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Properties

Label 1680.3.s.c
Level $1680$
Weight $3$
Character orbit 1680.s
Analytic conductor $45.777$
Analytic rank $0$
Dimension $12$
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(1441,1680)]
 
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, 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("1680.1441");
 
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.s (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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 21 x^{10} - 26 x^{9} + 295 x^{8} - 372 x^{7} + 1704 x^{6} - 1074 x^{5} + 5613 x^{4} + \cdots + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{8} q^{5} + ( - \beta_{10} + \beta_{8} + \beta_{3} + 1) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_{8} q^{5} + ( - \beta_{10} + \beta_{8} + \beta_{3} + 1) q^{7} - 3 q^{9} + ( - \beta_{11} + \beta_{8} - \beta_{5} + \cdots + 1) q^{11}+ \cdots + (3 \beta_{11} - 3 \beta_{8} + 3 \beta_{5} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{7} - 36 q^{9} + 16 q^{11} + 36 q^{21} + 64 q^{23} - 60 q^{25} + 104 q^{29} - 60 q^{35} + 32 q^{37} + 24 q^{39} - 152 q^{43} + 60 q^{49} - 24 q^{51} + 176 q^{53} - 240 q^{57} - 24 q^{63} - 240 q^{65} - 168 q^{67} - 32 q^{71} + 8 q^{77} - 120 q^{79} + 108 q^{81} + 120 q^{85} - 24 q^{91} + 48 q^{93} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 21 x^{10} - 26 x^{9} + 295 x^{8} - 372 x^{7} + 1704 x^{6} - 1074 x^{5} + 5613 x^{4} + \cdots + 441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1163071964 \nu^{11} + 179172244 \nu^{10} + 20630949354 \nu^{9} + 16671982336 \nu^{8} + \cdots + 2151653525805 ) / 5604734688861 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19870505890 \nu^{11} - 84005908649 \nu^{10} + 518349481172 \nu^{9} - 1428356575517 \nu^{8} + \cdots - 144857546720712 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7830116016 \nu^{11} + 14497160068 \nu^{10} - 164611608580 \nu^{9} + 182952067062 \nu^{8} + \cdots + 8249891025873 ) / 5604734688861 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28642872442 \nu^{11} - 134337649452 \nu^{10} + 684289167827 \nu^{9} + \cdots - 231105440938869 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 39871638046 \nu^{11} - 128034905585 \nu^{10} + 950291458850 \nu^{9} + \cdots - 159247976309163 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 420436 \nu^{11} - 161001 \nu^{10} + 7746906 \nu^{9} + 2597287 \nu^{8} + 112253475 \nu^{7} + \cdots + 794649726 ) / 133884909 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11187033068 \nu^{11} + 5210631312 \nu^{10} - 205981302354 \nu^{9} - 55032700892 \nu^{8} + \cdots - 18282636368199 ) / 2402029152369 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5888 \nu^{11} - 11357 \nu^{10} + 121362 \nu^{9} - 142983 \nu^{8} + 1697439 \nu^{7} + \cdots - 4306806 ) / 1130283 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10465136770 \nu^{11} - 21796565040 \nu^{10} + 217694819151 \nu^{9} - 281471772670 \nu^{8} + \cdots - 2470563074763 ) / 1528564006053 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4771832146 \nu^{11} + 7993408932 \nu^{10} - 97232846541 \nu^{9} + 92976169234 \nu^{8} + \cdots + 6309118974753 ) / 579800140227 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 151294336430 \nu^{11} + 350115328056 \nu^{10} - 3210228154956 \nu^{9} + \cdots + 212231462685480 ) / 16814204066583 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} + \beta_{10} + \beta_{9} - 3 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + \beta_{8} + 2\beta_{7} - \beta_{5} - 2\beta_{4} + \beta_{3} + 11\beta _1 - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 23 \beta_{11} + 15 \beta_{10} + 9 \beta_{9} + 59 \beta_{8} + 10 \beta_{7} + 32 \beta_{6} + 27 \beta_{5} + \cdots - 148 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 96 \beta_{11} - 48 \beta_{10} + 54 \beta_{9} + 16 \beta_{8} + 6 \beta_{7} - 8 \beta_{6} + 145 \beta_{5} + \cdots + 111 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 13 \beta_{11} - 144 \beta_{10} - 144 \beta_{9} + 13 \beta_{8} - 251 \beta_{7} - 452 \beta_{6} + \cdots + 1895 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 511 \beta_{11} + 381 \beta_{10} - 336 \beta_{9} - 541 \beta_{8} - 653 \beta_{7} - 118 \beta_{6} + \cdots + 1475 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5220 \beta_{11} + 1416 \beta_{10} + 2142 \beta_{9} - 13222 \beta_{8} + 3558 \beta_{7} + 6146 \beta_{6} + \cdots - 24261 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3194 \beta_{11} - 429 \beta_{10} - 429 \beta_{9} + 3194 \beta_{8} + 9158 \beta_{7} + 4808 \beta_{6} + \cdots - 33449 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 77615 \beta_{11} + 24843 \beta_{10} + 20103 \beta_{9} + 179477 \beta_{8} + 54352 \beta_{7} + \cdots - 323017 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 312327 \beta_{11} - 124221 \beta_{10} + 128205 \beta_{9} + 218377 \beta_{8} + 3984 \beta_{7} + \cdots + 362202 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1441.1
0.198184 − 0.343264i
−1.01714 + 1.76174i
1.31896 − 2.28450i
0.378061 − 0.654821i
1.86875 − 3.23677i
−1.74681 + 3.02556i
0.378061 + 0.654821i
1.86875 + 3.23677i
−1.74681 − 3.02556i
0.198184 + 0.343264i
−1.01714 − 1.76174i
1.31896 + 2.28450i
0 − 1.73205i 0 − 2.23607i 0 −4.15782 − 5.63139i 0 −3.00000 0
1441.2 0 − 1.73205i 0 − 2.23607i 0 3.33344 + 6.15534i 0 −3.00000 0
1441.3 0 − 1.73205i 0 − 2.23607i 0 6.69736 − 2.03600i 0 −3.00000 0
1441.4 0 − 1.73205i 0 2.23607i 0 −6.13981 + 3.36195i 0 −3.00000 0
1441.5 0 − 1.73205i 0 2.23607i 0 −2.44621 + 6.55866i 0 −3.00000 0
1441.6 0 − 1.73205i 0 2.23607i 0 6.71303 + 1.98374i 0 −3.00000 0
1441.7 0 1.73205i 0 − 2.23607i 0 −6.13981 − 3.36195i 0 −3.00000 0
1441.8 0 1.73205i 0 − 2.23607i 0 −2.44621 − 6.55866i 0 −3.00000 0
1441.9 0 1.73205i 0 − 2.23607i 0 6.71303 − 1.98374i 0 −3.00000 0
1441.10 0 1.73205i 0 2.23607i 0 −4.15782 + 5.63139i 0 −3.00000 0
1441.11 0 1.73205i 0 2.23607i 0 3.33344 − 6.15534i 0 −3.00000 0
1441.12 0 1.73205i 0 2.23607i 0 6.69736 + 2.03600i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1441.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.s.c 12
4.b odd 2 1 105.3.h.a ✓ 12
7.b odd 2 1 inner 1680.3.s.c 12
12.b even 2 1 315.3.h.d 12
20.d odd 2 1 525.3.h.d 12
20.e even 4 2 525.3.e.c 24
28.d even 2 1 105.3.h.a ✓ 12
84.h odd 2 1 315.3.h.d 12
140.c even 2 1 525.3.h.d 12
140.j odd 4 2 525.3.e.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.h.a ✓ 12 4.b odd 2 1
105.3.h.a ✓ 12 28.d even 2 1
315.3.h.d 12 12.b even 2 1
315.3.h.d 12 84.h odd 2 1
525.3.e.c 24 20.e even 4 2
525.3.e.c 24 140.j odd 4 2
525.3.h.d 12 20.d odd 2 1
525.3.h.d 12 140.c even 2 1
1680.3.s.c 12 1.a even 1 1 trivial
1680.3.s.c 12 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{6} - 8T_{11}^{5} - 528T_{11}^{4} + 3424T_{11}^{3} + 66832T_{11}^{2} - 336576T_{11} + 388416 \) acting on \(S_{3}^{\mathrm{new}}(1680, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{6} - 8 T^{5} + \cdots + 388416)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 1316818944 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 867491057664 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{6} - 32 T^{5} + \cdots - 11126976)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 52 T^{5} + \cdots + 172225344)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{6} - 16 T^{5} + \cdots + 82379584)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{6} + 76 T^{5} + \cdots + 44197696)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 57\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{6} - 88 T^{5} + \cdots + 19593854784)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{6} + 84 T^{5} + \cdots + 35588736064)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 16 T^{5} + \cdots - 12730697664)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{6} + 60 T^{5} + \cdots - 595422656)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 74\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
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