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Properties

Label 195.2.ba.a
Level $195$
Weight $2$
Character orbit 195.ba
Analytic conductor $1.557$
Analytic rank $0$
Dimension $24$
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] = [195,2,Mod(94,195)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(195, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("195.94");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.ba (of order \(6\), 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: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} + 4 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} + 4 q^{5} + 12 q^{9} - 4 q^{10} + 4 q^{11} + 24 q^{14} - 2 q^{15} + 16 q^{16} - 16 q^{19} - 16 q^{20} - 8 q^{21} - 16 q^{25} - 48 q^{26} - 12 q^{29} - 4 q^{30} + 8 q^{31} - 32 q^{34} + 10 q^{35} - 8 q^{36} + 8 q^{39} - 48 q^{40} - 40 q^{41} + 40 q^{44} + 2 q^{45} - 24 q^{46} - 16 q^{49} + 20 q^{50} - 24 q^{51} + 20 q^{55} - 24 q^{56} + 12 q^{59} + 48 q^{60} + 20 q^{61} + 48 q^{64} + 14 q^{65} - 56 q^{66} - 8 q^{69} - 56 q^{70} + 4 q^{71} - 12 q^{74} + 16 q^{75} + 8 q^{76} + 136 q^{79} - 4 q^{80} - 12 q^{81} - 16 q^{84} - 4 q^{85} + 48 q^{86} - 64 q^{89} - 8 q^{90} + 60 q^{91} - 48 q^{94} - 28 q^{95} + 40 q^{96} + 8 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1 −2.01317 + 1.16230i 0.866025 − 0.500000i 1.70191 − 2.94779i −0.174568 + 2.22924i −1.16230 + 2.01317i −0.473110 − 0.273150i 3.26331i 0.500000 − 0.866025i −2.23963 − 4.69075i
94.2 −1.85914 + 1.07337i −0.866025 + 0.500000i 1.30426 − 2.25904i −2.16557 + 0.557052i 1.07337 − 1.85914i −0.635729 − 0.367038i 1.30633i 0.500000 − 0.866025i 3.42817 − 3.36010i
94.3 −1.52669 + 0.881436i −0.866025 + 0.500000i 0.553860 − 0.959313i 1.52636 − 1.63408i 0.881436 − 1.52669i 1.92736 + 1.11276i − 1.57298i 0.500000 − 0.866025i −0.889946 + 3.84013i
94.4 −1.16430 + 0.672211i 0.866025 − 0.500000i −0.0962645 + 0.166735i −0.868136 − 2.06066i −0.672211 + 1.16430i −3.39681 − 1.96115i − 2.94768i 0.500000 − 0.866025i 2.39598 + 1.81567i
94.5 −0.729738 + 0.421315i 0.866025 − 0.500000i −0.644988 + 1.11715i 2.23540 − 0.0545741i −0.421315 + 0.729738i 0.347589 + 0.200681i − 2.77223i 0.500000 − 0.866025i −1.60827 + 0.981632i
94.6 −0.521384 + 0.301021i −0.866025 + 0.500000i −0.818772 + 1.41816i 0.446511 + 2.19103i 0.301021 − 0.521384i −3.08191 − 1.77934i − 2.18996i 0.500000 − 0.866025i −0.892352 − 1.00796i
94.7 0.521384 − 0.301021i 0.866025 − 0.500000i −0.818772 + 1.41816i 0.446511 − 2.19103i 0.301021 − 0.521384i 3.08191 + 1.77934i 2.18996i 0.500000 − 0.866025i −0.426744 − 1.27678i
94.8 0.729738 − 0.421315i −0.866025 + 0.500000i −0.644988 + 1.11715i 2.23540 + 0.0545741i −0.421315 + 0.729738i −0.347589 − 0.200681i 2.77223i 0.500000 − 0.866025i 1.65425 − 0.901983i
94.9 1.16430 − 0.672211i −0.866025 + 0.500000i −0.0962645 + 0.166735i −0.868136 + 2.06066i −0.672211 + 1.16430i 3.39681 + 1.96115i 2.94768i 0.500000 − 0.866025i 0.374428 + 2.98281i
94.10 1.52669 − 0.881436i 0.866025 − 0.500000i 0.553860 − 0.959313i 1.52636 + 1.63408i 0.881436 − 1.52669i −1.92736 − 1.11276i 1.57298i 0.500000 − 0.866025i 3.77062 + 1.14935i
94.11 1.85914 − 1.07337i 0.866025 − 0.500000i 1.30426 − 2.25904i −2.16557 − 0.557052i 1.07337 − 1.85914i 0.635729 + 0.367038i − 1.30633i 0.500000 − 0.866025i −4.62401 + 1.28883i
94.12 2.01317 − 1.16230i −0.866025 + 0.500000i 1.70191 − 2.94779i −0.174568 − 2.22924i −1.16230 + 2.01317i 0.473110 + 0.273150i − 3.26331i 0.500000 − 0.866025i −2.94250 − 4.28495i
139.1 −2.01317 − 1.16230i 0.866025 + 0.500000i 1.70191 + 2.94779i −0.174568 − 2.22924i −1.16230 − 2.01317i −0.473110 + 0.273150i − 3.26331i 0.500000 + 0.866025i −2.23963 + 4.69075i
139.2 −1.85914 − 1.07337i −0.866025 − 0.500000i 1.30426 + 2.25904i −2.16557 − 0.557052i 1.07337 + 1.85914i −0.635729 + 0.367038i − 1.30633i 0.500000 + 0.866025i 3.42817 + 3.36010i
139.3 −1.52669 − 0.881436i −0.866025 − 0.500000i 0.553860 + 0.959313i 1.52636 + 1.63408i 0.881436 + 1.52669i 1.92736 − 1.11276i 1.57298i 0.500000 + 0.866025i −0.889946 − 3.84013i
139.4 −1.16430 − 0.672211i 0.866025 + 0.500000i −0.0962645 − 0.166735i −0.868136 + 2.06066i −0.672211 − 1.16430i −3.39681 + 1.96115i 2.94768i 0.500000 + 0.866025i 2.39598 − 1.81567i
139.5 −0.729738 − 0.421315i 0.866025 + 0.500000i −0.644988 − 1.11715i 2.23540 + 0.0545741i −0.421315 − 0.729738i 0.347589 − 0.200681i 2.77223i 0.500000 + 0.866025i −1.60827 − 0.981632i
139.6 −0.521384 − 0.301021i −0.866025 − 0.500000i −0.818772 − 1.41816i 0.446511 − 2.19103i 0.301021 + 0.521384i −3.08191 + 1.77934i 2.18996i 0.500000 + 0.866025i −0.892352 + 1.00796i
139.7 0.521384 + 0.301021i 0.866025 + 0.500000i −0.818772 − 1.41816i 0.446511 + 2.19103i 0.301021 + 0.521384i 3.08191 − 1.77934i − 2.18996i 0.500000 + 0.866025i −0.426744 + 1.27678i
139.8 0.729738 + 0.421315i −0.866025 − 0.500000i −0.644988 − 1.11715i 2.23540 − 0.0545741i −0.421315 − 0.729738i −0.347589 + 0.200681i − 2.77223i 0.500000 + 0.866025i 1.65425 + 0.901983i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 94.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.ba.a ✓ 24
3.b odd 2 1 585.2.bs.b 24
5.b even 2 1 inner 195.2.ba.a ✓ 24
5.c odd 4 1 975.2.i.o 12
5.c odd 4 1 975.2.i.q 12
13.c even 3 1 inner 195.2.ba.a ✓ 24
15.d odd 2 1 585.2.bs.b 24
39.i odd 6 1 585.2.bs.b 24
65.n even 6 1 inner 195.2.ba.a ✓ 24
65.q odd 12 1 975.2.i.o 12
65.q odd 12 1 975.2.i.q 12
195.x odd 6 1 585.2.bs.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.ba.a ✓ 24 1.a even 1 1 trivial
195.2.ba.a ✓ 24 5.b even 2 1 inner
195.2.ba.a ✓ 24 13.c even 3 1 inner
195.2.ba.a ✓ 24 65.n even 6 1 inner
585.2.bs.b 24 3.b odd 2 1
585.2.bs.b 24 15.d odd 2 1
585.2.bs.b 24 39.i odd 6 1
585.2.bs.b 24 195.x odd 6 1
975.2.i.o 12 5.c odd 4 1
975.2.i.o 12 65.q odd 12 1
975.2.i.q 12 5.c odd 4 1
975.2.i.q 12 65.q odd 12 1

Hecke kernels

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