Properties

Label 946.2.q.c
Level $946$
Weight $2$
Character orbit 946.q
Analytic conductor $7.554$
Analytic rank $0$
Dimension $176$
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] = [946,2,Mod(49,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([12, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.q (of order \(15\), 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: \(7.55384803121\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(22\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176 q - 44 q^{2} + 7 q^{3} - 44 q^{4} + 9 q^{5} - 8 q^{6} + 8 q^{7} - 44 q^{8} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q - 44 q^{2} + 7 q^{3} - 44 q^{4} + 9 q^{5} - 8 q^{6} + 8 q^{7} - 44 q^{8} + 33 q^{9} - 6 q^{10} + 16 q^{11} + 2 q^{12} + 16 q^{13} + 8 q^{14} + 4 q^{15} - 44 q^{16} + 5 q^{17} + 18 q^{18} + 10 q^{19} + 9 q^{20} - 36 q^{21} + 26 q^{22} - 40 q^{23} - 8 q^{24} + 45 q^{25} - 9 q^{26} - 32 q^{27} - 7 q^{28} + 2 q^{29} + 4 q^{30} + 18 q^{31} + 176 q^{32} - 36 q^{33} - 86 q^{35} + 18 q^{36} - 5 q^{37} - 5 q^{38} - 36 q^{39} - 6 q^{40} + 22 q^{41} + 4 q^{42} - 36 q^{43} - 34 q^{44} - 32 q^{45} + 5 q^{46} - 32 q^{47} + 7 q^{48} + 62 q^{49} + 10 q^{50} - 54 q^{51} - 9 q^{52} + 23 q^{53} - 52 q^{54} - 70 q^{55} - 2 q^{56} + 10 q^{57} + 2 q^{58} - 76 q^{59} + 9 q^{60} - 12 q^{61} - 2 q^{62} - 46 q^{63} - 44 q^{64} - 20 q^{65} - 6 q^{66} - 22 q^{67} + 5 q^{68} - 101 q^{69} + 84 q^{70} - 8 q^{71} + 33 q^{72} + 53 q^{73} - 5 q^{74} + 76 q^{75} - 10 q^{76} + 28 q^{77} + 64 q^{78} + 17 q^{79} - 6 q^{80} + 73 q^{81} - 38 q^{82} + 10 q^{83} + 14 q^{84} - 24 q^{85} + 9 q^{86} - 12 q^{87} + 6 q^{88} - 64 q^{89} - 12 q^{90} - 23 q^{91} + 15 q^{92} + 100 q^{93} - 2 q^{94} + 27 q^{95} + 7 q^{96} + 60 q^{97} - 118 q^{98} - 25 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0.309017 + 0.951057i −2.96331 + 1.31935i −0.809017 + 0.587785i −1.76750 1.96300i −2.17049 2.41058i −2.81992 1.25551i −0.809017 0.587785i 5.03315 5.58987i 1.32074 2.28759i
49.2 0.309017 + 0.951057i −2.85768 + 1.27232i −0.809017 + 0.587785i −0.100024 0.111088i −2.09312 2.32465i 1.32624 + 0.590481i −0.809017 0.587785i 4.54016 5.04236i 0.0747421 0.129457i
49.3 0.309017 + 0.951057i −2.42670 + 1.08044i −0.809017 + 0.587785i 0.907052 + 1.00738i −1.77745 1.97406i −1.65777 0.738085i −0.809017 0.587785i 2.71415 3.01436i −0.677784 + 1.17396i
49.4 0.309017 + 0.951057i −2.36183 + 1.05155i −0.809017 + 0.587785i −2.01221 2.23479i −1.72993 1.92128i 4.17853 + 1.86040i −0.809017 0.587785i 2.46507 2.73774i 1.50360 2.60431i
49.5 0.309017 + 0.951057i −2.01424 + 0.896800i −0.809017 + 0.587785i 2.47080 + 2.74410i −1.47534 1.63853i −0.355681 0.158360i −0.809017 0.587785i 1.24554 1.38331i −1.84628 + 3.19785i
49.6 0.309017 + 0.951057i −1.18125 + 0.525927i −0.809017 + 0.587785i −0.922699 1.02476i −0.865214 0.960917i 2.70101 + 1.20257i −0.809017 0.587785i −0.888635 + 0.986929i 0.689476 1.19421i
49.7 0.309017 + 0.951057i −0.874270 + 0.389250i −0.809017 + 0.587785i 2.04125 + 2.26704i −0.640363 0.711195i −0.720918 0.320973i −0.809017 0.587785i −1.39456 + 1.54882i −1.52530 + 2.64190i
49.8 0.309017 + 0.951057i −0.634432 + 0.282467i −0.809017 + 0.587785i −1.26484 1.40474i −0.464693 0.516093i 2.16838 + 0.965424i −0.809017 0.587785i −1.68468 + 1.87102i 0.945134 1.63702i
49.9 0.309017 + 0.951057i −0.535900 + 0.238598i −0.809017 + 0.587785i 0.507379 + 0.563501i −0.392522 0.435940i −3.15031 1.40261i −0.809017 0.587785i −1.77713 + 1.97371i −0.379133 + 0.656677i
49.10 0.309017 + 0.951057i −0.149365 + 0.0665017i −0.809017 + 0.587785i −2.86850 3.18580i −0.109403 0.121505i −1.38069 0.614723i −0.809017 0.587785i −1.98950 + 2.20957i 2.14346 3.71257i
49.11 0.309017 + 0.951057i −0.0878619 + 0.0391186i −0.809017 + 0.587785i 2.11377 + 2.34758i −0.0643549 0.0714733i 3.80419 + 1.69373i −0.809017 0.587785i −2.00120 + 2.22256i −1.57949 + 2.73576i
49.12 0.309017 + 0.951057i 0.662206 0.294833i −0.809017 + 0.587785i 1.61748 + 1.79639i 0.485036 + 0.538687i −4.34265 1.93347i −0.809017 0.587785i −1.65580 + 1.83895i −1.20864 + 2.09343i
49.13 0.309017 + 0.951057i 0.747142 0.332649i −0.809017 + 0.587785i 0.275976 + 0.306503i 0.547247 + 0.607780i 1.10722 + 0.492968i −0.809017 0.587785i −1.55983 + 1.73236i −0.206220 + 0.357184i
49.14 0.309017 + 0.951057i 1.02548 0.456574i −0.809017 + 0.587785i −2.41561 2.68281i 0.751119 + 0.834203i 0.668118 + 0.297465i −0.809017 0.587785i −1.16424 + 1.29302i 1.80504 3.12642i
49.15 0.309017 + 0.951057i 1.05245 0.468579i −0.809017 + 0.587785i 0.468352 + 0.520157i 0.770869 + 0.856137i 2.03278 + 0.905052i −0.809017 0.587785i −1.11932 + 1.24313i −0.349970 + 0.606167i
49.16 0.309017 + 0.951057i 1.41592 0.630406i −0.809017 + 0.587785i −1.69936 1.88733i 1.03709 + 1.15181i −2.34952 1.04608i −0.809017 0.587785i −0.399987 + 0.444231i 1.26983 2.19941i
49.17 0.309017 + 0.951057i 1.66506 0.741331i −0.809017 + 0.587785i 2.29405 + 2.54780i 1.21958 + 1.35448i 0.933040 + 0.415416i −0.809017 0.587785i 0.215450 0.239281i −1.71420 + 2.96909i
49.18 0.309017 + 0.951057i 1.92784 0.858332i −0.809017 + 0.587785i 0.207088 + 0.229995i 1.41206 + 1.56825i 3.48351 + 1.55096i −0.809017 0.587785i 0.972459 1.08003i −0.154744 + 0.268025i
49.19 0.309017 + 0.951057i 2.10987 0.939377i −0.809017 + 0.587785i −2.33563 2.59398i 1.54539 + 1.71633i 3.92251 + 1.74641i −0.809017 0.587785i 1.56175 1.73450i 1.74527 3.02290i
49.20 0.309017 + 0.951057i 2.25002 1.00178i −0.809017 + 0.587785i −0.136830 0.151965i 1.64804 + 1.83034i −2.66085 1.18469i −0.809017 0.587785i 2.05167 2.27861i 0.102245 0.177093i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
43.c even 3 1 inner
473.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.q.c 176
11.c even 5 1 inner 946.2.q.c 176
43.c even 3 1 inner 946.2.q.c 176
473.q even 15 1 inner 946.2.q.c 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.q.c 176 1.a even 1 1 trivial
946.2.q.c 176 11.c even 5 1 inner
946.2.q.c 176 43.c even 3 1 inner
946.2.q.c 176 473.q even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{176} - 7 T_{3}^{175} - 25 T_{3}^{174} + 268 T_{3}^{173} + 104 T_{3}^{172} - 4299 T_{3}^{171} + 5220 T_{3}^{170} + 20101 T_{3}^{169} - 104534 T_{3}^{168} + 490505 T_{3}^{167} + 770506 T_{3}^{166} - 12277734 T_{3}^{165} + \cdots + 48\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display