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Properties

Label 840.2.z.d
Level $840$
Weight $2$
Character orbit 840.z
Analytic conductor $6.707$
Analytic rank $0$
Dimension $28$
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(811,840)]
 
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([1, 1, 0, 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("840.811");
 
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.z (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: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{2} - 2 q^{4} + 28 q^{5} + 4 q^{6} + 4 q^{7} - 10 q^{8} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{2} - 2 q^{4} + 28 q^{5} + 4 q^{6} + 4 q^{7} - 10 q^{8} - 28 q^{9} + 2 q^{10} - 8 q^{12} - 8 q^{13} + 2 q^{14} + 6 q^{16} - 2 q^{18} - 2 q^{20} - 4 q^{24} + 28 q^{25} + 16 q^{26} + 10 q^{28} + 4 q^{30} + 24 q^{31} + 2 q^{32} + 8 q^{34} + 4 q^{35} + 2 q^{36} + 36 q^{38} - 10 q^{40} + 24 q^{43} - 4 q^{44} - 28 q^{45} - 44 q^{46} - 24 q^{47} + 16 q^{48} + 28 q^{49} + 2 q^{50} + 16 q^{52} - 4 q^{54} - 22 q^{56} + 8 q^{57} - 12 q^{58} - 8 q^{60} - 16 q^{61} - 28 q^{62} - 4 q^{63} + 22 q^{64} - 8 q^{65} - 20 q^{66} - 8 q^{67} - 12 q^{68} - 24 q^{69} + 2 q^{70} + 10 q^{72} - 8 q^{74} - 16 q^{76} + 16 q^{77} + 20 q^{78} + 6 q^{80} + 28 q^{81} + 24 q^{82} - 12 q^{84} - 4 q^{86} + 16 q^{87} + 40 q^{88} - 2 q^{90} + 48 q^{91} + 40 q^{92} + 52 q^{94} - 16 q^{96} - 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
811.1 −1.41307 − 0.0568977i − 1.00000i 1.99353 + 0.160801i 1.00000 −0.0568977 + 1.41307i −2.57870 − 0.591873i −2.80784 − 0.340650i −1.00000 −1.41307 − 0.0568977i
811.2 −1.41307 + 0.0568977i 1.00000i 1.99353 − 0.160801i 1.00000 −0.0568977 − 1.41307i −2.57870 + 0.591873i −2.80784 + 0.340650i −1.00000 −1.41307 + 0.0568977i
811.3 −1.35569 − 0.402636i 1.00000i 1.67577 + 1.09170i 1.00000 0.402636 − 1.35569i 2.48628 + 0.904667i −1.83226 − 2.15472i −1.00000 −1.35569 − 0.402636i
811.4 −1.35569 + 0.402636i − 1.00000i 1.67577 − 1.09170i 1.00000 0.402636 + 1.35569i 2.48628 − 0.904667i −1.83226 + 2.15472i −1.00000 −1.35569 + 0.402636i
811.5 −1.22597 − 0.704981i − 1.00000i 1.00600 + 1.72857i 1.00000 −0.704981 + 1.22597i 2.37591 − 1.16407i −0.0147234 − 2.82839i −1.00000 −1.22597 − 0.704981i
811.6 −1.22597 + 0.704981i 1.00000i 1.00600 − 1.72857i 1.00000 −0.704981 − 1.22597i 2.37591 + 1.16407i −0.0147234 + 2.82839i −1.00000 −1.22597 + 0.704981i
811.7 −0.767567 − 1.18779i 1.00000i −0.821681 + 1.82341i 1.00000 1.18779 − 0.767567i −1.33527 + 2.28409i 2.79653 − 0.423611i −1.00000 −0.767567 − 1.18779i
811.8 −0.767567 + 1.18779i − 1.00000i −0.821681 − 1.82341i 1.00000 1.18779 + 0.767567i −1.33527 − 2.28409i 2.79653 + 0.423611i −1.00000 −0.767567 + 1.18779i
811.9 −0.718490 − 1.21810i 1.00000i −0.967544 + 1.75039i 1.00000 1.21810 − 0.718490i 1.92209 − 1.81812i 2.82732 − 0.0790702i −1.00000 −0.718490 − 1.21810i
811.10 −0.718490 + 1.21810i − 1.00000i −0.967544 − 1.75039i 1.00000 1.21810 + 0.718490i 1.92209 + 1.81812i 2.82732 + 0.0790702i −1.00000 −0.718490 + 1.21810i
811.11 −0.135580 − 1.40770i 1.00000i −1.96324 + 0.381712i 1.00000 1.40770 − 0.135580i −2.43116 − 1.04376i 0.803512 + 2.71189i −1.00000 −0.135580 − 1.40770i
811.12 −0.135580 + 1.40770i − 1.00000i −1.96324 − 0.381712i 1.00000 1.40770 + 0.135580i −2.43116 + 1.04376i 0.803512 − 2.71189i −1.00000 −0.135580 + 1.40770i
811.13 0.0653764 − 1.41270i − 1.00000i −1.99145 − 0.184715i 1.00000 −1.41270 − 0.0653764i 1.57556 + 2.12547i −0.391141 + 2.80125i −1.00000 0.0653764 − 1.41270i
811.14 0.0653764 + 1.41270i 1.00000i −1.99145 + 0.184715i 1.00000 −1.41270 + 0.0653764i 1.57556 − 2.12547i −0.391141 − 2.80125i −1.00000 0.0653764 + 1.41270i
811.15 0.403065 − 1.35556i 1.00000i −1.67508 − 1.09276i 1.00000 1.35556 + 0.403065i −0.527819 + 2.59257i −2.15646 + 1.83021i −1.00000 0.403065 − 1.35556i
811.16 0.403065 + 1.35556i − 1.00000i −1.67508 + 1.09276i 1.00000 1.35556 − 0.403065i −0.527819 − 2.59257i −2.15646 − 1.83021i −1.00000 0.403065 + 1.35556i
811.17 0.531410 − 1.31057i − 1.00000i −1.43521 − 1.39290i 1.00000 −1.31057 − 0.531410i −2.64060 − 0.165018i −2.58819 + 1.14074i −1.00000 0.531410 − 1.31057i
811.18 0.531410 + 1.31057i 1.00000i −1.43521 + 1.39290i 1.00000 −1.31057 + 0.531410i −2.64060 + 0.165018i −2.58819 − 1.14074i −1.00000 0.531410 + 1.31057i
811.19 0.818985 − 1.15294i 1.00000i −0.658528 − 1.88848i 1.00000 1.15294 + 0.818985i −0.864047 − 2.50068i −2.71662 − 0.787391i −1.00000 0.818985 − 1.15294i
811.20 0.818985 + 1.15294i − 1.00000i −0.658528 + 1.88848i 1.00000 1.15294 − 0.818985i −0.864047 + 2.50068i −2.71662 + 0.787391i −1.00000 0.818985 + 1.15294i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 811.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.e 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.z.d yes 28
4.b odd 2 1 3360.2.z.d 28
7.b odd 2 1 840.2.z.c ✓ 28
8.b even 2 1 3360.2.z.c 28
8.d odd 2 1 840.2.z.c ✓ 28
28.d even 2 1 3360.2.z.c 28
56.e even 2 1 inner 840.2.z.d yes 28
56.h odd 2 1 3360.2.z.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.z.c ✓ 28 7.b odd 2 1
840.2.z.c ✓ 28 8.d odd 2 1
840.2.z.d yes 28 1.a even 1 1 trivial
840.2.z.d yes 28 56.e even 2 1 inner
3360.2.z.c 28 8.b even 2 1
3360.2.z.c 28 28.d even 2 1
3360.2.z.d 28 4.b odd 2 1
3360.2.z.d 28 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11}^{14} - 84 T_{11}^{12} - 40 T_{11}^{11} + 2516 T_{11}^{10} + 2512 T_{11}^{9} - 33200 T_{11}^{8} + \cdots + 16384 \) Copy content Toggle raw display
\( T_{13}^{14} + 4 T_{13}^{13} - 68 T_{13}^{12} - 176 T_{13}^{11} + 1812 T_{13}^{10} + 2160 T_{13}^{9} + \cdots - 32768 \) Copy content Toggle raw display