Properties

Label 840.2.g.c
Level $840$
Weight $2$
Character orbit 840.g
Analytic conductor $6.707$
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] = [840,2,Mod(421,840)]
 
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, 1, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.421");
 
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.g (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: \(12\)
Coefficient field: 12.0.3058043990573056.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + x^{10} - 8x^{7} - 16x^{5} + 16x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_{4} q^{3} + \beta_{7} q^{4} - \beta_{4} q^{5} + \beta_{2} q^{6} + q^{7} + (\beta_{11} + \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + \beta_{4} q^{3} + \beta_{7} q^{4} - \beta_{4} q^{5} + \beta_{2} q^{6} + q^{7} + (\beta_{11} + \beta_1) q^{8} - q^{9} - \beta_{2} q^{10} + (\beta_{7} - \beta_{6} + \beta_{2}) q^{11} + \beta_{9} q^{12} + (\beta_{7} - \beta_{2} - \beta_1) q^{13} + \beta_{5} q^{14} + q^{15} + ( - \beta_{10} + \beta_{3}) q^{16} + (\beta_{11} + \beta_{5} + \beta_{3} + \beta_{2}) q^{17} - \beta_{5} q^{18} + (\beta_{10} + \beta_{9} + \beta_{5}) q^{19} - \beta_{9} q^{20} + \beta_{4} q^{21} + (\beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{4} + \beta_1) q^{22} + (\beta_{11} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1) q^{23} + (\beta_{10} + \beta_{3}) q^{24} - q^{25} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{4} + \beta_1) q^{26} - \beta_{4} q^{27} + \beta_{7} q^{28} + ( - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{29} + \beta_{5} q^{30} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 - 4) q^{31} + ( - \beta_{11} + 2 \beta_{6} - \beta_1 + 4) q^{32} + (\beta_{9} - \beta_{8} - \beta_{5}) q^{33} + ( - \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} + 2) q^{34} - \beta_{4} q^{35} - \beta_{7} q^{36} + ( - \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{37} + (\beta_{10} + \beta_{7} - \beta_{6} + \beta_{3} + \beta_1 - 2) q^{38} + ( - \beta_{10} + \beta_{9} + \beta_{5}) q^{39} + ( - \beta_{10} - \beta_{3}) q^{40} + (\beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{5} + \beta_{3} + 2 \beta_{2} - \beta_1) q^{41} + \beta_{2} q^{42} + (\beta_{11} - \beta_{10} + \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2}) q^{43} + (2 \beta_{3} + 2 \beta_{2} - 4) q^{44} + \beta_{4} q^{45} + ( - \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{4} + \beta_{3} + 2) q^{46} + ( - \beta_{11} - \beta_{10} + \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{47} + ( - \beta_{11} + \beta_1) q^{48} + q^{49} - \beta_{5} q^{50} + ( - \beta_{11} - \beta_{5} + \beta_{3} + \beta_{2}) q^{51} + ( - 2 \beta_{10} - 2 \beta_{2} - 4) q^{52} + (\beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 3 \beta_{2} - 2 \beta_1) q^{53} - \beta_{2} q^{54} + ( - \beta_{9} + \beta_{8} + \beta_{5}) q^{55} + (\beta_{11} + \beta_1) q^{56} + ( - \beta_{7} + \beta_{2} - \beta_1) q^{57} + (\beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{58} + ( - 2 \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 2 \beta_1) q^{59}+ \cdots + ( - \beta_{7} + \beta_{6} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{4} - 2 q^{6} + 12 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{4} - 2 q^{6} + 12 q^{7} - 12 q^{9} + 2 q^{10} + 12 q^{15} + 2 q^{16} + 8 q^{23} + 2 q^{24} - 12 q^{25} - 2 q^{28} - 40 q^{31} + 40 q^{32} + 20 q^{34} + 2 q^{36} - 20 q^{38} - 2 q^{40} - 2 q^{42} - 48 q^{44} + 20 q^{46} - 16 q^{47} + 12 q^{49} - 44 q^{52} + 2 q^{54} - 28 q^{58} - 2 q^{60} - 12 q^{63} - 2 q^{64} - 24 q^{66} - 4 q^{68} + 2 q^{70} + 56 q^{71} - 20 q^{74} + 20 q^{78} + 16 q^{79} + 12 q^{81} - 44 q^{82} + 28 q^{86} + 16 q^{87} + 40 q^{88} - 2 q^{90} - 16 q^{92} + 24 q^{94} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + x^{10} - 8x^{7} - 16x^{5} + 16x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + \nu^{5} - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + \nu^{8} - 8\nu^{3} - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + \nu^{8} - 8\nu^{5} + 16\nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + \nu^{9} - 8\nu^{4} - 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} - \nu^{9} + 8\nu^{6} + 16\nu^{4} - 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - \nu^{5} + 8\nu^{2} + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} - \nu^{8} + 8\nu^{5} + 16\nu^{3} - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} + \nu^{6} - 4\nu^{3} - 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + 5\nu^{9} + 4\nu^{7} - 8\nu^{6} - 16\nu^{4} - 32\nu^{2} + 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{8} + \nu^{6} - 4\nu^{3} - 12\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} - 3\nu^{9} - 12\nu^{7} - 8\nu^{6} - 8\nu^{5} + 16\nu^{4} + 64\nu^{2} + 16\nu + 64 ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{10} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{10} - \beta_{8} + 2\beta_{7} + 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{11} + 2\beta_{9} - \beta_{6} + 4\beta_{5} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{10} + \beta_{8} + 2\beta_{7} - 2\beta_{3} + 4\beta_{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{11} - 4\beta_{10} - 2\beta_{9} + 4\beta_{8} + \beta_{6} + 4\beta_{5} + 8\beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{10} - \beta_{8} - 2\beta_{7} + 2\beta_{3} - 4\beta_{2} + 8\beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 8 \beta_{7} - \beta_{6} - 4 \beta_{5} - 8 \beta_{4} + 8 \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -\beta_{10} + 16\beta_{9} + \beta_{8} + 2\beta_{7} + 8\beta_{6} + 16\beta_{5} - 2\beta_{3} + 4\beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} - 12 \beta_{8} + 8 \beta_{7} + \beta_{6} + 4 \beta_{5} + 8 \beta_{4} + 8 \beta_{3} + 32 \beta_{2} - \beta _1 + 32 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 16 \beta_{11} - 15 \beta_{10} + 15 \beta_{8} - 2 \beta_{7} - 16 \beta_{6} + 16 \beta_{5} + 64 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 8 \beta _1 + 8 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
421.1
−1.23149 + 0.695292i
−1.23149 0.695292i
−1.11568 + 0.869059i
−1.11568 0.869059i
−0.210663 + 1.39844i
−0.210663 1.39844i
0.422872 + 1.34951i
0.422872 1.34951i
0.723626 + 1.21506i
0.723626 1.21506i
1.41133 + 0.0902148i
1.41133 0.0902148i
−1.23149 0.695292i 1.00000i 1.03314 + 1.71249i 1.00000i −0.695292 + 1.23149i 1.00000 −0.0816198 2.82725i −1.00000 0.695292 1.23149i
421.2 −1.23149 + 0.695292i 1.00000i 1.03314 1.71249i 1.00000i −0.695292 1.23149i 1.00000 −0.0816198 + 2.82725i −1.00000 0.695292 + 1.23149i
421.3 −1.11568 0.869059i 1.00000i 0.489471 + 1.93918i 1.00000i 0.869059 1.11568i 1.00000 1.13917 2.58888i −1.00000 −0.869059 + 1.11568i
421.4 −1.11568 + 0.869059i 1.00000i 0.489471 1.93918i 1.00000i 0.869059 + 1.11568i 1.00000 1.13917 + 2.58888i −1.00000 −0.869059 1.11568i
421.5 −0.210663 1.39844i 1.00000i −1.91124 + 0.589197i 1.00000i −1.39844 + 0.210663i 1.00000 1.22658 + 2.54863i −1.00000 1.39844 0.210663i
421.6 −0.210663 + 1.39844i 1.00000i −1.91124 0.589197i 1.00000i −1.39844 0.210663i 1.00000 1.22658 2.54863i −1.00000 1.39844 + 0.210663i
421.7 0.422872 1.34951i 1.00000i −1.64236 1.14134i 1.00000i 1.34951 + 0.422872i 1.00000 −2.23476 + 1.73374i −1.00000 −1.34951 0.422872i
421.8 0.422872 + 1.34951i 1.00000i −1.64236 + 1.14134i 1.00000i 1.34951 0.422872i 1.00000 −2.23476 1.73374i −1.00000 −1.34951 + 0.422872i
421.9 0.723626 1.21506i 1.00000i −0.952732 1.75849i 1.00000i −1.21506 0.723626i 1.00000 −2.82609 0.114868i −1.00000 1.21506 + 0.723626i
421.10 0.723626 + 1.21506i 1.00000i −0.952732 + 1.75849i 1.00000i −1.21506 + 0.723626i 1.00000 −2.82609 + 0.114868i −1.00000 1.21506 0.723626i
421.11 1.41133 0.0902148i 1.00000i 1.98372 0.254646i 1.00000i 0.0902148 + 1.41133i 1.00000 2.77672 0.538352i −1.00000 −0.0902148 1.41133i
421.12 1.41133 + 0.0902148i 1.00000i 1.98372 + 0.254646i 1.00000i 0.0902148 1.41133i 1.00000 2.77672 + 0.538352i −1.00000 −0.0902148 + 1.41133i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b 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.g.c 12
4.b odd 2 1 3360.2.g.b 12
8.b even 2 1 inner 840.2.g.c 12
8.d odd 2 1 3360.2.g.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.g.c 12 1.a even 1 1 trivial
840.2.g.c 12 8.b even 2 1 inner
3360.2.g.b 12 4.b odd 2 1
3360.2.g.b 12 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{12} + 72T_{11}^{10} + 1664T_{11}^{8} + 14336T_{11}^{6} + 55360T_{11}^{4} + 98304T_{11}^{2} + 65536 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{10} - 8 T^{7} - 16 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 72 T^{10} + 1664 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( T^{12} + 64 T^{10} + 1232 T^{8} + \cdots + 9216 \) Copy content Toggle raw display
$17$ \( (T^{6} - 48 T^{4} + 64 T^{3} + 544 T^{2} + \cdots + 768)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 64 T^{10} + 1232 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T^{6} - 4 T^{5} - 52 T^{4} + 16 T^{3} + \cdots - 128)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 168 T^{10} + \cdots + 40144896 \) Copy content Toggle raw display
$31$ \( (T^{6} + 20 T^{5} + 100 T^{4} - 136 T^{3} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 264 T^{10} + \cdots + 373571584 \) Copy content Toggle raw display
$41$ \( (T^{6} - 156 T^{4} - 320 T^{3} + \cdots - 49408)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 352 T^{10} + \cdots + 967458816 \) Copy content Toggle raw display
$47$ \( (T^{6} + 8 T^{5} - 116 T^{4} - 864 T^{3} + \cdots - 26176)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 464 T^{10} + \cdots + 94745764864 \) Copy content Toggle raw display
$59$ \( T^{12} + 656 T^{10} + \cdots + 510504534016 \) Copy content Toggle raw display
$61$ \( T^{12} + 160 T^{10} + 5472 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$67$ \( T^{12} + 544 T^{10} + \cdots + 1882865664 \) Copy content Toggle raw display
$71$ \( (T^{6} - 28 T^{5} + 252 T^{4} - 744 T^{3} + \cdots - 8768)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 272 T^{4} - 1016 T^{3} + \cdots + 152416)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 8 T^{5} - 56 T^{4} + 384 T^{3} + \cdots + 512)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 672 T^{10} + \cdots + 84934656 \) Copy content Toggle raw display
$89$ \( (T^{6} - 44 T^{4} + 448 T^{2} + 256 T - 512)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 400 T^{4} + 1320 T^{3} + \cdots - 652576)^{2} \) Copy content Toggle raw display
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