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Properties

Label 1680.3.l.b
Level $1680$
Weight $3$
Character orbit 1680.l
Analytic conductor $45.777$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1680,3,Mod(1121,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [16,0,8,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(45.7766844125\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 37 x^{14} - 116 x^{13} + 298 x^{12} - 548 x^{11} + 267 x^{10} + 3264 x^{9} + \cdots + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{6} q^{5} + \beta_{3} q^{7} + (\beta_{6} + \beta_{4} + \beta_{3} + \cdots - 1) q^{9} + ( - \beta_{13} - \beta_{7} + \cdots - \beta_1) q^{11} + (\beta_{14} - \beta_{9} + \cdots - \beta_{3}) q^{13}+ \cdots + ( - 3 \beta_{15} - \beta_{14} + \cdots - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} - 10 q^{9} - 10 q^{15} - 48 q^{19} - 14 q^{21} - 80 q^{25} - 28 q^{27} - 24 q^{31} + 92 q^{33} - 40 q^{37} + 54 q^{39} + 168 q^{43} + 40 q^{45} + 112 q^{49} + 186 q^{51} + 80 q^{55} + 252 q^{57}+ \cdots - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 37 x^{14} - 116 x^{13} + 298 x^{12} - 548 x^{11} + 267 x^{10} + 3264 x^{9} + \cdots + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6617 \nu^{15} + 155095 \nu^{14} - 484166 \nu^{13} + 1283560 \nu^{12} - 2324684 \nu^{11} + \cdots + 495931706703 ) / 78517219104 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 542 \nu^{15} + 9619 \nu^{14} - 36236 \nu^{13} + 103633 \nu^{12} - 254000 \nu^{11} + \cdots + 32964222348 ) / 2066242608 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15895 \nu^{15} - 35884 \nu^{14} + 267092 \nu^{13} - 817993 \nu^{12} + 2164580 \nu^{11} + \cdots - 523969470981 ) / 39258609552 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} - 8 \nu^{14} + 37 \nu^{13} - 116 \nu^{12} + 298 \nu^{11} - 548 \nu^{10} + 267 \nu^{9} + \cdots - 36669429 ) / 1594323 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8731 \nu^{15} - 48959 \nu^{14} + 140464 \nu^{13} - 383678 \nu^{12} + 887140 \nu^{11} + \cdots - 21030714693 ) / 13086203184 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6079 \nu^{15} - 66812 \nu^{14} + 244327 \nu^{13} - 665510 \nu^{12} + 1847956 \nu^{11} + \cdots - 183948204771 ) / 6543101592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23951 \nu^{15} - 468376 \nu^{14} + 1916597 \nu^{13} - 5168698 \nu^{12} + 13553417 \nu^{11} + \cdots - 1716406689402 ) / 13086203184 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5825 \nu^{15} + 25801 \nu^{14} - 76997 \nu^{13} + 196117 \nu^{12} - 490457 \nu^{11} + \cdots + 7566656958 ) / 2066242608 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 111239 \nu^{15} - 625357 \nu^{14} + 2063474 \nu^{13} - 5431303 \nu^{12} + 12994082 \nu^{11} + \cdots - 892492449462 ) / 39258609552 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60619 \nu^{15} + 400505 \nu^{14} - 1419664 \nu^{13} + 3608051 \nu^{12} + \cdots + 796129973019 ) / 19629304776 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 146078 \nu^{15} - 1412830 \nu^{14} + 5906285 \nu^{13} - 14650633 \nu^{12} + \cdots - 4349531566251 ) / 39258609552 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 300875 \nu^{15} - 1251013 \nu^{14} + 3472646 \nu^{13} - 9313564 \nu^{12} + 19127444 \nu^{11} + \cdots - 34843929165 ) / 78517219104 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 151913 \nu^{15} + 1011040 \nu^{14} - 3625247 \nu^{13} + 9805183 \nu^{12} + \cdots + 1862579005011 ) / 39258609552 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 239504 \nu^{15} - 1774813 \nu^{14} + 6463841 \nu^{13} - 17796586 \nu^{12} + \cdots - 4122388368441 ) / 39258609552 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} + \beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{15} - 3 \beta_{14} + 6 \beta_{13} - 3 \beta_{11} - 3 \beta_{10} + 6 \beta_{9} + 3 \beta_{7} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{15} + 2 \beta_{14} - \beta_{13} - 7 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + \cdots - 46 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9 \beta_{15} + 15 \beta_{14} - 3 \beta_{13} + 9 \beta_{12} + 33 \beta_{10} + 42 \beta_{9} - 21 \beta_{8} + \cdots + 140 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 51 \beta_{15} - 82 \beta_{14} - 61 \beta_{13} - 54 \beta_{12} - 109 \beta_{11} + 98 \beta_{10} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 303 \beta_{15} - 309 \beta_{14} - 147 \beta_{13} + 9 \beta_{12} + 138 \beta_{11} + 141 \beta_{10} + \cdots - 1504 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 567 \beta_{15} + 239 \beta_{14} + 356 \beta_{13} + 540 \beta_{12} + 302 \beta_{11} - 265 \beta_{10} + \cdots + 1070 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 312 \beta_{15} - 1362 \beta_{14} - 129 \beta_{13} + 981 \beta_{12} + 2175 \beta_{11} - 2424 \beta_{10} + \cdots - 27952 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 96 \beta_{15} + 7004 \beta_{14} - 3160 \beta_{13} - 3186 \beta_{12} - 430 \beta_{11} - 664 \beta_{10} + \cdots - 18379 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1656 \beta_{15} + 30336 \beta_{14} + 6180 \beta_{13} + 14292 \beta_{12} - 4878 \beta_{11} + \cdots - 25177 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5136 \beta_{15} - 9892 \beta_{14} + 58808 \beta_{13} + 56970 \beta_{12} + 22346 \beta_{11} + \cdots + 724274 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 150312 \beta_{15} - 45912 \beta_{14} - 370668 \beta_{13} + 34272 \beta_{12} + 193818 \beta_{11} + \cdots + 983789 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 392112 \beta_{15} + 787307 \beta_{14} + 464279 \beta_{13} + 139374 \beta_{12} + 373649 \beta_{11} + \cdots - 1218655 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−2.88694 − 0.815838i
−2.88694 + 0.815838i
−1.67072 − 2.49172i
−1.67072 + 2.49172i
−0.895194 − 2.86332i
−0.895194 + 2.86332i
0.421042 − 2.97031i
0.421042 + 2.97031i
1.70253 − 2.47010i
1.70253 + 2.47010i
1.83120 − 2.37628i
1.83120 + 2.37628i
2.62806 − 1.44683i
2.62806 + 1.44683i
2.87002 − 0.873479i
2.87002 + 0.873479i
0 −2.88694 − 0.815838i 0 2.23607i 0 2.64575 0 7.66882 + 4.71055i 0
1121.2 0 −2.88694 + 0.815838i 0 − 2.23607i 0 2.64575 0 7.66882 − 4.71055i 0
1121.3 0 −1.67072 − 2.49172i 0 − 2.23607i 0 −2.64575 0 −3.41736 + 8.32596i 0
1121.4 0 −1.67072 + 2.49172i 0 2.23607i 0 −2.64575 0 −3.41736 − 8.32596i 0
1121.5 0 −0.895194 − 2.86332i 0 − 2.23607i 0 2.64575 0 −7.39726 + 5.12646i 0
1121.6 0 −0.895194 + 2.86332i 0 2.23607i 0 2.64575 0 −7.39726 − 5.12646i 0
1121.7 0 0.421042 − 2.97031i 0 2.23607i 0 −2.64575 0 −8.64545 − 2.50125i 0
1121.8 0 0.421042 + 2.97031i 0 − 2.23607i 0 −2.64575 0 −8.64545 + 2.50125i 0
1121.9 0 1.70253 − 2.47010i 0 − 2.23607i 0 −2.64575 0 −3.20276 − 8.41085i 0
1121.10 0 1.70253 + 2.47010i 0 2.23607i 0 −2.64575 0 −3.20276 + 8.41085i 0
1121.11 0 1.83120 − 2.37628i 0 2.23607i 0 2.64575 0 −2.29343 − 8.70288i 0
1121.12 0 1.83120 + 2.37628i 0 − 2.23607i 0 2.64575 0 −2.29343 + 8.70288i 0
1121.13 0 2.62806 − 1.44683i 0 − 2.23607i 0 2.64575 0 4.81337 − 7.60470i 0
1121.14 0 2.62806 + 1.44683i 0 2.23607i 0 2.64575 0 4.81337 + 7.60470i 0
1121.15 0 2.87002 − 0.873479i 0 2.23607i 0 −2.64575 0 7.47407 − 5.01381i 0
1121.16 0 2.87002 + 0.873479i 0 − 2.23607i 0 −2.64575 0 7.47407 + 5.01381i 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.b 16
3.b odd 2 1 inner 1680.3.l.b 16
4.b odd 2 1 420.3.g.a ✓ 16
12.b even 2 1 420.3.g.a ✓ 16
20.d odd 2 1 2100.3.g.d 16
20.e even 4 2 2100.3.e.c 32
60.h even 2 1 2100.3.g.d 16
60.l odd 4 2 2100.3.e.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.3.g.a ✓ 16 4.b odd 2 1
420.3.g.a ✓ 16 12.b even 2 1
1680.3.l.b 16 1.a even 1 1 trivial
1680.3.l.b 16 3.b odd 2 1 inner
2100.3.e.c 32 20.e even 4 2
2100.3.e.c 32 60.l odd 4 2
2100.3.g.d 16 20.d odd 2 1
2100.3.g.d 16 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{16} + 790 T_{11}^{14} + 213289 T_{11}^{12} + 24976304 T_{11}^{10} + 1389956960 T_{11}^{8} + \cdots + 1586874322944 \) 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
$11$ \( T^{16} + \cdots + 1586874322944 \) Copy content Toggle raw display
$13$ \( (T^{8} - 639 T^{6} + \cdots - 4591296)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{8} + 24 T^{7} + \cdots + 1003504896)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{8} + 12 T^{7} + \cdots + 1936267776)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 20 T^{7} + \cdots + 34943851776)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{8} - 84 T^{7} + \cdots - 588059056896)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 61\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{8} + 76 T^{7} + \cdots - 82454363136)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 132 T^{7} + \cdots - 375258686976)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 37362555561984)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 19525560488256)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 49\!\cdots\!64)^{2} \) Copy content Toggle raw display
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