Properties

Label 195.2.bd.a
Level $195$
Weight $2$
Character orbit 195.bd
Analytic conductor $1.557$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [195,2,Mod(67,195)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(195, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("195.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.bd (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: \(1.55708283941\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{2} - 28 q^{4} + 4 q^{5} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{2} - 28 q^{4} + 4 q^{5} - 24 q^{8} + 8 q^{11} - 16 q^{12} - 12 q^{13} + 4 q^{15} - 28 q^{16} + 8 q^{17} - 24 q^{19} - 12 q^{20} - 8 q^{21} - 28 q^{22} + 8 q^{23} + 20 q^{25} - 8 q^{31} + 8 q^{32} - 4 q^{33} + 4 q^{34} - 24 q^{37} + 8 q^{39} + 24 q^{40} - 4 q^{41} + 60 q^{42} + 12 q^{43} + 40 q^{44} - 16 q^{45} - 8 q^{46} + 16 q^{48} + 44 q^{49} - 52 q^{50} - 24 q^{52} + 4 q^{53} - 40 q^{55} + 72 q^{56} + 120 q^{58} - 64 q^{59} + 16 q^{61} + 36 q^{62} + 56 q^{64} - 4 q^{65} - 16 q^{66} + 64 q^{67} + 48 q^{68} + 8 q^{69} + 16 q^{70} - 16 q^{71} - 80 q^{73} + 36 q^{74} - 16 q^{75} + 112 q^{76} - 48 q^{77} - 40 q^{78} - 36 q^{80} + 28 q^{81} - 32 q^{82} - 32 q^{84} - 56 q^{85} - 64 q^{86} - 60 q^{87} - 120 q^{88} - 12 q^{89} + 12 q^{90} - 40 q^{91} - 64 q^{92} - 24 q^{94} - 40 q^{95} - 12 q^{97} + 108 q^{98} + 8 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.25215 2.16879i 0.965926 + 0.258819i −2.13577 + 3.69927i −0.256829 + 2.22127i −0.648162 2.41897i 2.27192 + 1.31169i 5.68865 0.866025 + 0.500000i 5.13906 2.22436i
67.2 −1.00587 1.74222i −0.965926 0.258819i −1.02355 + 1.77285i −1.62945 + 1.53131i 0.520677 + 1.94319i −0.165019 0.0952740i 0.0947706 0.866025 + 0.500000i 4.30689 + 1.29856i
67.3 −0.918199 1.59037i −0.965926 0.258819i −0.686179 + 1.18850i 1.86952 1.22674i 0.475295 + 1.77382i −2.27688 1.31456i −1.15260 0.866025 + 0.500000i −3.66756 1.84683i
67.4 −0.853152 1.47770i 0.965926 + 0.258819i −0.455736 + 0.789358i 1.35839 1.77617i −0.441624 1.64816i 4.26702 + 2.46357i −1.85736 0.866025 + 0.500000i −3.78356 0.491964i
67.5 −0.626089 1.08442i 0.965926 + 0.258819i 0.216026 0.374168i −1.65898 1.49926i −0.324087 1.20951i −2.38272 1.37566i −3.04536 0.866025 + 0.500000i −0.587155 + 2.73770i
67.6 −0.314928 0.545471i −0.965926 0.258819i 0.801641 1.38848i 1.08672 + 1.95424i 0.163019 + 0.608393i 3.19090 + 1.84227i −2.26955 0.866025 + 0.500000i 0.723739 1.20822i
67.7 0.0801292 + 0.138788i −0.965926 0.258819i 0.987159 1.70981i −2.23524 + 0.0607511i −0.0414779 0.154798i −3.84699 2.22106i 0.636918 0.866025 + 0.500000i −0.187540 0.305356i
67.8 0.334760 + 0.579821i 0.965926 + 0.258819i 0.775871 1.34385i −0.773477 + 2.09803i 0.173285 + 0.646707i 0.0515993 + 0.0297909i 2.37796 0.866025 + 0.500000i −1.47541 + 0.253859i
67.9 0.482143 + 0.835096i 0.965926 + 0.258819i 0.535077 0.926780i 0.137372 2.23184i 0.249575 + 0.931428i −1.17248 0.676930i 2.96050 0.866025 + 0.500000i 1.93004 0.961349i
67.10 0.563303 + 0.975669i −0.965926 0.258819i 0.365380 0.632856i −1.02941 1.98502i −0.291587 1.08822i 2.13490 + 1.23258i 3.07649 0.866025 + 0.500000i 1.35685 2.12254i
67.11 0.861480 + 1.49213i −0.965926 0.258819i −0.484294 + 0.838822i 2.23388 + 0.0988139i −0.445935 1.66425i 0.421075 + 0.243108i 1.77708 0.866025 + 0.500000i 1.77700 + 3.41836i
67.12 1.13669 + 1.96880i 0.965926 + 0.258819i −1.58411 + 2.74376i −2.22995 0.165292i 0.588392 + 2.19591i 1.84947 + 1.06779i −2.65581 0.866025 + 0.500000i −2.20933 4.57821i
67.13 1.23409 + 2.13750i −0.965926 0.258819i −2.04594 + 3.54367i −1.52807 + 1.63248i −0.638810 2.38407i −0.354557 0.204703i −5.16311 0.866025 + 0.500000i −5.37520 1.25162i
67.14 1.27780 + 2.21322i 0.965926 + 0.258819i −2.26557 + 3.92408i 2.19142 0.444615i 0.661440 + 2.46853i −3.98824 2.30261i −6.46858 0.866025 + 0.500000i 3.78424 + 4.28196i
97.1 −1.33341 + 2.30953i −0.258819 0.965926i −2.55595 4.42704i 1.09012 + 1.95234i 2.57595 + 0.690223i 3.66094 2.11365i 8.29887 −0.866025 + 0.500000i −5.96257 0.0856134i
97.2 −1.22286 + 2.11806i 0.258819 + 0.965926i −1.99078 3.44813i 2.19831 + 0.409196i −2.36239 0.633000i −3.85373 + 2.22495i 4.84635 −0.866025 + 0.500000i −3.55493 + 4.15576i
97.3 −0.856342 + 1.48323i −0.258819 0.965926i −0.466643 0.808249i −2.21901 + 0.275658i 1.65433 + 0.443275i −0.592129 + 0.341866i −1.82694 −0.866025 + 0.500000i 1.49137 3.52736i
97.4 −0.437238 + 0.757319i 0.258819 + 0.965926i 0.617645 + 1.06979i 2.23600 + 0.0167872i −0.844680 0.226331i 1.89201 1.09235i −2.82919 −0.866025 + 0.500000i −0.990380 + 1.68603i
97.5 −0.416908 + 0.722106i 0.258819 + 0.965926i 0.652375 + 1.12995i −2.02394 0.950609i −0.805405 0.215808i −4.10424 + 2.36958i −2.75555 −0.866025 + 0.500000i 1.53024 1.06518i
97.6 −0.278861 + 0.483002i −0.258819 0.965926i 0.844473 + 1.46267i 1.99255 + 1.01476i 0.538719 + 0.144349i −1.80160 + 1.04015i −2.05741 −0.866025 + 0.500000i −1.04578 + 0.679427i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.o even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.bd.a 56
3.b odd 2 1 585.2.cf.b 56
5.b even 2 1 975.2.bl.i 56
5.c odd 4 1 195.2.bm.a yes 56
5.c odd 4 1 975.2.bu.i 56
13.f odd 12 1 195.2.bm.a yes 56
15.e even 4 1 585.2.dp.c 56
39.k even 12 1 585.2.dp.c 56
65.o even 12 1 inner 195.2.bd.a 56
65.s odd 12 1 975.2.bu.i 56
65.t even 12 1 975.2.bl.i 56
195.bn odd 12 1 585.2.cf.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bd.a 56 1.a even 1 1 trivial
195.2.bd.a 56 65.o even 12 1 inner
195.2.bm.a yes 56 5.c odd 4 1
195.2.bm.a yes 56 13.f odd 12 1
585.2.cf.b 56 3.b odd 2 1
585.2.cf.b 56 195.bn odd 12 1
585.2.dp.c 56 15.e even 4 1
585.2.dp.c 56 39.k even 12 1
975.2.bl.i 56 5.b even 2 1
975.2.bl.i 56 65.t even 12 1
975.2.bu.i 56 5.c odd 4 1
975.2.bu.i 56 65.s odd 12 1

Hecke kernels

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