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Properties

Label 360.2.br.e
Level $360$
Weight $2$
Character orbit 360.br
Analytic conductor $2.875$
Analytic rank $0$
Dimension $256$
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] = [360,2,Mod(77,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 10, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.br (of order \(12\), degree \(4\), 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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(64\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q + 6 q^{2} - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q + 6 q^{2} - 8 q^{6} - 10 q^{12} + 28 q^{15} + 12 q^{16} - 28 q^{18} - 54 q^{20} + 14 q^{22} - 28 q^{25} - 32 q^{28} + 14 q^{30} - 32 q^{31} - 114 q^{32} + 4 q^{33} - 40 q^{36} - 30 q^{38} + 46 q^{40} + 24 q^{41} - 10 q^{42} - 16 q^{46} + 24 q^{47} - 2 q^{48} + 78 q^{50} + 38 q^{52} - 8 q^{55} - 96 q^{56} - 80 q^{57} - 18 q^{58} - 2 q^{60} - 144 q^{63} - 84 q^{65} - 4 q^{66} - 30 q^{68} - 30 q^{70} - 86 q^{72} + 64 q^{73} + 16 q^{76} - 82 q^{78} + 72 q^{81} - 64 q^{82} + 48 q^{86} - 4 q^{87} + 38 q^{88} + 78 q^{90} - 108 q^{92} - 24 q^{95} - 116 q^{96} + 36 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −1.41351 − 0.0447368i 0.131650 + 1.72704i 1.99600 + 0.126472i −0.531426 − 2.17200i −0.108826 − 2.44707i 0.193986 − 0.723964i −2.81570 − 0.268063i −2.96534 + 0.454729i 0.654006 + 3.09391i
77.2 −1.40492 + 0.161860i 1.19421 − 1.25454i 1.94760 − 0.454802i −1.72814 + 1.41899i −1.47470 + 1.95582i 0.183802 − 0.685960i −2.66261 + 0.954200i −0.147746 − 2.99636i 2.19822 − 2.27328i
77.3 −1.40115 + 0.191780i 0.925695 + 1.46393i 1.92644 − 0.537425i −0.274171 + 2.21920i −1.57779 − 1.87365i 0.362508 − 1.35290i −2.59616 + 1.12247i −1.28618 + 2.71030i −0.0414427 − 3.16201i
77.4 −1.38126 − 0.303534i 1.72383 + 0.168545i 1.81573 + 0.838517i −0.461790 − 2.18786i −2.32989 − 0.756046i −0.976425 + 3.64407i −2.25347 − 1.70934i 2.94318 + 0.581088i −0.0262424 + 3.16217i
77.5 −1.38022 + 0.308205i −1.08549 − 1.34971i 1.81002 − 0.850781i −2.00950 − 0.980772i 1.91421 + 1.52834i −0.563716 + 2.10382i −2.23601 + 1.73212i −0.643411 + 2.93019i 3.07583 + 0.734345i
77.6 −1.37718 − 0.321532i 1.63984 − 0.557615i 1.79323 + 0.885613i 2.22426 + 0.229476i −2.43764 + 0.240675i 0.712972 − 2.66085i −2.18485 − 1.79623i 2.37813 − 1.82880i −2.98942 − 1.03120i
77.7 −1.35344 − 0.410134i −1.63984 + 0.557615i 1.66358 + 1.11018i −2.22426 − 0.229476i 2.44811 − 0.0821442i 0.712972 − 2.66085i −1.79623 − 2.18485i 2.37813 − 1.82880i 2.91628 + 1.22283i
77.8 −1.34797 − 0.427759i −1.72383 − 0.168545i 1.63404 + 1.15321i 0.461790 + 2.18786i 2.25157 + 0.964579i −0.976425 + 3.64407i −1.70934 − 2.25347i 2.94318 + 0.581088i 0.313401 − 3.14671i
77.9 −1.24650 − 0.668010i −0.131650 − 1.72704i 1.10753 + 1.66535i 0.531426 + 2.17200i −0.989578 + 2.24070i 0.193986 − 0.723964i −0.268063 − 2.81570i −2.96534 + 0.454729i 0.788494 − 3.06240i
77.10 −1.15671 + 0.813649i 0.674078 − 1.59550i 0.675951 − 1.88231i 0.576180 − 2.16056i 0.518465 + 2.39399i 0.331473 − 1.23707i 0.749662 + 2.72727i −2.09124 − 2.15098i 1.09146 + 2.96795i
77.11 −1.14969 + 0.823530i −1.51198 + 0.844934i 0.643596 − 1.89362i −0.944749 + 2.02668i 1.04249 − 2.21658i 0.661416 − 2.46844i 0.819513 + 2.70710i 1.57217 − 2.55505i −0.582864 − 3.10810i
77.12 −1.13577 − 0.842635i −1.19421 + 1.25454i 0.579931 + 1.91407i 1.72814 − 1.41899i 2.41346 − 0.418586i 0.183802 − 0.685960i 0.954200 − 2.66261i −0.147746 − 2.99636i −3.15845 + 0.155449i
77.13 −1.11797 + 0.866111i 1.51581 − 0.838041i 0.499705 − 1.93657i 1.47125 + 1.68387i −0.968794 + 2.24976i −1.06946 + 3.99129i 1.11863 + 2.59782i 1.59537 − 2.54063i −3.10323 − 0.608245i
77.14 −1.11754 − 0.866661i −0.925695 − 1.46393i 0.497797 + 1.93706i 0.274171 − 2.21920i −0.234228 + 2.43827i 0.362508 − 1.35290i 1.12247 − 2.59616i −1.28618 + 2.71030i −2.22969 + 2.24243i
77.15 −1.04120 − 0.957024i 1.08549 + 1.34971i 0.168211 + 1.99291i 2.00950 + 0.980772i 0.161481 − 2.44416i −0.563716 + 2.10382i 1.73212 − 2.23601i −0.643411 + 2.93019i −1.15368 − 2.94432i
77.16 −0.967822 + 1.03117i 1.22306 + 1.22643i −0.126641 − 1.99599i −2.20381 + 0.378455i −2.44836 + 0.0742232i −1.11197 + 4.14991i 2.18078 + 1.80117i −0.00825140 + 2.99999i 1.74264 − 2.63879i
77.17 −0.922014 + 1.07233i −1.71827 + 0.218017i −0.299780 − 1.97741i 0.420988 − 2.19608i 1.35049 − 2.04357i −0.688838 + 2.57078i 2.39683 + 1.50173i 2.90494 − 0.749227i 1.96676 + 2.47626i
77.18 −0.890307 + 1.09880i −0.176815 + 1.72300i −0.414706 − 1.95653i −1.48043 − 1.67581i −1.73581 − 1.72829i 0.827385 − 3.08784i 2.51905 + 1.28624i −2.93747 − 0.609306i 3.15941 − 0.134701i
77.19 −0.745090 + 1.20202i −0.272060 − 1.71055i −0.889682 − 1.79122i −1.63914 + 1.52092i 2.25882 + 0.947494i 0.619613 − 2.31243i 2.81597 + 0.265208i −2.85197 + 0.930744i −0.606863 − 3.10350i
77.20 −0.613736 + 1.27410i −0.853956 − 1.50690i −1.24666 − 1.56392i 2.16728 + 0.550360i 2.44405 − 0.163184i −0.556805 + 2.07802i 2.75771 − 0.628530i −1.54152 + 2.57366i −2.03135 + 2.42355i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.b even 2 1 inner
9.d odd 6 1 inner
40.i odd 4 1 inner
45.l even 12 1 inner
72.j odd 6 1 inner
360.br even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.br.e ✓ 256
5.c odd 4 1 inner 360.2.br.e ✓ 256
8.b even 2 1 inner 360.2.br.e ✓ 256
9.d odd 6 1 inner 360.2.br.e ✓ 256
40.i odd 4 1 inner 360.2.br.e ✓ 256
45.l even 12 1 inner 360.2.br.e ✓ 256
72.j odd 6 1 inner 360.2.br.e ✓ 256
360.br even 12 1 inner 360.2.br.e ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.br.e ✓ 256 1.a even 1 1 trivial
360.2.br.e ✓ 256 5.c odd 4 1 inner
360.2.br.e ✓ 256 8.b even 2 1 inner
360.2.br.e ✓ 256 9.d odd 6 1 inner
360.2.br.e ✓ 256 40.i odd 4 1 inner
360.2.br.e ✓ 256 45.l even 12 1 inner
360.2.br.e ✓ 256 72.j odd 6 1 inner
360.2.br.e ✓ 256 360.br even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\):

\( T_{7}^{128} - 24 T_{7}^{125} - 1636 T_{7}^{124} + 48 T_{7}^{123} + 288 T_{7}^{122} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
\( T_{11}^{128} + 382 T_{11}^{126} + 77185 T_{11}^{124} + 10759586 T_{11}^{122} + 1150747136 T_{11}^{120} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display