Properties

Label 374.2.r.b
Level $374$
Weight $2$
Character orbit 374.r
Analytic conductor $2.986$
Analytic rank $0$
Dimension $160$
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] = [374,2,Mod(9,374)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(374, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([24, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("374.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 374 = 2 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 374.r (of order \(40\), degree \(16\), 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: \(2.98640503560\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(10\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 8 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 8 q^{5} + 8 q^{9} + 8 q^{10} - 4 q^{11} - 8 q^{12} + 4 q^{14} - 8 q^{15} + 40 q^{16} - 12 q^{17} + 32 q^{18} + 12 q^{19} - 8 q^{22} + 32 q^{23} + 12 q^{24} - 8 q^{25} + 36 q^{27} - 4 q^{28} + 8 q^{29} - 40 q^{31} + 48 q^{33} + 8 q^{34} + 16 q^{35} - 12 q^{36} + 8 q^{37} - 16 q^{39} + 28 q^{41} + 60 q^{42} - 56 q^{43} - 8 q^{44} + 28 q^{46} - 116 q^{49} - 80 q^{50} + 8 q^{52} + 8 q^{53} - 8 q^{54} - 148 q^{57} + 8 q^{58} - 8 q^{59} + 8 q^{60} - 12 q^{61} - 40 q^{62} - 72 q^{63} + 144 q^{65} - 28 q^{66} - 256 q^{67} - 8 q^{69} + 8 q^{70} - 60 q^{71} - 8 q^{73} - 8 q^{74} - 116 q^{75} + 8 q^{76} - 60 q^{77} - 16 q^{78} + 40 q^{79} + 12 q^{80} - 16 q^{82} + 76 q^{83} - 56 q^{84} + 36 q^{85} - 8 q^{86} - 144 q^{87} - 4 q^{88} - 40 q^{90} + 44 q^{91} - 32 q^{92} - 40 q^{93} - 4 q^{94} + 36 q^{95} - 124 q^{97} - 176 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0.453990 + 0.891007i −2.94097 + 0.706064i −0.587785 + 0.809017i −2.43559 0.191685i −1.96428 2.29987i 0.337215 1.40460i −0.987688 0.156434i 5.47774 2.79105i −0.934943 2.25715i
9.2 0.453990 + 0.891007i −2.59262 + 0.622433i −0.587785 + 0.809017i 3.64557 + 0.286913i −1.73162 2.02746i 0.957546 3.98846i −0.987688 0.156434i 3.66123 1.86549i 1.39941 + 3.37848i
9.3 0.453990 + 0.891007i −2.56251 + 0.615205i −0.587785 + 0.809017i 0.301675 + 0.0237423i −1.71151 2.00392i −1.12606 + 4.69038i −0.987688 0.156434i 3.51498 1.79097i 0.115803 + 0.279573i
9.4 0.453990 + 0.891007i −0.983879 + 0.236209i −0.587785 + 0.809017i −0.705643 0.0555353i −0.657135 0.769407i 0.0706580 0.294312i −0.987688 0.156434i −1.76080 + 0.897170i −0.270873 0.653945i
9.5 0.453990 + 0.891007i 0.0291666 0.00700227i −0.587785 + 0.809017i −3.50736 0.276036i 0.0194804 + 0.0228086i 1.19587 4.98117i −0.987688 0.156434i −2.67222 + 1.36156i −1.34636 3.25040i
9.6 0.453990 + 0.891007i 0.786517 0.188826i −0.587785 + 0.809017i 2.58893 + 0.203753i 0.525317 + 0.615067i −0.931434 + 3.87970i −0.987688 0.156434i −2.09007 + 1.06494i 0.993805 + 2.39926i
9.7 0.453990 + 0.891007i 1.17879 0.283004i −0.587785 + 0.809017i −2.86147 0.225203i 0.787320 + 0.921833i −0.890507 + 3.70923i −0.987688 0.156434i −1.36355 + 0.694765i −1.09842 2.65183i
9.8 0.453990 + 0.891007i 1.66749 0.400329i −0.587785 + 0.809017i 2.96047 + 0.232994i 1.11372 + 1.30400i −0.0178879 + 0.0745085i −0.987688 0.156434i −0.0527593 + 0.0268822i 1.13643 + 2.74358i
9.9 0.453990 + 0.891007i 1.87910 0.451133i −0.587785 + 0.809017i 2.19917 + 0.173078i 1.25506 + 1.46948i 0.891322 3.71262i −0.987688 0.156434i 0.654489 0.333479i 0.844188 + 2.03805i
9.10 0.453990 + 0.891007i 3.07895 0.739191i −0.587785 + 0.809017i −0.951174 0.0748590i 2.05644 + 2.40778i −0.0553730 + 0.230645i −0.987688 0.156434i 6.26053 3.18990i −0.365124 0.881488i
15.1 −0.156434 0.987688i −3.14661 + 0.247644i −0.951057 + 0.309017i 0.983379 1.60473i 0.736834 + 3.06913i 0.233197 2.96305i 0.453990 + 0.891007i 6.87678 1.08918i −1.73881 0.720237i
15.2 −0.156434 0.987688i −2.22135 + 0.174824i −0.951057 + 0.309017i −1.78241 + 2.90863i 0.520166 + 2.16665i 0.244646 3.10853i 0.453990 + 0.891007i 1.94075 0.307385i 3.15166 + 1.30546i
15.3 −0.156434 0.987688i −2.00102 + 0.157484i −0.951057 + 0.309017i 1.44748 2.36207i 0.468573 + 1.95175i −0.369987 + 4.70113i 0.453990 + 0.891007i 1.01621 0.160952i −2.55942 1.06015i
15.4 −0.156434 0.987688i −0.586497 + 0.0461583i −0.951057 + 0.309017i −0.493290 + 0.804977i 0.137338 + 0.572056i 0.0756830 0.961643i 0.453990 + 0.891007i −2.62122 + 0.415160i 0.872234 + 0.361291i
15.5 −0.156434 0.987688i 0.454529 0.0357722i −0.951057 + 0.309017i 1.53012 2.49693i −0.106436 0.443337i 0.0968258 1.23029i 0.453990 + 0.891007i −2.75775 + 0.436784i −2.70555 1.12068i
15.6 −0.156434 0.987688i 0.497702 0.0391700i −0.951057 + 0.309017i −1.07627 + 1.75632i −0.116545 0.485447i −0.117981 + 1.49909i 0.453990 + 0.891007i −2.71689 + 0.430313i 1.90306 + 0.788272i
15.7 −0.156434 0.987688i 1.60107 0.126007i −0.951057 + 0.309017i −0.423785 + 0.691555i −0.374918 1.56165i 0.193717 2.46140i 0.453990 + 0.891007i −0.415518 + 0.0658115i 0.749335 + 0.310385i
15.8 −0.156434 0.987688i 2.43763 0.191846i −0.951057 + 0.309017i 1.89159 3.08680i −0.570813 2.37761i −0.346575 + 4.40365i 0.453990 + 0.891007i 2.94216 0.465993i −3.34470 1.38542i
15.9 −0.156434 0.987688i 2.89846 0.228114i −0.951057 + 0.309017i −1.94335 + 3.17125i −0.678724 2.82709i −0.302950 + 3.84934i 0.453990 + 0.891007i 5.38596 0.853052i 3.43621 + 1.42333i
15.10 −0.156434 0.987688i 3.04661 0.239773i −0.951057 + 0.309017i 0.463220 0.755906i −0.713416 2.97159i 0.353474 4.49131i 0.453990 + 0.891007i 6.26127 0.991688i −0.819063 0.339267i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
17.d even 8 1 inner
187.r even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 374.2.r.b 160
11.c even 5 1 inner 374.2.r.b 160
17.d even 8 1 inner 374.2.r.b 160
187.r even 40 1 inner 374.2.r.b 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
374.2.r.b 160 1.a even 1 1 trivial
374.2.r.b 160 11.c even 5 1 inner
374.2.r.b 160 17.d even 8 1 inner
374.2.r.b 160 187.r even 40 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{160} - 4 T_{3}^{158} - 36 T_{3}^{157} + 8 T_{3}^{156} - 16 T_{3}^{155} + 692 T_{3}^{154} + 1864 T_{3}^{153} - 11606 T_{3}^{152} - 22080 T_{3}^{151} - 1216 T_{3}^{150} + 459300 T_{3}^{149} + 149368 T_{3}^{148} + \cdots + 70\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(374, [\chi])\). Copy content Toggle raw display