Properties

Label 960.2.ci.a
Level $960$
Weight $2$
Character orbit 960.ci
Analytic conductor $7.666$
Analytic rank $0$
Dimension $240$
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] = [960,2,Mod(61,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.ci (of order \(16\), degree \(8\), 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: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(30\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 48 q^{8} + 16 q^{22} - 112 q^{26} + 80 q^{28} + 80 q^{32} + 80 q^{34} + 80 q^{38} - 48 q^{40} + 96 q^{44} + 48 q^{51} - 48 q^{52} - 16 q^{54} - 16 q^{56} + 160 q^{58} + 64 q^{59} - 96 q^{62} + 32 q^{63} - 48 q^{64} - 48 q^{66} + 32 q^{67} - 96 q^{68} + 64 q^{71} - 16 q^{74} - 128 q^{76} - 48 q^{78} + 144 q^{79} + 160 q^{86} + 80 q^{88} + 64 q^{93} + 16 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −1.40862 0.125697i 0.195090 0.980785i 1.96840 + 0.354118i −0.831470 0.555570i −0.398089 + 1.35703i 3.09010 1.27996i −2.72821 0.746240i −0.923880 0.382683i 1.10139 + 0.887099i
61.2 −1.40592 0.152965i −0.195090 + 0.980785i 1.95320 + 0.430112i 0.831470 + 0.555570i 0.424306 1.34906i −3.59279 + 1.48818i −2.68025 0.903472i −0.923880 0.382683i −1.08399 0.908271i
61.3 −1.38078 + 0.305685i −0.195090 + 0.980785i 1.81311 0.844167i 0.831470 + 0.555570i −0.0304340 1.41389i 2.99051 1.23871i −2.24546 + 1.71985i −0.923880 0.382683i −1.31791 0.512953i
61.4 −1.32750 + 0.487578i 0.195090 0.980785i 1.52454 1.29452i −0.831470 0.555570i 0.219226 + 1.39712i −2.84484 + 1.17837i −1.39265 + 2.46182i −0.923880 0.382683i 1.37466 + 0.332115i
61.5 −1.24393 0.672791i −0.195090 + 0.980785i 1.09471 + 1.67380i 0.831470 + 0.555570i 0.902541 1.08877i 0.288579 0.119533i −0.235613 2.81860i −0.923880 0.382683i −0.660504 1.25049i
61.6 −1.14486 + 0.830238i −0.195090 + 0.980785i 0.621409 1.90101i 0.831470 + 0.555570i −0.590934 1.28483i −0.863258 + 0.357573i 0.866867 + 2.69231i −0.923880 0.382683i −1.41317 + 0.0542677i
61.7 −1.09658 + 0.893032i 0.195090 0.980785i 0.404986 1.95857i −0.831470 0.555570i 0.661940 + 1.24973i 0.347892 0.144101i 1.30496 + 2.50940i −0.923880 0.382683i 1.40792 0.133301i
61.8 −1.00537 0.994599i 0.195090 0.980785i 0.0215467 + 1.99988i −0.831470 0.555570i −1.17163 + 0.792018i −0.499068 + 0.206721i 1.96742 2.03206i −0.923880 0.382683i 0.283367 + 1.38553i
61.9 −0.947680 1.04972i −0.195090 + 0.980785i −0.203805 + 1.98959i 0.831470 + 0.555570i 1.21443 0.724681i −2.30204 + 0.953536i 2.28164 1.67156i −0.923880 0.382683i −0.204776 1.39931i
61.10 −0.680390 + 1.23979i −0.195090 + 0.980785i −1.07414 1.68708i 0.831470 + 0.555570i −1.08323 0.909186i −2.54761 + 1.05526i 2.82245 0.183836i −0.923880 0.382683i −1.25451 + 0.652840i
61.11 −0.651744 + 1.25508i 0.195090 0.980785i −1.15046 1.63598i −0.831470 0.555570i 1.10382 + 0.884075i 1.90330 0.788373i 2.80310 0.377676i −0.923880 0.382683i 1.23919 0.681472i
61.12 −0.481268 1.32980i 0.195090 0.980785i −1.53676 + 1.27999i −0.831470 0.555570i −1.39814 + 0.212589i 0.354154 0.146695i 2.44173 + 1.42758i −0.923880 0.382683i −0.338640 + 1.37307i
61.13 −0.334080 + 1.37419i 0.195090 0.980785i −1.77678 0.918178i −0.831470 0.555570i 1.28261 + 0.595752i −4.83516 + 2.00279i 1.85534 2.13488i −0.923880 0.382683i 1.04124 0.956990i
61.14 −0.192400 1.40106i −0.195090 + 0.980785i −1.92596 + 0.539131i 0.831470 + 0.555570i 1.41168 + 0.0846308i 0.179560 0.0743761i 1.12591 + 2.59467i −0.923880 0.382683i 0.618415 1.27183i
61.15 −0.0932669 + 1.41113i −0.195090 + 0.980785i −1.98260 0.263224i 0.831470 + 0.555570i −1.36582 0.366774i 0.812877 0.336705i 0.556356 2.77317i −0.923880 0.382683i −0.861533 + 1.12150i
61.16 0.138497 + 1.40742i 0.195090 0.980785i −1.96164 + 0.389844i −0.831470 0.555570i 1.40739 + 0.138738i 2.52673 1.04660i −0.820353 2.70685i −0.923880 0.382683i 0.666763 1.24717i
61.17 0.329243 1.37535i 0.195090 0.980785i −1.78320 0.905651i −0.831470 0.555570i −1.28470 0.591235i −2.55159 + 1.05690i −1.83270 + 2.15435i −0.923880 0.382683i −1.03786 + 0.960648i
61.18 0.425634 1.34864i −0.195090 + 0.980785i −1.63767 1.14806i 0.831470 + 0.555570i 1.23969 + 0.680563i −0.967492 + 0.400748i −2.24537 + 1.71998i −0.923880 0.382683i 1.10317 0.884885i
61.19 0.454075 + 1.33933i −0.195090 + 0.980785i −1.58763 + 1.21632i 0.831470 + 0.555570i −1.40218 + 0.184059i 0.344497 0.142695i −2.34996 1.57407i −0.923880 0.382683i −0.366545 + 1.36589i
61.20 0.587652 + 1.28634i 0.195090 0.980785i −1.30933 + 1.51184i −0.831470 0.555570i 1.37627 0.325408i −0.956612 + 0.396242i −2.71417 0.795807i −0.923880 0.382683i 0.226036 1.39603i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
64.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.ci.a 240
64.i even 16 1 inner 960.2.ci.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.ci.a 240 1.a even 1 1 trivial
960.2.ci.a 240 64.i even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{240} - 800 T_{7}^{235} + 4864 T_{7}^{234} + 12416 T_{7}^{233} + 1081920 T_{7}^{232} + 188672 T_{7}^{231} + 1602944 T_{7}^{230} - 25090560 T_{7}^{229} + 1896448 T_{7}^{228} - 268509888 T_{7}^{227} + \cdots + 28\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\). Copy content Toggle raw display