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Properties

Label 770.2.m.f
Level $770$
Weight $2$
Character orbit 770.m
Analytic conductor $6.148$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [770,2,Mod(43,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.m (of order \(4\), degree \(2\), 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: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 4 q^{3} + 12 q^{11} + 4 q^{12} - 4 q^{15} - 36 q^{16} - 12 q^{20} - 12 q^{22} + 4 q^{23} + 12 q^{25} + 24 q^{26} + 56 q^{27} + 8 q^{31} - 44 q^{33} - 44 q^{36} - 28 q^{37} + 16 q^{38} + 4 q^{42} - 44 q^{45} + 12 q^{47} + 4 q^{48} + 28 q^{53} + 40 q^{55} - 36 q^{56} - 24 q^{58} + 12 q^{60} + 24 q^{66} + 12 q^{67} - 12 q^{70} - 112 q^{71} - 52 q^{75} - 12 q^{77} + 48 q^{78} + 4 q^{81} + 40 q^{82} + 32 q^{86} - 12 q^{88} + 24 q^{91} - 4 q^{92} - 80 q^{93} + 100 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −0.707107 − 0.707107i −2.29790 − 2.29790i 1.00000i 2.01966 − 0.959678i 3.24972i −0.707107 − 0.707107i 0.707107 − 0.707107i 7.56071i −2.10671 − 0.749519i
43.2 −0.707107 − 0.707107i −1.98542 − 1.98542i 1.00000i −1.96244 + 1.07183i 2.80781i −0.707107 − 0.707107i 0.707107 − 0.707107i 4.88379i 2.14556 + 0.629754i
43.3 −0.707107 − 0.707107i −1.12640 − 1.12640i 1.00000i −0.277104 + 2.21883i 1.59297i −0.707107 − 0.707107i 0.707107 − 0.707107i − 0.462431i 1.76489 − 1.37301i
43.4 −0.707107 − 0.707107i −0.888243 − 0.888243i 1.00000i 1.38828 + 1.75290i 1.25616i −0.707107 − 0.707107i 0.707107 − 0.707107i − 1.42205i 0.257823 − 2.22115i
43.5 −0.707107 − 0.707107i 0.129076 + 0.129076i 1.00000i −1.76845 − 1.36842i − 0.182542i −0.707107 − 0.707107i 0.707107 − 0.707107i − 2.96668i 0.282869 + 2.21810i
43.6 −0.707107 − 0.707107i 0.436316 + 0.436316i 1.00000i 0.925572 − 2.03551i − 0.617044i −0.707107 − 0.707107i 0.707107 − 0.707107i − 2.61926i −2.09380 + 0.784847i
43.7 −0.707107 − 0.707107i 1.47574 + 1.47574i 1.00000i −2.23579 − 0.0355240i − 2.08701i −0.707107 − 0.707107i 0.707107 − 0.707107i 1.35562i 1.55582 + 1.60606i
43.8 −0.707107 − 0.707107i 1.50262 + 1.50262i 1.00000i 2.23151 + 0.142694i − 2.12502i −0.707107 − 0.707107i 0.707107 − 0.707107i 1.51571i −1.47702 − 1.67882i
43.9 −0.707107 − 0.707107i 1.75422 + 1.75422i 1.00000i −0.321241 + 2.21287i − 2.48084i −0.707107 − 0.707107i 0.707107 − 0.707107i 3.15458i 1.79189 − 1.33759i
43.10 0.707107 + 0.707107i −2.29790 − 2.29790i 1.00000i 2.01966 − 0.959678i − 3.24972i 0.707107 + 0.707107i −0.707107 + 0.707107i 7.56071i 2.10671 + 0.749519i
43.11 0.707107 + 0.707107i −1.98542 − 1.98542i 1.00000i −1.96244 + 1.07183i − 2.80781i 0.707107 + 0.707107i −0.707107 + 0.707107i 4.88379i −2.14556 − 0.629754i
43.12 0.707107 + 0.707107i −1.12640 − 1.12640i 1.00000i −0.277104 + 2.21883i − 1.59297i 0.707107 + 0.707107i −0.707107 + 0.707107i − 0.462431i −1.76489 + 1.37301i
43.13 0.707107 + 0.707107i −0.888243 − 0.888243i 1.00000i 1.38828 + 1.75290i − 1.25616i 0.707107 + 0.707107i −0.707107 + 0.707107i − 1.42205i −0.257823 + 2.22115i
43.14 0.707107 + 0.707107i 0.129076 + 0.129076i 1.00000i −1.76845 − 1.36842i 0.182542i 0.707107 + 0.707107i −0.707107 + 0.707107i − 2.96668i −0.282869 − 2.21810i
43.15 0.707107 + 0.707107i 0.436316 + 0.436316i 1.00000i 0.925572 − 2.03551i 0.617044i 0.707107 + 0.707107i −0.707107 + 0.707107i − 2.61926i 2.09380 − 0.784847i
43.16 0.707107 + 0.707107i 1.47574 + 1.47574i 1.00000i −2.23579 − 0.0355240i 2.08701i 0.707107 + 0.707107i −0.707107 + 0.707107i 1.35562i −1.55582 − 1.60606i
43.17 0.707107 + 0.707107i 1.50262 + 1.50262i 1.00000i 2.23151 + 0.142694i 2.12502i 0.707107 + 0.707107i −0.707107 + 0.707107i 1.51571i 1.47702 + 1.67882i
43.18 0.707107 + 0.707107i 1.75422 + 1.75422i 1.00000i −0.321241 + 2.21287i 2.48084i 0.707107 + 0.707107i −0.707107 + 0.707107i 3.15458i −1.79189 + 1.33759i
197.1 −0.707107 + 0.707107i −2.29790 + 2.29790i − 1.00000i 2.01966 + 0.959678i − 3.24972i −0.707107 + 0.707107i 0.707107 + 0.707107i − 7.56071i −2.10671 + 0.749519i
197.2 −0.707107 + 0.707107i −1.98542 + 1.98542i − 1.00000i −1.96244 − 1.07183i − 2.80781i −0.707107 + 0.707107i 0.707107 + 0.707107i − 4.88379i 2.14556 − 0.629754i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.m.f ✓ 36
5.c odd 4 1 inner 770.2.m.f ✓ 36
11.b odd 2 1 inner 770.2.m.f ✓ 36
55.e even 4 1 inner 770.2.m.f ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.m.f ✓ 36 1.a even 1 1 trivial
770.2.m.f ✓ 36 5.c odd 4 1 inner
770.2.m.f ✓ 36 11.b odd 2 1 inner
770.2.m.f ✓ 36 55.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\):

\( T_{3}^{18} + 2 T_{3}^{17} + 2 T_{3}^{16} - 12 T_{3}^{15} + 104 T_{3}^{14} + 128 T_{3}^{13} + 120 T_{3}^{12} + \cdots + 512 \) Copy content Toggle raw display
\( T_{17}^{36} + 3008 T_{17}^{32} + 3388256 T_{17}^{28} + 1844692736 T_{17}^{24} + 517828534528 T_{17}^{20} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display