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Properties

Label 1680.3.l.c
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: \( x^{16} - 8 x^{15} + 34 x^{14} - 80 x^{13} + 97 x^{12} - 80 x^{11} + 498 x^{10} - 3288 x^{9} + \cdots + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} + 1) q^{3} - \beta_{9} q^{5} - \beta_{5} q^{7} + ( - \beta_{11} - \beta_{9} + \cdots + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + 1) q^{3} - \beta_{9} q^{5} - \beta_{5} q^{7} + ( - \beta_{11} - \beta_{9} + \cdots + \beta_{4}) q^{9}+ \cdots + ( - 2 \beta_{15} + 4 \beta_{14} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{3} - 4 q^{9} + 20 q^{15} - 48 q^{19} + 28 q^{21} - 80 q^{25} - 64 q^{27} - 88 q^{33} + 80 q^{37} - 156 q^{39} + 336 q^{43} - 80 q^{45} + 112 q^{49} - 84 q^{51} + 80 q^{55} - 264 q^{57} + 112 q^{61} - 112 q^{63} - 240 q^{67} + 8 q^{69} + 48 q^{73} - 40 q^{75} - 8 q^{79} - 124 q^{81} + 120 q^{85} + 120 q^{87} + 56 q^{91} + 104 q^{93} - 192 q^{97} + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 34 x^{14} - 80 x^{13} + 97 x^{12} - 80 x^{11} + 498 x^{10} - 3288 x^{9} + \cdots + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 19601 \nu^{15} + 3306295 \nu^{14} - 1766369 \nu^{13} - 7909577 \nu^{12} + \cdots - 6480506876697 ) / 2344496612544 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 136511 \nu^{15} + 15079807 \nu^{14} - 54118775 \nu^{13} + 172692235 \nu^{12} + \cdots + 63868338637227 ) / 4688993225088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 52123 \nu^{15} - 1700114 \nu^{14} + 6007501 \nu^{13} - 12749018 \nu^{12} + \cdots - 4939018146594 ) / 586124153136 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 688951 \nu^{15} - 6907355 \nu^{14} + 21882463 \nu^{13} - 41780087 \nu^{12} + \cdots - 17172996697143 ) / 4688993225088 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13307 \nu^{15} + 179536 \nu^{14} - 640907 \nu^{13} + 1208056 \nu^{12} - 207224 \nu^{11} + \cdots + 532478372832 ) / 63364773312 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1020649 \nu^{15} - 5303425 \nu^{14} + 33095159 \nu^{13} - 97930957 \nu^{12} + \cdots - 42547326423741 ) / 4688993225088 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 44557 \nu^{15} - 325551 \nu^{14} + 1313733 \nu^{13} - 1257939 \nu^{12} + 1966944 \nu^{11} + \cdots - 1108213385859 ) / 173666415744 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1300615 \nu^{15} - 22366253 \nu^{14} + 115318111 \nu^{13} - 277285313 \nu^{12} + \cdots - 132861662038737 ) / 4688993225088 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 33544 \nu^{15} + 152423 \nu^{14} - 301840 \nu^{13} + 248939 \nu^{12} + 144488 \nu^{11} + \cdots + 49412852739 ) / 101934635328 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2704391 \nu^{15} - 32419405 \nu^{14} + 97959071 \nu^{13} - 178968529 \nu^{12} + \cdots - 77735610224865 ) / 4688993225088 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2758439 \nu^{15} + 11651983 \nu^{14} - 28131143 \nu^{13} + 5641387 \nu^{12} + \cdots - 269209410165 ) / 4688993225088 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3680753 \nu^{15} + 20895007 \nu^{14} - 65751713 \nu^{13} + 56632675 \nu^{12} + \cdots + 16377244578675 ) / 4688993225088 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3999643 \nu^{15} + 47452169 \nu^{14} - 189052195 \nu^{13} + 332759837 \nu^{12} + \cdots + 171642108613869 ) / 4688993225088 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5119673 \nu^{15} - 15714157 \nu^{14} + 8784881 \nu^{13} + 24934415 \nu^{12} + \cdots + 25526241605007 ) / 4688993225088 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 3192499 \nu^{15} + 26230607 \nu^{14} - 83470435 \nu^{13} + 129281687 \nu^{12} + \cdots + 58284796286007 ) / 2344496612544 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{14} + 3\beta_{9} - 3\beta_{5} - 5\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 8 \beta_{15} + 2 \beta_{14} - 4 \beta_{12} - 2 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + 4 \beta_{8} + \cdots - 7 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{15} + \beta_{14} + 6 \beta_{13} - 6 \beta_{12} - 12 \beta_{11} + 6 \beta_{10} + 18 \beta_{9} + \cdots - 30 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 26 \beta_{15} + 8 \beta_{14} - 4 \beta_{12} - 2 \beta_{11} + 22 \beta_{10} + 16 \beta_{9} + 4 \beta_{8} + \cdots - 43 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 60 \beta_{15} + 67 \beta_{14} - 12 \beta_{13} + 12 \beta_{12} + 60 \beta_{11} - 84 \beta_{10} + \cdots + 222 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4 \beta_{15} - 47 \beta_{14} + 72 \beta_{13} - 20 \beta_{12} + 116 \beta_{11} - 16 \beta_{10} + \cdots + 127 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1284 \beta_{15} + 61 \beta_{14} + 924 \beta_{13} + 156 \beta_{12} - 1236 \beta_{11} - 84 \beta_{10} + \cdots + 1248 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 568 \beta_{15} + 308 \beta_{14} + 2016 \beta_{13} + 284 \beta_{12} - 758 \beta_{11} - 1364 \beta_{10} + \cdots - 4273 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7494 \beta_{15} + 2089 \beta_{14} - 5106 \beta_{13} - 2490 \beta_{12} - 6816 \beta_{11} - 750 \beta_{10} + \cdots - 13422 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 6130 \beta_{15} - 16174 \beta_{14} - 11232 \beta_{13} + 15764 \beta_{12} - 4430 \beta_{11} + \cdots + 16427 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 106104 \beta_{15} - 44933 \beta_{14} - 38136 \beta_{13} + 67080 \beta_{12} + 61080 \beta_{11} + \cdots + 334482 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 32072 \beta_{15} - 45722 \beta_{14} - 36000 \beta_{13} - 22160 \beta_{12} + 8864 \beta_{11} + \cdots - 578105 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 675096 \beta_{15} - 419699 \beta_{14} + 37752 \beta_{13} - 980376 \beta_{12} - 240456 \beta_{11} + \cdots + 2976000 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1745576 \beta_{15} + 1034102 \beta_{14} + 703584 \beta_{13} - 1266340 \beta_{12} - 2823554 \beta_{11} + \cdots + 4189097 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 8179854 \beta_{15} + 8049205 \beta_{14} + 540078 \beta_{13} - 2241222 \beta_{12} - 1215012 \beta_{11} + \cdots - 11350182 ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.53169 − 1.60951i
2.53169 + 1.60951i
1.86110 + 2.35293i
1.86110 − 2.35293i
2.97371 + 0.396304i
2.97371 − 0.396304i
1.51079 + 2.59182i
1.51079 − 2.59182i
0.812085 + 2.88800i
0.812085 − 2.88800i
−2.18398 + 2.05675i
−2.18398 − 2.05675i
−2.85457 − 0.922735i
−2.85457 + 0.922735i
−0.650833 + 2.92855i
−0.650833 − 2.92855i
0 −2.85457 − 0.922735i 0 2.23607i 0 −2.64575 0 7.29712 + 5.26802i 0
1121.2 0 −2.85457 + 0.922735i 0 − 2.23607i 0 −2.64575 0 7.29712 − 5.26802i 0
1121.3 0 −2.18398 − 2.05675i 0 2.23607i 0 −2.64575 0 0.539533 + 8.98381i 0
1121.4 0 −2.18398 + 2.05675i 0 − 2.23607i 0 −2.64575 0 0.539533 − 8.98381i 0
1121.5 0 −0.650833 − 2.92855i 0 2.23607i 0 2.64575 0 −8.15283 + 3.81200i 0
1121.6 0 −0.650833 + 2.92855i 0 − 2.23607i 0 2.64575 0 −8.15283 − 3.81200i 0
1121.7 0 0.812085 − 2.88800i 0 − 2.23607i 0 2.64575 0 −7.68104 − 4.69059i 0
1121.8 0 0.812085 + 2.88800i 0 2.23607i 0 2.64575 0 −7.68104 + 4.69059i 0
1121.9 0 1.51079 − 2.59182i 0 2.23607i 0 2.64575 0 −4.43502 − 7.83139i 0
1121.10 0 1.51079 + 2.59182i 0 − 2.23607i 0 2.64575 0 −4.43502 + 7.83139i 0
1121.11 0 1.86110 − 2.35293i 0 − 2.23607i 0 −2.64575 0 −2.07259 − 8.75810i 0
1121.12 0 1.86110 + 2.35293i 0 2.23607i 0 −2.64575 0 −2.07259 + 8.75810i 0
1121.13 0 2.53169 − 1.60951i 0 2.23607i 0 −2.64575 0 3.81894 − 8.14958i 0
1121.14 0 2.53169 + 1.60951i 0 − 2.23607i 0 −2.64575 0 3.81894 + 8.14958i 0
1121.15 0 2.97371 − 0.396304i 0 − 2.23607i 0 2.64575 0 8.68589 − 2.35699i 0
1121.16 0 2.97371 + 0.396304i 0 2.23607i 0 2.64575 0 8.68589 + 2.35699i 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.c 16
3.b odd 2 1 inner 1680.3.l.c 16
4.b odd 2 1 210.3.e.a ✓ 16
12.b even 2 1 210.3.e.a ✓ 16
20.d odd 2 1 1050.3.e.d 16
20.e even 4 2 1050.3.c.c 32
60.h even 2 1 1050.3.e.d 16
60.l odd 4 2 1050.3.c.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.e.a ✓ 16 4.b odd 2 1
210.3.e.a ✓ 16 12.b even 2 1
1050.3.c.c 32 20.e even 4 2
1050.3.c.c 32 60.l odd 4 2
1050.3.e.d 16 20.d odd 2 1
1050.3.e.d 16 60.h even 2 1
1680.3.l.c 16 1.a even 1 1 trivial
1680.3.l.c 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{16} + 1348 T_{11}^{14} + 718702 T_{11}^{12} + 193917500 T_{11}^{10} + 28186215185 T_{11}^{8} + \cdots + 33\!\cdots\!76 \) 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
$3$ \( T^{16} - 8 T^{15} + \cdots + 43046721 \) 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
$11$ \( T^{16} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{8} - 858 T^{6} + \cdots - 64451196)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{8} + 24 T^{7} + \cdots + 1552481856)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{8} - 4004 T^{6} + \cdots + 212472592896)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 40 T^{7} + \cdots + 74680094976)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{8} - 168 T^{7} + \cdots + 729854118144)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 28344025168896)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 331910403784704)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 535924119269376)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 148620892321344)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 53\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 1371263608636)^{2} \) Copy content Toggle raw display
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