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Properties

Label 605.2.e.b
Level $605$
Weight $2$
Character orbit 605.e
Analytic conductor $4.831$
Analytic rank $0$
Dimension $32$
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] = [605,2,Mod(362,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.362");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.e (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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 55)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{3} + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{3} + 8 q^{5} + 12 q^{12} - 36 q^{15} - 8 q^{16} - 64 q^{20} - 24 q^{23} + 16 q^{25} - 16 q^{27} - 8 q^{31} + 24 q^{36} + 32 q^{37} - 40 q^{38} + 60 q^{42} - 28 q^{45} - 28 q^{47} + 56 q^{48} + 116 q^{53} - 80 q^{56} - 80 q^{58} + 104 q^{60} - 8 q^{67} - 80 q^{70} + 24 q^{71} - 76 q^{75} + 60 q^{78} + 8 q^{80} + 8 q^{81} - 20 q^{82} - 40 q^{86} + 80 q^{91} + 52 q^{92} + 32 q^{93} + 92 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
362.1 −1.79273 + 1.79273i −1.41767 + 1.41767i − 4.42777i 1.69208 − 1.46180i − 5.08300i −1.27846 + 1.27846i 4.35233 + 4.35233i − 1.01958i −0.412838 + 5.65406i
362.2 −1.72010 + 1.72010i 0.422864 − 0.422864i − 3.91750i −0.759172 − 2.10325i 1.45474i −1.82555 + 1.82555i 3.29830 + 3.29830i 2.64237i 4.92366 + 2.31195i
362.3 −1.07485 + 1.07485i 0.563723 − 0.563723i − 0.310591i 2.23083 + 0.152980i 1.21183i 0.135390 − 0.135390i −1.81586 − 1.81586i 2.36443i −2.56223 + 2.23337i
362.4 −1.03659 + 1.03659i 0.588647 − 0.588647i − 0.149021i −2.18706 + 0.465567i 1.22037i 2.98069 − 2.98069i −1.91870 − 1.91870i 2.30699i 1.78448 − 2.74968i
362.5 −0.875624 + 0.875624i −1.79897 + 1.79897i 0.466567i 1.70992 + 1.44089i − 3.15044i −2.45241 + 2.45241i −2.15978 − 2.15978i − 3.47259i −2.75893 + 0.235572i
362.6 −0.738792 + 0.738792i 1.99135 − 1.99135i 0.908372i 0.742178 − 2.10931i 2.94239i −0.388787 + 0.388787i −2.14868 − 2.14868i − 4.93096i 1.01002 + 2.10665i
362.7 −0.407176 + 0.407176i −0.544295 + 0.544295i 1.66842i 0.752803 + 2.10554i − 0.443248i 0.843711 − 0.843711i −1.49369 − 1.49369i 2.40749i −1.16385 − 0.550801i
362.8 −0.345308 + 0.345308i −0.805651 + 0.805651i 1.76152i −2.18158 − 0.490620i − 0.556396i −2.06222 + 2.06222i −1.29889 − 1.29889i 1.70185i 0.922733 − 0.583903i
362.9 0.345308 − 0.345308i −0.805651 + 0.805651i 1.76152i −2.18158 − 0.490620i 0.556396i 2.06222 − 2.06222i 1.29889 + 1.29889i 1.70185i −0.922733 + 0.583903i
362.10 0.407176 − 0.407176i −0.544295 + 0.544295i 1.66842i 0.752803 + 2.10554i 0.443248i −0.843711 + 0.843711i 1.49369 + 1.49369i 2.40749i 1.16385 + 0.550801i
362.11 0.738792 − 0.738792i 1.99135 − 1.99135i 0.908372i 0.742178 − 2.10931i − 2.94239i 0.388787 − 0.388787i 2.14868 + 2.14868i − 4.93096i −1.01002 − 2.10665i
362.12 0.875624 − 0.875624i −1.79897 + 1.79897i 0.466567i 1.70992 + 1.44089i 3.15044i 2.45241 − 2.45241i 2.15978 + 2.15978i − 3.47259i 2.75893 − 0.235572i
362.13 1.03659 − 1.03659i 0.588647 − 0.588647i − 0.149021i −2.18706 + 0.465567i − 1.22037i −2.98069 + 2.98069i 1.91870 + 1.91870i 2.30699i −1.78448 + 2.74968i
362.14 1.07485 − 1.07485i 0.563723 − 0.563723i − 0.310591i 2.23083 + 0.152980i − 1.21183i −0.135390 + 0.135390i 1.81586 + 1.81586i 2.36443i 2.56223 − 2.23337i
362.15 1.72010 − 1.72010i 0.422864 − 0.422864i − 3.91750i −0.759172 − 2.10325i − 1.45474i 1.82555 − 1.82555i −3.29830 − 3.29830i 2.64237i −4.92366 − 2.31195i
362.16 1.79273 − 1.79273i −1.41767 + 1.41767i − 4.42777i 1.69208 − 1.46180i 5.08300i 1.27846 − 1.27846i −4.35233 − 4.35233i − 1.01958i 0.412838 − 5.65406i
483.1 −1.79273 − 1.79273i −1.41767 − 1.41767i 4.42777i 1.69208 + 1.46180i 5.08300i −1.27846 − 1.27846i 4.35233 − 4.35233i 1.01958i −0.412838 − 5.65406i
483.2 −1.72010 − 1.72010i 0.422864 + 0.422864i 3.91750i −0.759172 + 2.10325i − 1.45474i −1.82555 − 1.82555i 3.29830 − 3.29830i − 2.64237i 4.92366 − 2.31195i
483.3 −1.07485 − 1.07485i 0.563723 + 0.563723i 0.310591i 2.23083 − 0.152980i − 1.21183i 0.135390 + 0.135390i −1.81586 + 1.81586i − 2.36443i −2.56223 − 2.23337i
483.4 −1.03659 − 1.03659i 0.588647 + 0.588647i 0.149021i −2.18706 − 0.465567i − 1.22037i 2.98069 + 2.98069i −1.91870 + 1.91870i − 2.30699i 1.78448 + 2.74968i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 362.16
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 605.2.e.b 32
5.c odd 4 1 inner 605.2.e.b 32
11.b odd 2 1 inner 605.2.e.b 32
11.c even 5 1 55.2.l.a ✓ 32
11.c even 5 1 605.2.m.c 32
11.c even 5 1 605.2.m.d 32
11.c even 5 1 605.2.m.e 32
11.d odd 10 1 55.2.l.a ✓ 32
11.d odd 10 1 605.2.m.c 32
11.d odd 10 1 605.2.m.d 32
11.d odd 10 1 605.2.m.e 32
33.f even 10 1 495.2.bj.a 32
33.h odd 10 1 495.2.bj.a 32
44.g even 10 1 880.2.cm.a 32
44.h odd 10 1 880.2.cm.a 32
55.e even 4 1 inner 605.2.e.b 32
55.h odd 10 1 275.2.bm.b 32
55.j even 10 1 275.2.bm.b 32
55.k odd 20 1 55.2.l.a ✓ 32
55.k odd 20 1 275.2.bm.b 32
55.k odd 20 1 605.2.m.c 32
55.k odd 20 1 605.2.m.d 32
55.k odd 20 1 605.2.m.e 32
55.l even 20 1 55.2.l.a ✓ 32
55.l even 20 1 275.2.bm.b 32
55.l even 20 1 605.2.m.c 32
55.l even 20 1 605.2.m.d 32
55.l even 20 1 605.2.m.e 32
165.u odd 20 1 495.2.bj.a 32
165.v even 20 1 495.2.bj.a 32
220.v even 20 1 880.2.cm.a 32
220.w odd 20 1 880.2.cm.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.l.a ✓ 32 11.c even 5 1
55.2.l.a ✓ 32 11.d odd 10 1
55.2.l.a ✓ 32 55.k odd 20 1
55.2.l.a ✓ 32 55.l even 20 1
275.2.bm.b 32 55.h odd 10 1
275.2.bm.b 32 55.j even 10 1
275.2.bm.b 32 55.k odd 20 1
275.2.bm.b 32 55.l even 20 1
495.2.bj.a 32 33.f even 10 1
495.2.bj.a 32 33.h odd 10 1
495.2.bj.a 32 165.u odd 20 1
495.2.bj.a 32 165.v even 20 1
605.2.e.b 32 1.a even 1 1 trivial
605.2.e.b 32 5.c odd 4 1 inner
605.2.e.b 32 11.b odd 2 1 inner
605.2.e.b 32 55.e even 4 1 inner
605.2.m.c 32 11.c even 5 1
605.2.m.c 32 11.d odd 10 1
605.2.m.c 32 55.k odd 20 1
605.2.m.c 32 55.l even 20 1
605.2.m.d 32 11.c even 5 1
605.2.m.d 32 11.d odd 10 1
605.2.m.d 32 55.k odd 20 1
605.2.m.d 32 55.l even 20 1
605.2.m.e 32 11.c even 5 1
605.2.m.e 32 11.d odd 10 1
605.2.m.e 32 55.k odd 20 1
605.2.m.e 32 55.l even 20 1
880.2.cm.a 32 44.g even 10 1
880.2.cm.a 32 44.h odd 10 1
880.2.cm.a 32 220.v even 20 1
880.2.cm.a 32 220.w odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 90 T_{2}^{28} + 2555 T_{2}^{24} + 24860 T_{2}^{20} + 103725 T_{2}^{16} + 188800 T_{2}^{12} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\). Copy content Toggle raw display