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Properties

Label 840.2.e.d
Level $840$
Weight $2$
Character orbit 840.e
Analytic conductor $6.707$
Analytic rank $0$
Dimension $44$
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(491,840)]
 
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, 1, 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("840.491");
 
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.e (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: \(44\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{2} + 4 q^{3} - 2 q^{4} - 44 q^{5} - 6 q^{6} - 22 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{2} + 4 q^{3} - 2 q^{4} - 44 q^{5} - 6 q^{6} - 22 q^{8} + 4 q^{9} - 2 q^{10} + 6 q^{12} + 4 q^{14} - 4 q^{15} + 22 q^{16} + 2 q^{18} + 16 q^{19} + 2 q^{20} + 16 q^{23} + 10 q^{24} + 44 q^{25} - 16 q^{26} + 4 q^{27} + 16 q^{29} + 6 q^{30} - 18 q^{32} + 24 q^{33} + 20 q^{34} - 46 q^{36} + 8 q^{38} - 16 q^{39} + 22 q^{40} - 4 q^{42} + 24 q^{43} + 32 q^{44} - 4 q^{45} - 24 q^{47} + 18 q^{48} - 44 q^{49} + 2 q^{50} - 20 q^{52} - 32 q^{53} + 2 q^{54} + 8 q^{56} - 4 q^{58} - 6 q^{60} + 32 q^{62} - 50 q^{64} - 8 q^{66} + 24 q^{67} - 28 q^{68} - 8 q^{69} - 4 q^{70} + 8 q^{71} + 2 q^{72} + 8 q^{73} - 48 q^{74} + 4 q^{75} + 12 q^{76} + 8 q^{77} - 84 q^{78} - 22 q^{80} + 20 q^{81} - 8 q^{82} + 4 q^{84} - 4 q^{86} - 16 q^{87} - 28 q^{88} - 2 q^{90} - 16 q^{91} + 76 q^{92} + 16 q^{93} - 16 q^{95} + 38 q^{96} + 24 q^{97} - 2 q^{98} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
491.1 −1.40537 − 0.157878i 0.663478 + 1.59994i 1.95015 + 0.443756i −1.00000 −0.679840 − 2.35326i 1.00000i −2.67063 − 0.931529i −2.11959 + 2.12305i 1.40537 + 0.157878i
491.2 −1.40537 + 0.157878i 0.663478 − 1.59994i 1.95015 − 0.443756i −1.00000 −0.679840 + 2.35326i − 1.00000i −2.67063 + 0.931529i −2.11959 − 2.12305i 1.40537 − 0.157878i
491.3 −1.39982 − 0.201241i −1.38582 + 1.03899i 1.91900 + 0.563404i −1.00000 2.14899 − 1.17551i 1.00000i −2.57288 − 1.17485i 0.841016 − 2.87970i 1.39982 + 0.201241i
491.4 −1.39982 + 0.201241i −1.38582 − 1.03899i 1.91900 − 0.563404i −1.00000 2.14899 + 1.17551i − 1.00000i −2.57288 + 1.17485i 0.841016 + 2.87970i 1.39982 − 0.201241i
491.5 −1.34976 − 0.422085i −0.393935 − 1.68666i 1.64369 + 1.13942i −1.00000 −0.180195 + 2.44285i 1.00000i −1.73765 − 2.23172i −2.68963 + 1.32887i 1.34976 + 0.422085i
491.6 −1.34976 + 0.422085i −0.393935 + 1.68666i 1.64369 − 1.13942i −1.00000 −0.180195 − 2.44285i − 1.00000i −1.73765 + 2.23172i −2.68963 − 1.32887i 1.34976 − 0.422085i
491.7 −1.34484 − 0.437500i 1.72227 − 0.183849i 1.61719 + 1.17673i −1.00000 −2.39661 − 0.506243i − 1.00000i −1.66004 − 2.29004i 2.93240 − 0.633274i 1.34484 + 0.437500i
491.8 −1.34484 + 0.437500i 1.72227 + 0.183849i 1.61719 − 1.17673i −1.00000 −2.39661 + 0.506243i 1.00000i −1.66004 + 2.29004i 2.93240 + 0.633274i 1.34484 − 0.437500i
491.9 −1.17505 − 0.786923i 1.39892 + 1.02128i 0.761505 + 1.84935i −1.00000 −0.840138 − 2.30091i − 1.00000i 0.560490 − 2.77234i 0.913962 + 2.85739i 1.17505 + 0.786923i
491.10 −1.17505 + 0.786923i 1.39892 − 1.02128i 0.761505 − 1.84935i −1.00000 −0.840138 + 2.30091i 1.00000i 0.560490 + 2.77234i 0.913962 − 2.85739i 1.17505 − 0.786923i
491.11 −1.15075 − 0.822050i 1.20864 − 1.24064i 0.648469 + 1.89195i −1.00000 −2.41071 + 0.434112i 1.00000i 0.809051 − 2.71025i −0.0783833 − 2.99898i 1.15075 + 0.822050i
491.12 −1.15075 + 0.822050i 1.20864 + 1.24064i 0.648469 − 1.89195i −1.00000 −2.41071 − 0.434112i − 1.00000i 0.809051 + 2.71025i −0.0783833 + 2.99898i 1.15075 − 0.822050i
491.13 −0.537216 − 1.30820i 1.51567 + 0.838298i −1.42280 + 1.40558i −1.00000 0.282422 − 2.43315i 1.00000i 2.60313 + 1.10621i 1.59451 + 2.54117i 0.537216 + 1.30820i
491.14 −0.537216 + 1.30820i 1.51567 − 0.838298i −1.42280 − 1.40558i −1.00000 0.282422 + 2.43315i − 1.00000i 2.60313 − 1.10621i 1.59451 − 2.54117i 0.537216 − 1.30820i
491.15 −0.369265 − 1.36515i 0.467013 + 1.66790i −1.72729 + 1.00821i −1.00000 2.10449 − 1.25344i − 1.00000i 2.01418 + 1.98572i −2.56380 + 1.55787i 0.369265 + 1.36515i
491.16 −0.369265 + 1.36515i 0.467013 − 1.66790i −1.72729 − 1.00821i −1.00000 2.10449 + 1.25344i 1.00000i 2.01418 − 1.98572i −2.56380 − 1.55787i 0.369265 − 1.36515i
491.17 −0.258552 − 1.39038i −1.73079 − 0.0660561i −1.86630 + 0.718970i −1.00000 0.355656 + 2.42353i 1.00000i 1.48218 + 2.40897i 2.99127 + 0.228659i 0.258552 + 1.39038i
491.18 −0.258552 + 1.39038i −1.73079 + 0.0660561i −1.86630 − 0.718970i −1.00000 0.355656 − 2.42353i − 1.00000i 1.48218 − 2.40897i 2.99127 − 0.228659i 0.258552 − 1.39038i
491.19 −0.0543167 − 1.41317i −1.27193 − 1.17567i −1.99410 + 0.153518i −1.00000 −1.59233 + 1.86131i − 1.00000i 0.325259 + 2.80966i 0.235608 + 2.99073i 0.0543167 + 1.41317i
491.20 −0.0543167 + 1.41317i −1.27193 + 1.17567i −1.99410 − 0.153518i −1.00000 −1.59233 − 1.86131i 1.00000i 0.325259 − 2.80966i 0.235608 − 2.99073i 0.0543167 − 1.41317i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 491.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f 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.e.d yes 44
3.b odd 2 1 840.2.e.c ✓ 44
4.b odd 2 1 3360.2.e.c 44
8.b even 2 1 3360.2.e.d 44
8.d odd 2 1 840.2.e.c ✓ 44
12.b even 2 1 3360.2.e.d 44
24.f even 2 1 inner 840.2.e.d yes 44
24.h odd 2 1 3360.2.e.c 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.e.c ✓ 44 3.b odd 2 1
840.2.e.c ✓ 44 8.d odd 2 1
840.2.e.d yes 44 1.a even 1 1 trivial
840.2.e.d yes 44 24.f even 2 1 inner
3360.2.e.c 44 4.b odd 2 1
3360.2.e.c 44 24.h odd 2 1
3360.2.e.d 44 8.b even 2 1
3360.2.e.d 44 12.b even 2 1

Hecke kernels

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

\( T_{11}^{44} + 264 T_{11}^{42} + 31844 T_{11}^{40} + 2327928 T_{11}^{38} + 115382518 T_{11}^{36} + \cdots + 202012780134400 \) Copy content Toggle raw display
\( T_{23}^{22} - 8 T_{23}^{21} - 216 T_{23}^{20} + 1776 T_{23}^{19} + 18240 T_{23}^{18} - 153728 T_{23}^{17} + \cdots - 527433728 \) Copy content Toggle raw display