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Properties

Label 840.2.f.a
Level $840$
Weight $2$
Character orbit 840.f
Analytic conductor $6.707$
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] = [840,2,Mod(41,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.f (of order \(2\), degree \(1\), 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: \(6.70743376979\)
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: \( x^{16} - 2 x^{15} + x^{14} - 4 x^{13} + 10 x^{12} - 32 x^{11} + 71 x^{10} - 70 x^{9} + 74 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - q^{5} + \beta_{4} q^{7} - \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - q^{5} + \beta_{4} q^{7} - \beta_1 q^{9} + \beta_{8} q^{11} + ( - \beta_{15} + \beta_{9} + \cdots - \beta_1) q^{13}+ \cdots + ( - \beta_{15} + \beta_{13} - \beta_{12} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{5} + 2 q^{7} - 2 q^{9} + 2 q^{21} + 16 q^{25} - 6 q^{27} - 6 q^{33} - 2 q^{35} + 12 q^{37} + 6 q^{39} - 32 q^{41} + 32 q^{43} + 2 q^{45} - 4 q^{47} - 4 q^{49} + 6 q^{51} + 24 q^{59} - 4 q^{63} - 8 q^{69} + 32 q^{77} - 4 q^{79} - 6 q^{81} - 20 q^{83} - 6 q^{87} + 24 q^{89} + 20 q^{91} - 32 q^{93} - 58 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + x^{14} - 4 x^{13} + 10 x^{12} - 32 x^{11} + 71 x^{10} - 70 x^{9} + 74 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13 \nu^{15} + 55 \nu^{14} - 50 \nu^{13} + 74 \nu^{12} - 176 \nu^{11} + 160 \nu^{10} - 517 \nu^{9} + \cdots - 37179 ) / 46656 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17 \nu^{15} - 17 \nu^{14} + 22 \nu^{13} + 114 \nu^{12} - 48 \nu^{11} - 152 \nu^{10} + 135 \nu^{9} + \cdots + 9477 ) / 46656 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{15} - 19 \nu^{14} + 14 \nu^{13} - 30 \nu^{12} + 84 \nu^{11} - 220 \nu^{10} + 333 \nu^{9} + \cdots + 8019 ) / 15552 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17 \nu^{15} - 5 \nu^{14} - 182 \nu^{13} + 218 \nu^{12} - 392 \nu^{11} + 1072 \nu^{10} + \cdots + 111537 ) / 34992 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19 \nu^{15} + 45 \nu^{14} - 126 \nu^{13} + 10 \nu^{12} - 184 \nu^{11} - 276 \nu^{10} - 683 \nu^{9} + \cdots - 102789 ) / 23328 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2 \nu^{15} - \nu^{14} - 4 \nu^{13} - 5 \nu^{12} + 8 \nu^{11} - 34 \nu^{10} + 46 \nu^{9} + 73 \nu^{8} + \cdots - 6561 ) / 2187 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 71 \nu^{15} + 5 \nu^{14} + 182 \nu^{13} - 218 \nu^{12} + 284 \nu^{11} - 1936 \nu^{10} + \cdots - 181521 ) / 69984 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 75 \nu^{15} - 47 \nu^{14} + 22 \nu^{13} - 62 \nu^{12} + 68 \nu^{11} - 1892 \nu^{10} + 1849 \nu^{9} + \cdots - 280665 ) / 46656 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 131 \nu^{15} + 19 \nu^{14} - 140 \nu^{13} + 704 \nu^{12} - 914 \nu^{11} + 2446 \nu^{10} + \cdots + 194643 ) / 69984 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15 \nu^{15} - 13 \nu^{14} + 24 \nu^{13} - 72 \nu^{12} - 34 \nu^{11} - 362 \nu^{10} + 651 \nu^{9} + \cdots - 60021 ) / 7776 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6 \nu^{15} - 4 \nu^{14} - \nu^{13} - 16 \nu^{12} + 28 \nu^{11} - 184 \nu^{10} + 215 \nu^{9} + \cdots - 26244 ) / 2916 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 57 \nu^{15} + 7 \nu^{14} + 70 \nu^{13} - 302 \nu^{12} + 440 \nu^{11} - 2048 \nu^{10} + 1951 \nu^{9} + \cdots - 305451 ) / 23328 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 35 \nu^{15} - 13 \nu^{14} + 11 \nu^{13} - 209 \nu^{12} + 185 \nu^{11} - 775 \nu^{10} + 940 \nu^{9} + \cdots - 131220 ) / 11664 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 427 \nu^{15} - 491 \nu^{14} - 2 \nu^{13} - 1102 \nu^{12} + 2332 \nu^{11} - 9980 \nu^{10} + \cdots - 1047573 ) / 139968 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{12} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{14} - 2 \beta_{12} - 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 4 \beta_{7} + \cdots + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - 2\beta_{12} + \beta_{10} + 2\beta_{8} + 2\beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{15} - \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 14 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} + \cdots + 32 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{15} - 3 \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{9} + 2 \beta_{8} - 4 \beta_{7} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5 \beta_{15} + 8 \beta_{14} + 21 \beta_{12} - 2 \beta_{11} + 6 \beta_{10} - 34 \beta_{9} + \cdots + 66 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4 \beta_{15} + 16 \beta_{14} - 7 \beta_{13} - 2 \beta_{12} - 8 \beta_{11} + 13 \beta_{10} + 8 \beta_{9} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 72 \beta_{15} - 20 \beta_{14} - 30 \beta_{12} + 24 \beta_{11} + 58 \beta_{10} - 108 \beta_{9} + \cdots + 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8 \beta_{15} + 16 \beta_{14} - 16 \beta_{13} + 16 \beta_{12} - 5 \beta_{11} - 13 \beta_{10} + 5 \beta_{9} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 165 \beta_{15} + 136 \beta_{14} - 128 \beta_{13} - 123 \beta_{12} - 280 \beta_{11} + 112 \beta_{10} + \cdots - 800 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 20 \beta_{15} - 16 \beta_{14} + 80 \beta_{13} + 32 \beta_{12} - 35 \beta_{11} + 65 \beta_{10} + \cdots - 123 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 447 \beta_{15} + 88 \beta_{14} - 384 \beta_{13} - 161 \beta_{12} - 56 \beta_{11} - 224 \beta_{10} + \cdots + 160 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 184 \beta_{15} - 256 \beta_{14} - 160 \beta_{13} + 208 \beta_{12} + 39 \beta_{11} - 169 \beta_{10} + \cdots + 208 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 568 \beta_{15} + 364 \beta_{14} - 1408 \beta_{13} + 2214 \beta_{12} + 230 \beta_{11} - 2300 \beta_{10} + \cdots - 558 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
0.0964469 − 1.72936i
0.0964469 + 1.72936i
1.14188 − 1.30235i
1.14188 + 1.30235i
−1.34935 + 1.08593i
−1.34935 − 1.08593i
1.71703 − 0.227581i
1.71703 + 0.227581i
1.66912 + 0.462633i
1.66912 − 0.462633i
−1.49826 − 0.869033i
−1.49826 + 0.869033i
−1.12510 − 1.31688i
−1.12510 + 1.31688i
0.348228 + 1.69668i
0.348228 − 1.69668i
0 −1.72936 − 0.0964469i 0 −1.00000 0 0.208829 + 2.63750i 0 2.98140 + 0.333584i 0
41.2 0 −1.72936 + 0.0964469i 0 −1.00000 0 0.208829 − 2.63750i 0 2.98140 − 0.333584i 0
41.3 0 −1.30235 − 1.14188i 0 −1.00000 0 −1.35345 − 2.27336i 0 0.392236 + 2.97425i 0
41.4 0 −1.30235 + 1.14188i 0 −1.00000 0 −1.35345 + 2.27336i 0 0.392236 − 2.97425i 0
41.5 0 −1.08593 − 1.34935i 0 −1.00000 0 2.53123 + 0.769995i 0 −0.641511 + 2.93061i 0
41.6 0 −1.08593 + 1.34935i 0 −1.00000 0 2.53123 − 0.769995i 0 −0.641511 − 2.93061i 0
41.7 0 −0.227581 − 1.71703i 0 −1.00000 0 −1.22074 + 2.34729i 0 −2.89641 + 0.781528i 0
41.8 0 −0.227581 + 1.71703i 0 −1.00000 0 −1.22074 − 2.34729i 0 −2.89641 − 0.781528i 0
41.9 0 0.462633 − 1.66912i 0 −1.00000 0 1.62879 − 2.08496i 0 −2.57194 − 1.54438i 0
41.10 0 0.462633 + 1.66912i 0 −1.00000 0 1.62879 + 2.08496i 0 −2.57194 + 1.54438i 0
41.11 0 0.869033 − 1.49826i 0 −1.00000 0 −0.807952 + 2.51937i 0 −1.48956 − 2.60407i 0
41.12 0 0.869033 + 1.49826i 0 −1.00000 0 −0.807952 − 2.51937i 0 −1.48956 + 2.60407i 0
41.13 0 1.31688 − 1.12510i 0 −1.00000 0 2.64497 + 0.0644212i 0 0.468322 − 2.96322i 0
41.14 0 1.31688 + 1.12510i 0 −1.00000 0 2.64497 − 0.0644212i 0 0.468322 + 2.96322i 0
41.15 0 1.69668 − 0.348228i 0 −1.00000 0 −2.63166 − 0.272689i 0 2.75748 − 1.18166i 0
41.16 0 1.69668 + 0.348228i 0 −1.00000 0 −2.63166 + 0.272689i 0 2.75748 + 1.18166i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.f.a ✓ 16
3.b odd 2 1 840.2.f.b yes 16
4.b odd 2 1 1680.2.f.k 16
7.b odd 2 1 840.2.f.b yes 16
12.b even 2 1 1680.2.f.l 16
21.c even 2 1 inner 840.2.f.a ✓ 16
28.d even 2 1 1680.2.f.l 16
84.h odd 2 1 1680.2.f.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.f.a ✓ 16 1.a even 1 1 trivial
840.2.f.a ✓ 16 21.c even 2 1 inner
840.2.f.b yes 16 3.b odd 2 1
840.2.f.b yes 16 7.b odd 2 1
1680.2.f.k 16 4.b odd 2 1
1680.2.f.k 16 84.h odd 2 1
1680.2.f.l 16 12.b even 2 1
1680.2.f.l 16 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{8} - 79T_{17}^{6} - 98T_{17}^{5} + 1612T_{17}^{4} + 4048T_{17}^{3} - 1232T_{17}^{2} - 4640T_{17} + 1856 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 78 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 256000000 \) Copy content Toggle raw display
$17$ \( (T^{8} - 79 T^{6} + \cdots + 1856)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 112 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$23$ \( T^{16} + 192 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 2316304384 \) Copy content Toggle raw display
$31$ \( T^{16} + 260 T^{14} + \cdots + 89718784 \) Copy content Toggle raw display
$37$ \( (T^{8} - 6 T^{7} + \cdots + 1573888)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 16 T^{7} + \cdots - 80896)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 16 T^{7} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 2 T^{7} + \cdots + 696832)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 2083195715584 \) Copy content Toggle raw display
$59$ \( (T^{8} - 12 T^{7} + \cdots - 495616)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1073741824 \) Copy content Toggle raw display
$67$ \( (T^{8} - 316 T^{6} + \cdots + 6326272)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 83534872576 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 3761489575936 \) Copy content Toggle raw display
$79$ \( (T^{8} + 2 T^{7} + \cdots + 352256)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 10 T^{7} + \cdots + 1067008)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 12 T^{7} + \cdots - 2134016)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 895060875034624 \) Copy content Toggle raw display
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