c.b 16
60.l odd 4 2 525.3.f.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.c.a 16 4.b odd 2 1
105.3.c.a 16 12.b even 2 1
525.3.c.b 16 20.d odd 2 1
525.3.c.b 16 60.h even 2 1
525.3.f.b 32 20.e even 4 2
525.3.f.b 32 60.l odd 4 2
1680.3.l.a 16 1.a even 1 1 trivial
1680.3.l.a 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{16} + 1302 T_{11}^{14} + 695913 T_{11}^{12} + 201895536 T_{11}^{10} + 35000873568 T_{11}^{8} + 3726118349952 T_{11}^{6} + 238332652195072 T_{11}^{4} + \cdots + 12\!\cdots\!76 \) acting on \(S_{3}^{\mathrm{new}}(1680, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 8 T^{15} + 21 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + 1302 T^{14} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{8} - 607 T^{6} + 1452 T^{5} + \cdots + 187600704)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 2706 T^{14} + \cdots + 60\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{8} - 8 T^{7} - 2028 T^{6} + \cdots + 18126732544)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 3984 T^{14} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{16} + 8078 T^{14} + \cdots + 53\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{8} - 36 T^{7} - 2100 T^{6} + \cdots - 949999104)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 20 T^{7} + \cdots - 722645680896)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 9352 T^{14} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{8} + 140 T^{7} + \cdots + 104284349696)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 18850 T^{14} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{16} + 18536 T^{14} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} + 40896 T^{14} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{8} + 28 T^{7} + \cdots - 58902532217856)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 60 T^{7} + \cdots + 49453479094784)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 63416 T^{14} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{8} + 104 T^{7} + \cdots - 1772600244224)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 102 T^{7} + \cdots + 110155963598144)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 67984 T^{14} + \cdots + 77\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + 93704 T^{14} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{8} - 364 T^{7} + \cdots - 99213562086336)^{2} \) Copy content Toggle raw display
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