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Properties

Label 946.2.r.d
Level $946$
Weight $2$
Character orbit 946.r
Analytic conductor $7.554$
Analytic rank $0$
Dimension $120$
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(23,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.r (of order \(21\), degree \(12\), 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: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 20 q^{2} - 9 q^{3} - 20 q^{4} - q^{5} - 5 q^{6} + 11 q^{7} + 20 q^{8} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 20 q^{2} - 9 q^{3} - 20 q^{4} - q^{5} - 5 q^{6} + 11 q^{7} + 20 q^{8} + 47 q^{9} + q^{10} - 20 q^{11} + 12 q^{12} + 7 q^{13} + 10 q^{14} - 28 q^{15} - 20 q^{16} + 22 q^{17} - 12 q^{18} - 15 q^{19} - q^{20} - 58 q^{21} + 20 q^{22} - 15 q^{23} + 2 q^{24} + 35 q^{25} - 7 q^{26} + 12 q^{27} + 11 q^{28} + 24 q^{29} + 42 q^{30} - 16 q^{31} + 20 q^{32} - 2 q^{33} - 8 q^{34} - 6 q^{35} - 79 q^{36} - q^{37} - 20 q^{38} - 6 q^{39} + q^{40} - 59 q^{41} - 82 q^{42} - 6 q^{43} + 120 q^{44} + 46 q^{45} - 62 q^{46} - 8 q^{47} - 2 q^{48} - 67 q^{49} + 70 q^{50} + 53 q^{51} - 7 q^{52} + 37 q^{53} + 2 q^{54} - q^{55} - 4 q^{56} + 40 q^{57} - 17 q^{58} - 40 q^{59} + 63 q^{60} + 14 q^{61} + 9 q^{62} - 167 q^{63} - 20 q^{64} - 14 q^{65} + 2 q^{66} + 22 q^{67} + 8 q^{68} - 38 q^{69} + 6 q^{70} + 106 q^{71} - 12 q^{72} + 20 q^{73} + q^{74} - 37 q^{75} - q^{76} - 10 q^{77} + 62 q^{78} - 29 q^{79} + 6 q^{80} + 49 q^{81} + 31 q^{82} - 8 q^{83} - 58 q^{84} - 18 q^{85} - 64 q^{86} + 176 q^{87} + 20 q^{88} - 37 q^{89} + 17 q^{90} + 8 q^{91} - 36 q^{92} - 62 q^{93} + 29 q^{94} + 35 q^{95} + 2 q^{96} + 48 q^{97} - 10 q^{98} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.623490 − 0.781831i −1.18552 − 3.02065i −0.222521 + 0.974928i 0.0114658 − 0.153000i −1.62248 + 2.81022i 1.44290 + 2.49918i 0.900969 − 0.433884i −5.51973 + 5.12156i −0.126769 + 0.0864297i
23.2 −0.623490 − 0.781831i −0.908748 − 2.31545i −0.222521 + 0.974928i −0.256096 + 3.41737i −1.24370 + 2.15415i −1.29646 − 2.24553i 0.900969 − 0.433884i −2.33634 + 2.16781i 2.83148 − 1.93047i
23.3 −0.623490 − 0.781831i −0.812745 − 2.07084i −0.222521 + 0.974928i 0.0468429 − 0.625076i −1.11231 + 1.92658i −0.738508 − 1.27913i 0.900969 − 0.433884i −1.42867 + 1.32561i −0.517910 + 0.353105i
23.4 −0.623490 − 0.781831i −0.282813 − 0.720597i −0.222521 + 0.974928i 0.151716 − 2.02451i −0.387054 + 0.670397i 0.837486 + 1.45057i 0.900969 − 0.433884i 1.75988 − 1.63293i −1.67742 + 1.14364i
23.5 −0.623490 − 0.781831i −0.186967 − 0.476385i −0.222521 + 0.974928i 0.190685 − 2.54452i −0.255881 + 0.443198i −2.44040 − 4.22689i 0.900969 − 0.433884i 2.00717 − 1.86238i −2.10827 + 1.43740i
23.6 −0.623490 − 0.781831i −0.0907222 − 0.231156i −0.222521 + 0.974928i −0.205144 + 2.73745i −0.124161 + 0.215053i 1.52149 + 2.63529i 0.900969 − 0.433884i 2.15395 − 1.99858i 2.26813 − 1.54639i
23.7 −0.623490 − 0.781831i −0.00178946 − 0.00455948i −0.222521 + 0.974928i 0.247717 − 3.30555i −0.00244903 + 0.00424185i 1.25995 + 2.18229i 0.900969 − 0.433884i 2.19914 − 2.04050i −2.73883 + 1.86730i
23.8 −0.623490 − 0.781831i 0.565102 + 1.43986i −0.222521 + 0.974928i −0.0575391 + 0.767807i 0.773390 − 1.33955i −0.879733 − 1.52374i 0.900969 − 0.433884i 0.445308 − 0.413186i 0.636170 − 0.433734i
23.9 −0.623490 − 0.781831i 0.674685 + 1.71907i −0.222521 + 0.974928i −0.0987280 + 1.31743i 0.923364 − 1.59931i −2.16690 − 3.75318i 0.900969 − 0.433884i −0.300845 + 0.279143i 1.09157 − 0.744218i
23.10 −0.623490 − 0.781831i 1.14165 + 2.90889i −0.222521 + 0.974928i −0.105650 + 1.40980i 1.56245 − 2.70624i 0.980146 + 1.69766i 0.900969 − 0.433884i −4.95911 + 4.60138i 1.16809 − 0.796393i
67.1 −0.623490 + 0.781831i −3.20029 + 0.482366i −0.222521 − 0.974928i 2.31836 − 1.58063i 1.61822 − 2.80284i −0.546995 − 0.947423i 0.900969 + 0.433884i 7.14247 − 2.20316i −0.209686 + 2.79807i
67.2 −0.623490 + 0.781831i −2.51089 + 0.378456i −0.222521 − 0.974928i −2.14143 + 1.46000i 1.26962 − 2.19905i 1.12406 + 1.94693i 0.900969 + 0.433884i 3.29461 − 1.01625i 0.193684 − 2.58453i
67.3 −0.623490 + 0.781831i −1.66427 + 0.250849i −0.222521 − 0.974928i 2.97422 − 2.02779i 0.841536 − 1.45758i −1.11974 − 1.93945i 0.900969 + 0.433884i −0.159834 + 0.0493023i −0.269007 + 3.58965i
67.4 −0.623490 + 0.781831i −1.66255 + 0.250589i −0.222521 − 0.974928i 1.14896 − 0.783349i 0.840665 − 1.45607i 1.62679 + 2.81769i 0.900969 + 0.433884i −0.165441 + 0.0510318i −0.103919 + 1.38671i
67.5 −0.623490 + 0.781831i −0.972923 + 0.146644i −0.222521 − 0.974928i −2.33496 + 1.59195i 0.491956 − 0.852093i −0.211359 − 0.366085i 0.900969 + 0.433884i −1.94164 + 0.598918i 0.211188 − 2.81811i
67.6 −0.623490 + 0.781831i 0.928187 − 0.139902i −0.222521 − 0.974928i −0.491483 + 0.335087i −0.469336 + 0.812913i −1.80982 − 3.13470i 0.900969 + 0.433884i −2.02476 + 0.624556i 0.0444527 − 0.593180i
67.7 −0.623490 + 0.781831i 1.73090 − 0.260891i −0.222521 − 0.974928i −2.53459 + 1.72805i −0.875225 + 1.51593i 1.78950 + 3.09951i 0.900969 + 0.433884i 0.0612294 − 0.0188868i 0.229244 − 3.05905i
67.8 −0.623490 + 0.781831i 2.06806 − 0.311710i −0.222521 − 0.974928i 1.73965 − 1.18607i −1.04571 + 1.81122i 1.55979 + 2.70164i 0.900969 + 0.433884i 1.31299 − 0.405005i −0.157345 + 2.09962i
67.9 −0.623490 + 0.781831i 2.20378 − 0.332166i −0.222521 − 0.974928i 0.659149 − 0.449400i −1.11434 + 1.93009i −1.95318 − 3.38301i 0.900969 + 0.433884i 1.87959 − 0.579778i −0.0596175 + 0.795540i
67.10 −0.623490 + 0.781831i 3.34631 − 0.504375i −0.222521 − 0.974928i −2.16412 + 1.47547i −1.69205 + 2.93072i −0.250453 − 0.433798i 0.900969 + 0.433884i 8.07667 − 2.49132i 0.195736 − 2.61192i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.r.d ✓ 120
43.g even 21 1 inner 946.2.r.d ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.r.d ✓ 120 1.a even 1 1 trivial
946.2.r.d ✓ 120 43.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{120} + 9 T_{3}^{119} + 2 T_{3}^{118} - 237 T_{3}^{117} - 737 T_{3}^{116} + 1672 T_{3}^{115} + \cdots + 33\!\cdots\!84 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display