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Properties

Label 195.2.n.a
Level $195$
Weight $2$
Character orbit 195.n
Analytic conductor $1.557$
Analytic rank $0$
Dimension $48$
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] = [195,2,Mod(44,195)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(195, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("195.44");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.n (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: \(1.55708283941\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{6} - 8 q^{9} - 16 q^{15} - 24 q^{16} - 24 q^{19} + 4 q^{21} - 8 q^{24} - 8 q^{31} + 32 q^{34} + 16 q^{39} + 8 q^{40} + 20 q^{45} - 40 q^{46} - 24 q^{54} + 8 q^{55} + 76 q^{60} - 120 q^{61} + 32 q^{66} - 48 q^{70} + 104 q^{76} + 8 q^{79} - 48 q^{81} + 108 q^{84} + 56 q^{85} + 32 q^{91} - 192 q^{94} - 12 q^{96} - 60 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −1.72658 + 1.72658i 1.24653 − 1.20257i − 3.96217i −2.19956 − 0.402404i −0.0759004 + 4.22856i −0.495582 + 0.495582i 3.38784 + 3.38784i 0.107662 − 2.99807i 4.49251 − 3.10294i
44.2 −1.72658 + 1.72658i 1.24653 + 1.20257i − 3.96217i 0.402404 + 2.19956i −4.22856 + 0.0759004i 0.495582 − 0.495582i 3.38784 + 3.38784i 0.107662 + 2.99807i −4.49251 − 3.10294i
44.3 −1.69452 + 1.69452i −1.14037 − 1.30367i − 3.74279i 0.990671 − 2.00464i 4.14147 + 0.276713i −2.57529 + 2.57529i 2.95318 + 2.95318i −0.399109 + 2.97333i 1.71818 + 5.07561i
44.4 −1.69452 + 1.69452i −1.14037 + 1.30367i − 3.74279i 2.00464 − 0.990671i −0.276713 − 4.14147i 2.57529 − 2.57529i 2.95318 + 2.95318i −0.399109 − 2.97333i −1.71818 + 5.07561i
44.5 −1.26780 + 1.26780i 0.428519 − 1.67820i − 1.21464i 1.71852 + 1.43063i 1.58435 + 2.67091i 1.40105 − 1.40105i −0.995685 − 0.995685i −2.63274 − 1.43829i −3.99249 + 0.364987i
44.6 −1.26780 + 1.26780i 0.428519 + 1.67820i − 1.21464i −1.43063 − 1.71852i −2.67091 − 1.58435i −1.40105 + 1.40105i −0.995685 − 0.995685i −2.63274 + 1.43829i 3.99249 + 0.364987i
44.7 −0.800574 + 0.800574i 1.71512 − 0.241578i 0.718163i 0.0148595 − 2.23602i −1.17968 + 1.56648i 3.41858 − 3.41858i −2.17609 − 2.17609i 2.88328 − 0.828672i 1.77820 + 1.80199i
44.8 −0.800574 + 0.800574i 1.71512 + 0.241578i 0.718163i 2.23602 − 0.0148595i −1.56648 + 1.17968i −3.41858 + 3.41858i −2.17609 − 2.17609i 2.88328 + 0.828672i −1.77820 + 1.80199i
44.9 −0.590281 + 0.590281i −0.502466 − 1.65757i 1.30314i −2.08713 + 0.802426i 1.27503 + 0.681834i −0.315643 + 0.315643i −1.94978 − 1.94978i −2.49506 + 1.66574i 0.758337 − 1.70565i
44.10 −0.590281 + 0.590281i −0.502466 + 1.65757i 1.30314i −0.802426 + 2.08713i −0.681834 − 1.27503i 0.315643 − 0.315643i −1.94978 − 1.94978i −2.49506 − 1.66574i −0.758337 − 1.70565i
44.11 −0.225510 + 0.225510i −1.50598 − 0.855580i 1.89829i 0.293103 − 2.21677i 0.532556 − 0.146672i 0.934339 − 0.934339i −0.879103 − 0.879103i 1.53596 + 2.57698i 0.433807 + 0.566002i
44.12 −0.225510 + 0.225510i −1.50598 + 0.855580i 1.89829i 2.21677 − 0.293103i 0.146672 − 0.532556i −0.934339 + 0.934339i −0.879103 − 0.879103i 1.53596 − 2.57698i −0.433807 + 0.566002i
44.13 0.225510 − 0.225510i 1.50598 − 0.855580i 1.89829i −0.293103 + 2.21677i 0.146672 − 0.532556i 0.934339 − 0.934339i 0.879103 + 0.879103i 1.53596 − 2.57698i 0.433807 + 0.566002i
44.14 0.225510 − 0.225510i 1.50598 + 0.855580i 1.89829i −2.21677 + 0.293103i 0.532556 − 0.146672i −0.934339 + 0.934339i 0.879103 + 0.879103i 1.53596 + 2.57698i −0.433807 + 0.566002i
44.15 0.590281 − 0.590281i 0.502466 − 1.65757i 1.30314i 2.08713 − 0.802426i −0.681834 − 1.27503i −0.315643 + 0.315643i 1.94978 + 1.94978i −2.49506 − 1.66574i 0.758337 − 1.70565i
44.16 0.590281 − 0.590281i 0.502466 + 1.65757i 1.30314i 0.802426 − 2.08713i 1.27503 + 0.681834i 0.315643 − 0.315643i 1.94978 + 1.94978i −2.49506 + 1.66574i −0.758337 − 1.70565i
44.17 0.800574 − 0.800574i −1.71512 − 0.241578i 0.718163i −0.0148595 + 2.23602i −1.56648 + 1.17968i 3.41858 − 3.41858i 2.17609 + 2.17609i 2.88328 + 0.828672i 1.77820 + 1.80199i
44.18 0.800574 − 0.800574i −1.71512 + 0.241578i 0.718163i −2.23602 + 0.0148595i −1.17968 + 1.56648i −3.41858 + 3.41858i 2.17609 + 2.17609i 2.88328 − 0.828672i −1.77820 + 1.80199i
44.19 1.26780 − 1.26780i −0.428519 − 1.67820i − 1.21464i −1.71852 − 1.43063i −2.67091 − 1.58435i 1.40105 − 1.40105i 0.995685 + 0.995685i −2.63274 + 1.43829i −3.99249 + 0.364987i
44.20 1.26780 − 1.26780i −0.428519 + 1.67820i − 1.21464i 1.43063 + 1.71852i 1.58435 + 2.67091i −1.40105 + 1.40105i 0.995685 + 0.995685i −2.63274 − 1.43829i 3.99249 + 0.364987i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
13.d odd 4 1 inner
15.d odd 2 1 inner
39.f even 4 1 inner
65.g odd 4 1 inner
195.n even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.n.a ✓ 48
3.b odd 2 1 inner 195.2.n.a ✓ 48
5.b even 2 1 inner 195.2.n.a ✓ 48
5.c odd 4 2 975.2.o.q 48
13.d odd 4 1 inner 195.2.n.a ✓ 48
15.d odd 2 1 inner 195.2.n.a ✓ 48
15.e even 4 2 975.2.o.q 48
39.f even 4 1 inner 195.2.n.a ✓ 48
65.f even 4 1 975.2.o.q 48
65.g odd 4 1 inner 195.2.n.a ✓ 48
65.k even 4 1 975.2.o.q 48
195.j odd 4 1 975.2.o.q 48
195.n even 4 1 inner 195.2.n.a ✓ 48
195.u odd 4 1 975.2.o.q 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.n.a ✓ 48 1.a even 1 1 trivial
195.2.n.a ✓ 48 3.b odd 2 1 inner
195.2.n.a ✓ 48 5.b even 2 1 inner
195.2.n.a ✓ 48 13.d odd 4 1 inner
195.2.n.a ✓ 48 15.d odd 2 1 inner
195.2.n.a ✓ 48 39.f even 4 1 inner
195.2.n.a ✓ 48 65.g odd 4 1 inner
195.2.n.a ✓ 48 195.n even 4 1 inner
975.2.o.q 48 5.c odd 4 2
975.2.o.q 48 15.e even 4 2
975.2.o.q 48 65.f even 4 1
975.2.o.q 48 65.k even 4 1
975.2.o.q 48 195.j odd 4 1
975.2.o.q 48 195.u odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(195, [\chi])\).