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Properties

Label 968.2.o.j
Level $968$
Weight $2$
Character orbit 968.o
Analytic conductor $7.730$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [968,2,Mod(245,968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(968, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.245");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 968.o (of order \(10\), degree \(4\), not 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: \(7.72951891566\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 88)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 5 q^{2} - q^{4} + 8 q^{6} + 5 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 5 q^{2} - q^{4} + 8 q^{6} + 5 q^{8} + 10 q^{9} - 20 q^{10} - 6 q^{12} + 2 q^{14} + 22 q^{15} - 25 q^{16} + 4 q^{17} + 25 q^{18} + 8 q^{20} - 8 q^{23} - 40 q^{24} + 6 q^{25} - 20 q^{26} - 18 q^{28} + 12 q^{30} + 4 q^{31} + 20 q^{32} + 10 q^{34} - 27 q^{36} - 33 q^{38} + 6 q^{39} - 20 q^{40} + 16 q^{41} - 20 q^{42} - 42 q^{46} + 4 q^{47} + 23 q^{48} + 6 q^{49} - 39 q^{50} + 108 q^{54} - 32 q^{56} + 4 q^{57} - 16 q^{58} + 94 q^{60} - 52 q^{62} - 30 q^{63} + 11 q^{64} - 36 q^{65} + 63 q^{68} + 52 q^{70} + 12 q^{71} + 44 q^{72} + 4 q^{73} - 26 q^{74} - 134 q^{76} - 12 q^{78} - 16 q^{79} + 16 q^{80} + 16 q^{81} + 54 q^{82} + 72 q^{84} - 10 q^{86} + 68 q^{87} - 16 q^{89} + 126 q^{90} - 64 q^{92} + 40 q^{94} + 26 q^{95} - 5 q^{96} - 20 q^{97} - 144 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1 −1.30039 + 0.555854i 1.78795 − 0.580940i 1.38205 − 1.44566i 0.289841 − 0.398931i −2.00212 + 1.74929i 0.0540070 − 0.166217i −0.993642 + 2.64815i 0.432225 − 0.314030i −0.155160 + 0.679877i
245.2 −1.07161 − 0.922853i 0.513977 − 0.167001i 0.296684 + 1.97787i 1.70510 − 2.34687i −0.704899 − 0.295365i 0.804837 − 2.47703i 1.50736 − 2.39330i −2.19077 + 1.59169i −3.99302 + 0.941367i
245.3 −0.930492 + 1.06498i −1.78795 + 0.580940i −0.268368 − 1.98191i −0.289841 + 0.398931i 1.04498 − 2.44469i 0.0540070 − 0.166217i 2.36041 + 1.55835i 0.432225 − 0.314030i −0.155160 − 0.679877i
245.4 −0.528001 − 1.31195i −2.71697 + 0.882798i −1.44243 + 1.38542i −1.53289 + 2.10984i 2.59275 + 3.09841i 0.0606792 − 0.186751i 2.57921 + 1.16089i 4.17555 − 3.03372i 3.57737 + 0.897076i
245.5 0.322293 − 1.37700i 0.305500 − 0.0992629i −1.79225 − 0.887594i 0.756671 − 1.04147i −0.0382245 − 0.452665i −1.02182 + 3.14484i −1.79985 + 2.18187i −2.34357 + 1.70271i −1.19023 − 1.37759i
245.6 0.546541 + 1.30434i −0.513977 + 0.167001i −1.40259 + 1.42575i −1.70510 + 2.34687i −0.498735 − 0.579126i 0.804837 − 2.47703i −2.62622 − 1.05022i −2.19077 + 1.59169i −3.99302 − 0.941367i
245.7 1.05579 − 0.940911i 2.42981 − 0.789492i 0.229373 − 1.98680i −1.58432 + 2.18063i 1.82252 − 3.11977i 1.22033 − 3.75580i −1.62724 − 2.31346i 2.85361 − 2.07327i 0.379074 + 3.79299i
245.8 1.08458 + 0.907574i 2.71697 − 0.882798i 0.352619 + 1.96867i 1.53289 − 2.10984i 3.74797 + 1.50839i 0.0606792 − 0.186751i −1.40427 + 2.45520i 4.17555 − 3.03372i 3.57737 − 0.897076i
245.9 1.22112 − 0.713356i −2.42981 + 0.789492i 0.982247 − 1.74218i 1.58432 − 2.18063i −2.40389 + 2.69738i 1.22033 − 3.75580i −0.0433576 − 2.82809i 2.85361 − 2.07327i 0.379074 − 3.79299i
245.10 1.40920 + 0.118997i −0.305500 + 0.0992629i 1.97168 + 0.335382i −0.756671 + 1.04147i −0.442322 + 0.103527i −1.02182 + 3.14484i 2.73858 + 0.707244i −2.34357 + 1.70271i −1.19023 + 1.37759i
269.1 −1.40911 − 0.120047i −0.359846 + 0.495286i 1.97118 + 0.338317i 1.38882 + 0.451256i 0.566520 − 0.654713i −3.58243 + 2.60279i −2.73699 − 0.713359i 0.811232 + 2.49672i −1.90283 − 0.802592i
269.2 −0.992238 + 1.00770i −1.43886 + 1.98041i −0.0309261 − 1.99976i 0.190662 + 0.0619499i −0.567980 − 3.41498i 1.89808 − 1.37903i 2.04585 + 1.95308i −0.924686 − 2.84589i −0.251609 + 0.130662i
269.3 −0.915193 − 1.07816i −0.756086 + 1.04066i −0.324844 + 1.97344i −1.50450 − 0.488843i 1.81396 − 0.137228i 1.51144 − 1.09813i 2.42498 − 1.45585i 0.415738 + 1.27951i 0.849862 + 2.06948i
269.4 −0.730005 + 1.21124i 1.62101 − 2.23113i −0.934186 − 1.76842i 2.76352 + 0.897923i 1.51908 + 3.59217i −1.91213 + 1.38924i 2.82393 + 0.159432i −1.42322 − 4.38022i −3.10498 + 2.69179i
269.5 0.106682 − 1.41018i 0.756086 − 1.04066i −1.97724 − 0.300882i 1.50450 + 0.488843i −1.38687 − 1.17724i 1.51144 − 1.09813i −0.635235 + 2.75617i 0.415738 + 1.27951i 0.849862 − 2.06948i
269.6 0.227030 + 1.39587i 0.510574 − 0.702745i −1.89691 + 0.633809i −3.49435 − 1.13538i 1.09686 + 0.553152i 0.967003 − 0.702569i −1.31537 − 2.50396i 0.693886 + 2.13556i 0.791527 − 5.13543i
269.7 0.636802 + 1.26273i −0.510574 + 0.702745i −1.18897 + 1.60822i 3.49435 + 1.13538i −1.21251 − 0.197208i 0.967003 − 0.702569i −2.78788 − 0.477229i 0.693886 + 2.13556i 0.791527 + 5.13543i
269.8 1.06943 − 0.925373i 0.359846 − 0.495286i 0.287368 − 1.97925i −1.38882 − 0.451256i −0.0734933 − 0.862666i −3.58243 + 2.60279i −1.52422 − 2.38259i 0.811232 + 2.49672i −1.90283 + 0.802592i
269.9 1.30253 + 0.550825i −1.62101 + 2.23113i 1.39318 + 1.43493i −2.76352 − 0.897923i −3.34039 + 2.01323i −1.91213 + 1.38924i 1.02427 + 2.63645i −1.42322 − 4.38022i −3.10498 − 2.69179i
269.10 1.39505 + 0.232025i 1.43886 − 1.98041i 1.89233 + 0.647373i −0.190662 − 0.0619499i 2.46678 − 2.42893i 1.89808 − 1.37903i 2.48969 + 1.34218i −0.924686 − 2.84589i −0.251609 − 0.130662i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
11.c even 5 1 inner
88.o even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.o.j 40
8.b even 2 1 inner 968.2.o.j 40
11.b odd 2 1 968.2.o.d 40
11.c even 5 2 88.2.o.a ✓ 40
11.c even 5 1 968.2.c.h 20
11.c even 5 1 inner 968.2.o.j 40
11.d odd 10 1 968.2.c.i 20
11.d odd 10 1 968.2.o.d 40
11.d odd 10 2 968.2.o.i 40
33.h odd 10 2 792.2.br.b 40
44.g even 10 1 3872.2.c.i 20
44.h odd 10 2 352.2.w.a 40
44.h odd 10 1 3872.2.c.h 20
88.b odd 2 1 968.2.o.d 40
88.k even 10 1 3872.2.c.i 20
88.l odd 10 2 352.2.w.a 40
88.l odd 10 1 3872.2.c.h 20
88.o even 10 2 88.2.o.a ✓ 40
88.o even 10 1 968.2.c.h 20
88.o even 10 1 inner 968.2.o.j 40
88.p odd 10 1 968.2.c.i 20
88.p odd 10 1 968.2.o.d 40
88.p odd 10 2 968.2.o.i 40
264.t odd 10 2 792.2.br.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.o.a ✓ 40 11.c even 5 2
88.2.o.a ✓ 40 88.o even 10 2
352.2.w.a 40 44.h odd 10 2
352.2.w.a 40 88.l odd 10 2
792.2.br.b 40 33.h odd 10 2
792.2.br.b 40 264.t odd 10 2
968.2.c.h 20 11.c even 5 1
968.2.c.h 20 88.o even 10 1
968.2.c.i 20 11.d odd 10 1
968.2.c.i 20 88.p odd 10 1
968.2.o.d 40 11.b odd 2 1
968.2.o.d 40 11.d odd 10 1
968.2.o.d 40 88.b odd 2 1
968.2.o.d 40 88.p odd 10 1
968.2.o.i 40 11.d odd 10 2
968.2.o.i 40 88.p odd 10 2
968.2.o.j 40 1.a even 1 1 trivial
968.2.o.j 40 8.b even 2 1 inner
968.2.o.j 40 11.c even 5 1 inner
968.2.o.j 40 88.o even 10 1 inner
3872.2.c.h 20 44.h odd 10 1
3872.2.c.h 20 88.l odd 10 1
3872.2.c.i 20 44.g even 10 1
3872.2.c.i 20 88.k even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):

\( T_{3}^{40} - 20 T_{3}^{38} + 241 T_{3}^{36} - 2331 T_{3}^{34} + 21135 T_{3}^{32} - 145733 T_{3}^{30} + \cdots + 14641 \) Copy content Toggle raw display
\( T_{5}^{40} - 28 T_{5}^{38} + 450 T_{5}^{36} - 5347 T_{5}^{34} + 65807 T_{5}^{32} - 595668 T_{5}^{30} + \cdots + 16777216 \) Copy content Toggle raw display
\( T_{7}^{20} + 16 T_{7}^{18} + 7 T_{7}^{17} + 251 T_{7}^{16} - 834 T_{7}^{15} + 4255 T_{7}^{14} + \cdots + 4096 \) Copy content Toggle raw display