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Properties

Label 195.2.k.a
Level $195$
Weight $2$
Character orbit 195.k
Analytic conductor $1.557$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [195,2,Mod(112,195)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(195, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("195.112"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.55708283941\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 28 q^{4} + 8 q^{5} - 8 q^{11} + 8 q^{12} - 12 q^{13} + 4 q^{15} + 28 q^{16} - 28 q^{17} - 4 q^{18} + 8 q^{21} - 32 q^{22} + 8 q^{23} - 4 q^{25} - 16 q^{31} + 28 q^{34} + 32 q^{37} + 8 q^{39} - 48 q^{40}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
112.1 − 2.73623i −0.707107 + 0.707107i −5.48694 −1.07675 + 1.95975i 1.93480 + 1.93480i −2.18253 9.54105i − 1.00000i 5.36231 + 2.94624i
112.2 − 2.56480i 0.707107 − 0.707107i −4.57821 1.71342 − 1.43673i −1.81359 − 1.81359i −1.73944 6.61261i − 1.00000i −3.68494 − 4.39458i
112.3 − 1.94332i −0.707107 + 0.707107i −1.77647 2.16194 − 0.570984i 1.37413 + 1.37413i 2.33552 − 0.434380i − 1.00000i −1.10960 − 4.20133i
112.4 − 1.67997i 0.707107 − 0.707107i −0.822299 1.30099 + 1.81863i −1.18792 − 1.18792i 2.35789 − 1.97850i − 1.00000i 3.05525 − 2.18562i
112.5 − 1.38150i −0.707107 + 0.707107i 0.0914500 −0.843366 − 2.07093i 0.976870 + 0.976870i −3.94352 − 2.88934i − 1.00000i −2.86099 + 1.16511i
112.6 − 0.792814i 0.707107 − 0.707107i 1.37145 −2.23324 + 0.112444i −0.560604 − 0.560604i 1.67222 − 2.67293i − 1.00000i 0.0891476 + 1.77054i
112.7 − 0.470635i 0.707107 − 0.707107i 1.77850 0.0206478 − 2.23597i −0.332789 − 0.332789i −1.17941 − 1.77829i − 1.00000i −1.05233 − 0.00971759i
112.8 − 0.147953i −0.707107 + 0.707107i 1.97811 2.11069 + 0.738223i 0.104618 + 0.104618i −1.74764 − 0.588572i − 1.00000i 0.109222 − 0.312283i
112.9 0.709689i 0.707107 − 0.707107i 1.49634 2.19848 + 0.408252i 0.501826 + 0.501826i −3.37036 2.48132i − 1.00000i −0.289732 + 1.56024i
112.10 0.750656i −0.707107 + 0.707107i 1.43651 −0.411007 − 2.19797i −0.530794 − 0.530794i 3.56892 2.57964i − 1.00000i 1.64992 − 0.308525i
112.11 1.58074i 0.707107 − 0.707107i −0.498726 −1.18571 + 1.89581i 1.11775 + 1.11775i 0.974287 2.37312i − 1.00000i −2.99677 − 1.87430i
112.12 1.97160i −0.707107 + 0.707107i −1.88719 0.644677 + 2.14112i −1.39413 − 1.39413i −0.616758 0.222418i − 1.00000i −4.22142 + 1.27104i
112.13 2.21780i 0.707107 − 0.707107i −2.91862 1.59963 − 1.56243i 1.56822 + 1.56822i 4.11325 − 2.03732i − 1.00000i 3.46515 + 3.54765i
112.14 2.48675i −0.707107 + 0.707107i −4.18390 −2.00040 − 0.999208i −1.75839 − 1.75839i −0.242414 − 5.43081i − 1.00000i 2.48478 − 4.97448i
148.1 − 2.48675i −0.707107 − 0.707107i −4.18390 −2.00040 + 0.999208i −1.75839 + 1.75839i −0.242414 5.43081i 1.00000i 2.48478 + 4.97448i
148.2 − 2.21780i 0.707107 + 0.707107i −2.91862 1.59963 + 1.56243i 1.56822 − 1.56822i 4.11325 2.03732i 1.00000i 3.46515 − 3.54765i
148.3 − 1.97160i −0.707107 − 0.707107i −1.88719 0.644677 − 2.14112i −1.39413 + 1.39413i −0.616758 − 0.222418i 1.00000i −4.22142 − 1.27104i
148.4 − 1.58074i 0.707107 + 0.707107i −0.498726 −1.18571 − 1.89581i 1.11775 − 1.11775i 0.974287 − 2.37312i 1.00000i −2.99677 + 1.87430i
148.5 − 0.750656i −0.707107 − 0.707107i 1.43651 −0.411007 + 2.19797i −0.530794 + 0.530794i 3.56892 − 2.57964i 1.00000i 1.64992 + 0.308525i
148.6 − 0.709689i 0.707107 + 0.707107i 1.49634 2.19848 − 0.408252i 0.501826 − 0.501826i −3.37036 − 2.48132i 1.00000i −0.289732 − 1.56024i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 112.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f 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.k.a ✓ 28
3.b odd 2 1 585.2.n.g 28
5.b even 2 1 975.2.k.d 28
5.c odd 4 1 195.2.t.a yes 28
5.c odd 4 1 975.2.t.d 28
13.d odd 4 1 195.2.t.a yes 28
15.e even 4 1 585.2.w.g 28
39.f even 4 1 585.2.w.g 28
65.f even 4 1 inner 195.2.k.a ✓ 28
65.g odd 4 1 975.2.t.d 28
65.k even 4 1 975.2.k.d 28
195.u odd 4 1 585.2.n.g 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.k.a ✓ 28 1.a even 1 1 trivial
195.2.k.a ✓ 28 65.f even 4 1 inner
195.2.t.a yes 28 5.c odd 4 1
195.2.t.a yes 28 13.d odd 4 1
585.2.n.g 28 3.b odd 2 1
585.2.n.g 28 195.u odd 4 1
585.2.w.g 28 15.e even 4 1
585.2.w.g 28 39.f even 4 1
975.2.k.d 28 5.b even 2 1
975.2.k.d 28 65.k even 4 1
975.2.t.d 28 5.c odd 4 1
975.2.t.d 28 65.g odd 4 1

Hecke kernels

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