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Properties

Label 946.2.bf.a
Level $946$
Weight $2$
Character orbit 946.bf
Analytic conductor $7.554$
Analytic rank $0$
Dimension $1056$
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] = [946,2,Mod(19,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(210))
 
chi = DirichletCharacter(H, H._module([63, 95]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.bf (of order \(210\), degree \(48\), 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: \(1056\)
Relative dimension: \(22\) over \(\Q(\zeta_{210})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{210}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1056 q - 44 q^{2} + 44 q^{4} + 5 q^{6} + 15 q^{7} - 44 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1056 q - 44 q^{2} + 44 q^{4} + 5 q^{6} + 15 q^{7} - 44 q^{8} + 20 q^{9} + 5 q^{11} + 12 q^{13} - 20 q^{14} + 77 q^{15} + 44 q^{16} + 36 q^{17} - 25 q^{18} + 31 q^{19} - 9 q^{22} - 24 q^{23} + 2 q^{24} + 24 q^{25} - 43 q^{26} - 60 q^{27} + 11 q^{29} + 41 q^{30} + 50 q^{31} + 176 q^{32} - 28 q^{33} - 45 q^{34} + 60 q^{35} - 129 q^{36} - 98 q^{37} - 157 q^{38} - 86 q^{39} + 21 q^{40} + 28 q^{41} - 5 q^{43} + 24 q^{44} - 98 q^{45} - 34 q^{46} + 4 q^{47} + 55 q^{49} + 100 q^{50} + 38 q^{51} + 20 q^{52} + 27 q^{53} - 42 q^{54} - 42 q^{55} + 132 q^{57} + 3 q^{58} - 49 q^{59} - 6 q^{60} - 21 q^{61} - 7 q^{62} - 51 q^{63} + 44 q^{64} + 68 q^{65} - 118 q^{66} - 25 q^{67} - 37 q^{68} + 66 q^{69} + 43 q^{71} - 20 q^{72} - 103 q^{73} - 22 q^{74} - 217 q^{75} - 33 q^{76} - 69 q^{77} - 92 q^{78} + 52 q^{79} - 69 q^{81} + 23 q^{82} - 86 q^{83} - 82 q^{85} + 107 q^{86} + 12 q^{88} + 69 q^{89} + 30 q^{91} + 8 q^{92} + 126 q^{93} - 43 q^{94} + 12 q^{95} - 103 q^{97} - 106 q^{98} - 494 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 0.0448648 − 0.998993i −2.74167 + 1.37122i −0.995974 − 0.0896393i 0.0359607 − 1.20154i 1.24683 + 2.80043i 0.915252 − 0.194543i −0.134233 + 0.990950i 3.83706 − 5.11846i −1.19872 − 0.0898314i
19.2 0.0448648 − 0.998993i −2.64672 + 1.32373i −0.995974 − 0.0896393i 0.114677 − 3.83166i 1.20365 + 2.70345i −2.45459 + 0.521740i −0.134233 + 0.990950i 3.45342 − 4.60670i −3.82266 − 0.286469i
19.3 0.0448648 − 0.998993i −2.53533 + 1.26802i −0.995974 − 0.0896393i −0.0818476 + 2.73474i 1.15299 + 2.58966i 1.71284 − 0.364076i −0.134233 + 0.990950i 3.02055 − 4.02927i 2.72831 + 0.204459i
19.4 0.0448648 − 0.998993i −2.00338 + 1.00197i −0.995974 − 0.0896393i −0.0968852 + 3.23718i 0.911076 + 2.04631i −4.39390 + 0.933952i −0.134233 + 0.990950i 1.21011 − 1.61423i 3.22958 + 0.242023i
19.5 0.0448648 − 0.998993i −1.68036 + 0.840414i −0.995974 − 0.0896393i 0.00707986 − 0.236556i 0.764179 + 1.71637i 3.94629 − 0.838809i −0.134233 + 0.990950i 0.317849 − 0.423995i −0.236001 − 0.0176858i
19.6 0.0448648 − 0.998993i −1.38856 + 0.694472i −0.995974 − 0.0896393i 0.00884738 − 0.295613i 0.631475 + 1.41832i −2.71112 + 0.576266i −0.134233 + 0.990950i −0.353666 + 0.471773i −0.294919 − 0.0221011i
19.7 0.0448648 − 0.998993i −1.37126 + 0.685823i −0.995974 − 0.0896393i 0.0108198 − 0.361517i 0.623611 + 1.40065i −3.91186 + 0.831491i −0.134233 + 0.990950i −0.389453 + 0.519511i −0.360667 − 0.0270283i
19.8 0.0448648 − 0.998993i −1.32098 + 0.660673i −0.995974 − 0.0896393i 0.0810958 − 2.70962i 0.600742 + 1.34929i 0.698881 − 0.148552i −0.134233 + 0.990950i −0.490972 + 0.654933i −2.70325 − 0.202581i
19.9 0.0448648 − 0.998993i −0.699586 + 0.349890i −0.995974 − 0.0896393i −0.115801 + 3.86920i 0.318151 + 0.714579i 4.15699 − 0.883595i −0.134233 + 0.990950i −1.43247 + 1.91084i 3.86011 + 0.289275i
19.10 0.0448648 − 0.998993i 0.0353026 − 0.0176562i −0.995974 − 0.0896393i 0.123942 − 4.14122i −0.0160546 − 0.0360591i 2.63774 − 0.560668i −0.134233 + 0.990950i −1.79853 + 2.39915i −4.13149 − 0.309613i
19.11 0.0448648 − 0.998993i 0.188383 − 0.0942175i −0.995974 − 0.0896393i −0.0275146 + 0.919334i −0.0856709 − 0.192420i −0.931486 + 0.197994i −0.134233 + 0.990950i −1.77285 + 2.36490i 0.917174 + 0.0687327i
19.12 0.0448648 − 0.998993i 0.361471 − 0.180786i −0.995974 − 0.0896393i −0.114175 + 3.81487i −0.164386 − 0.369217i −1.31468 + 0.279445i −0.134233 + 0.990950i −1.70149 + 2.26970i 3.80590 + 0.285213i
19.13 0.0448648 − 0.998993i 0.388918 − 0.194513i −0.995974 − 0.0896393i 0.0793599 − 2.65162i −0.176869 − 0.397253i −2.50517 + 0.532491i −0.134233 + 0.990950i −1.68604 + 2.24910i −2.64539 − 0.198244i
19.14 0.0448648 − 0.998993i 0.765068 − 0.382641i −0.995974 − 0.0896393i −0.0302355 + 1.01025i −0.347931 − 0.781465i 3.90110 − 0.829204i −0.134233 + 0.990950i −1.36055 + 1.81491i 1.00787 + 0.0755295i
19.15 0.0448648 − 0.998993i 0.890379 − 0.445314i −0.995974 − 0.0896393i −0.0198343 + 0.662716i −0.404918 − 0.909462i 1.76747 − 0.375687i −0.134233 + 0.990950i −1.20499 + 1.60740i 0.661159 + 0.0495470i
19.16 0.0448648 − 0.998993i 1.12595 − 0.563133i −0.995974 − 0.0896393i −0.0545429 + 1.82242i −0.512050 − 1.15008i −1.59118 + 0.338215i −0.134233 + 0.990950i −0.848815 + 1.13228i 1.81814 + 0.136250i
19.17 0.0448648 − 0.998993i 1.73030 − 0.865392i −0.995974 − 0.0896393i −0.0352887 + 1.17908i −0.786891 − 1.76739i −3.27691 + 0.696528i −0.134233 + 0.990950i 0.445576 − 0.594377i 1.17631 + 0.0881526i
19.18 0.0448648 − 0.998993i 2.24014 − 1.12038i −0.995974 − 0.0896393i 0.0753680 − 2.51824i −1.01875 − 2.28815i 1.62617 − 0.345654i −0.134233 + 0.990950i 1.96351 − 2.61922i −2.51232 − 0.188273i
19.19 0.0448648 − 0.998993i 2.34011 − 1.17038i −0.995974 − 0.0896393i 0.105462 − 3.52375i −1.06421 − 2.39026i −2.27160 + 0.482843i −0.134233 + 0.990950i 2.30685 − 3.07722i −3.51547 − 0.263448i
19.20 0.0448648 − 0.998993i 2.46246 − 1.23157i −0.995974 − 0.0896393i 0.0469793 − 1.56970i −1.11985 − 2.51523i 1.40846 − 0.299377i −0.134233 + 0.990950i 2.74747 − 3.66500i −1.56601 − 0.117356i
See next 80 embeddings (of 1056 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
473.bf even 210 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.bf.a ✓ 1056
11.d odd 10 1 946.2.bf.b yes 1056
43.h odd 42 1 946.2.bf.b yes 1056
473.bf even 210 1 inner 946.2.bf.a ✓ 1056
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.bf.a ✓ 1056 1.a even 1 1 trivial
946.2.bf.a ✓ 1056 473.bf even 210 1 inner
946.2.bf.b yes 1056 11.d odd 10 1
946.2.bf.b yes 1056 43.h odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{1056} - 43 T_{3}^{1054} + 40 T_{3}^{1053} + 1034 T_{3}^{1052} - 1636 T_{3}^{1051} + \cdots + 48\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display