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Properties

Label 260.2.p.d
Level $260$
Weight $2$
Character orbit 260.p
Analytic conductor $2.076$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [260,2,Mod(103,260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(260, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("260.103");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 260.p (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: \(2.07611045255\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 24 q^{12} - 4 q^{16} - 40 q^{17} - 36 q^{22} + 44 q^{26} + 24 q^{30} + 28 q^{36} + 16 q^{38} - 44 q^{40} + 8 q^{42} - 44 q^{48} + 56 q^{52} - 48 q^{53} - 64 q^{56} + 80 q^{61} + 20 q^{62} - 72 q^{65} - 24 q^{66} - 76 q^{68} - 112 q^{77} - 20 q^{78} + 80 q^{81} + 52 q^{82} - 152 q^{88} - 64 q^{90} + 56 q^{92}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −1.41384 − 0.0324895i −2.32949 + 2.32949i 1.99789 + 0.0918700i 2.23599 + 0.0191440i 3.36921 − 3.21784i 1.97121 − 1.97121i −2.82171 − 0.194800i − 7.85303i −3.16071 − 0.0997126i
103.2 −1.36804 − 0.358430i 1.80624 − 1.80624i 1.74306 + 0.980692i 1.12997 − 1.92955i −3.11841 + 1.82359i −0.163803 + 0.163803i −2.03306 − 1.96639i − 3.52499i −2.23745 + 2.23468i
103.3 −1.35124 + 0.417327i 0.561614 − 0.561614i 1.65168 − 1.12781i −2.09189 + 0.789927i −0.524497 + 0.993250i 1.48711 − 1.48711i −1.76114 + 2.21323i 2.36918i 2.49698 − 1.94038i
103.4 −1.31170 + 0.528615i −0.834468 + 0.834468i 1.44113 − 1.38677i −0.462357 − 2.18774i 0.653463 − 1.53569i −1.21162 + 1.21162i −1.15727 + 2.58084i 1.60733i 1.76295 + 2.62526i
103.5 −1.27985 + 0.601658i 1.55255 − 1.55255i 1.27602 − 1.54006i 1.95736 + 1.08109i −1.05293 + 2.92114i 3.02953 − 3.02953i −0.706517 + 2.73876i − 1.82084i −3.15556 − 0.205972i
103.6 −1.23237 − 0.693735i −0.828304 + 0.828304i 1.03746 + 1.70987i −1.31937 + 1.80534i 1.59540 − 0.446152i 2.94084 − 2.94084i −0.0923349 − 2.82692i 1.62783i 2.87838 − 1.30955i
103.7 −1.22720 − 0.702829i −1.63423 + 1.63423i 1.01206 + 1.72503i −1.85457 − 1.24923i 3.15411 − 0.856948i −1.80181 + 1.80181i −0.0296052 − 2.82827i − 2.34139i 1.39793 + 2.83651i
103.8 −1.10269 + 0.885485i −1.59124 + 1.59124i 0.431834 − 1.95282i −0.349775 + 2.20854i 0.345619 − 3.16365i −1.15539 + 1.15539i 1.25302 + 2.53573i − 2.06407i −1.56994 − 2.74505i
103.9 −0.885485 + 1.10269i 1.59124 − 1.59124i −0.431834 − 1.95282i 0.349775 − 2.20854i 0.345619 + 3.16365i −1.15539 + 1.15539i 2.53573 + 1.25302i − 2.06407i 2.12561 + 2.34132i
103.10 −0.702829 − 1.22720i 1.63423 − 1.63423i −1.01206 + 1.72503i −1.85457 − 1.24923i −3.15411 − 0.856948i 1.80181 − 1.80181i 2.82827 + 0.0296052i − 2.34139i −0.229622 + 3.15393i
103.11 −0.693735 − 1.23237i 0.828304 − 0.828304i −1.03746 + 1.70987i −1.31937 + 1.80534i −1.59540 − 0.446152i −2.94084 + 2.94084i 2.82692 + 0.0923349i 1.62783i 3.14014 + 0.373519i
103.12 −0.601658 + 1.27985i −1.55255 + 1.55255i −1.27602 − 1.54006i −1.95736 − 1.08109i −1.05293 − 2.92114i 3.02953 − 3.02953i 2.73876 − 0.706517i − 1.82084i 2.56129 − 1.85467i
103.13 −0.528615 + 1.31170i 0.834468 − 0.834468i −1.44113 − 1.38677i 0.462357 + 2.18774i 0.653463 + 1.53569i −1.21162 + 1.21162i 2.58084 − 1.15727i 1.60733i −3.11408 − 0.549999i
103.14 −0.417327 + 1.35124i −0.561614 + 0.561614i −1.65168 − 1.12781i 2.09189 − 0.789927i −0.524497 − 0.993250i 1.48711 − 1.48711i 2.21323 − 1.76114i 2.36918i 0.194375 + 3.15630i
103.15 −0.358430 − 1.36804i −1.80624 + 1.80624i −1.74306 + 0.980692i 1.12997 − 1.92955i 3.11841 + 1.82359i 0.163803 − 0.163803i 1.96639 + 2.03306i − 3.52499i −3.04471 − 0.854237i
103.16 −0.0324895 − 1.41384i 2.32949 − 2.32949i −1.99789 + 0.0918700i 2.23599 + 0.0191440i −3.36921 − 3.21784i −1.97121 + 1.97121i 0.194800 + 2.82171i − 7.85303i −0.0455796 − 3.16195i
103.17 0.0324895 + 1.41384i 2.32949 − 2.32949i −1.99789 + 0.0918700i −2.23599 − 0.0191440i 3.36921 + 3.21784i 1.97121 − 1.97121i −0.194800 − 2.82171i − 7.85303i −0.0455796 − 3.16195i
103.18 0.358430 + 1.36804i −1.80624 + 1.80624i −1.74306 + 0.980692i −1.12997 + 1.92955i −3.11841 − 1.82359i −0.163803 + 0.163803i −1.96639 − 2.03306i − 3.52499i −3.04471 − 0.854237i
103.19 0.417327 − 1.35124i −0.561614 + 0.561614i −1.65168 − 1.12781i −2.09189 + 0.789927i 0.524497 + 0.993250i −1.48711 + 1.48711i −2.21323 + 1.76114i 2.36918i 0.194375 + 3.15630i
103.20 0.528615 − 1.31170i 0.834468 − 0.834468i −1.44113 − 1.38677i −0.462357 − 2.18774i −0.653463 − 1.53569i 1.21162 − 1.21162i −2.58084 + 1.15727i 1.60733i −3.11408 − 0.549999i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
13.b even 2 1 inner
20.e even 4 1 inner
52.b odd 2 1 inner
65.h odd 4 1 inner
260.p even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.p.d ✓ 64
4.b odd 2 1 inner 260.2.p.d ✓ 64
5.c odd 4 1 inner 260.2.p.d ✓ 64
13.b even 2 1 inner 260.2.p.d ✓ 64
20.e even 4 1 inner 260.2.p.d ✓ 64
52.b odd 2 1 inner 260.2.p.d ✓ 64
65.h odd 4 1 inner 260.2.p.d ✓ 64
260.p even 4 1 inner 260.2.p.d ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.p.d ✓ 64 1.a even 1 1 trivial
260.2.p.d ✓ 64 4.b odd 2 1 inner
260.2.p.d ✓ 64 5.c odd 4 1 inner
260.2.p.d ✓ 64 13.b even 2 1 inner
260.2.p.d ✓ 64 20.e even 4 1 inner
260.2.p.d ✓ 64 52.b odd 2 1 inner
260.2.p.d ✓ 64 65.h odd 4 1 inner
260.2.p.d ✓ 64 260.p 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 \(S_{2}^{\mathrm{new}}(260, [\chi])\):

\( T_{3}^{32} + 242 T_{3}^{28} + 20429 T_{3}^{24} + 807668 T_{3}^{20} + 15868452 T_{3}^{16} + \cdots + 123921424 \) Copy content Toggle raw display
\( T_{7}^{32} + 774 T_{7}^{28} + 195049 T_{7}^{24} + 18186004 T_{7}^{20} + 742170832 T_{7}^{16} + \cdots + 888516864 \) Copy content Toggle raw display
\( T_{37}^{32} + 8258 T_{37}^{28} + 11169457 T_{37}^{24} + 5348990192 T_{37}^{20} + 1048121669152 T_{37}^{16} + \cdots + 98\!\cdots\!56 \) Copy content Toggle raw display