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Properties

Label 946.2.v.a
Level $946$
Weight $2$
Character orbit 946.v
Analytic conductor $7.554$
Analytic rank $0$
Dimension $504$
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(47,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([56, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.v (of order \(35\), degree \(24\), 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: \(504\)
Relative dimension: \(21\) over \(\Q(\zeta_{35})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 504 q - 21 q^{2} + 21 q^{4} + 4 q^{5} - 6 q^{7} - 21 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 504 q - 21 q^{2} + 21 q^{4} + 4 q^{5} - 6 q^{7} - 21 q^{8} + 19 q^{9} - 4 q^{10} - 4 q^{11} - 18 q^{13} - 15 q^{14} - 6 q^{15} + 21 q^{16} + 14 q^{17} - 19 q^{18} + 11 q^{19} + 4 q^{20} + 16 q^{21} - 8 q^{22} + 62 q^{23} - q^{25} - 37 q^{26} + 6 q^{27} + 4 q^{28} + 14 q^{29} - 22 q^{30} + 30 q^{31} + 84 q^{32} - 78 q^{33} - 24 q^{34} - 44 q^{35} - 114 q^{36} + 72 q^{37} + 11 q^{38} + 90 q^{39} - 15 q^{40} + 26 q^{41} + 4 q^{42} - 80 q^{43} - 6 q^{44} - 76 q^{45} - q^{46} + 27 q^{47} - 84 q^{49} + 76 q^{50} - 30 q^{51} - 12 q^{52} + 22 q^{53} + 24 q^{54} + 47 q^{55} - 4 q^{56} + 92 q^{57} - 84 q^{59} - 28 q^{60} + 53 q^{61} - 22 q^{62} - 80 q^{63} + 21 q^{64} - 6 q^{65} - 51 q^{66} + 48 q^{67} + 21 q^{68} + 12 q^{69} - 46 q^{70} - 15 q^{71} - 19 q^{72} - 90 q^{73} + 26 q^{74} - 195 q^{75} + 14 q^{76} + 33 q^{77} - 32 q^{78} - 76 q^{79} - 6 q^{80} + 65 q^{81} - 36 q^{82} - 88 q^{83} - 4 q^{84} + 250 q^{85} - 27 q^{86} - 204 q^{87} - 8 q^{88} - 78 q^{89} + 14 q^{90} + 64 q^{91} + 10 q^{92} - 368 q^{93} + 24 q^{94} - 4 q^{95} + 46 q^{97} + 88 q^{98} + 318 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0.0448648 − 0.998993i −1.72338 + 2.61081i −0.995974 − 0.0896393i −0.589180 − 1.09488i 2.53086 + 1.83878i −0.350552 + 1.07889i −0.134233 + 0.990950i −2.66721 − 6.24024i −1.12021 + 0.539465i
47.2 0.0448648 − 0.998993i −1.59405 + 2.41488i −0.995974 − 0.0896393i −1.50402 − 2.79493i 2.34093 + 1.70079i 1.50198 − 4.62261i −0.134233 + 0.990950i −2.11158 − 4.94029i −2.85960 + 1.37711i
47.3 0.0448648 − 0.998993i −1.39977 + 2.12056i −0.995974 − 0.0896393i 0.353655 + 0.657201i 2.05562 + 1.49350i 0.318185 − 0.979273i −0.134233 + 0.990950i −1.35834 − 3.17800i 0.672406 − 0.323814i
47.4 0.0448648 − 0.998993i −1.24373 + 1.88417i −0.995974 − 0.0896393i 0.795189 + 1.47771i 1.82647 + 1.32701i −1.42933 + 4.39903i −0.134233 + 0.990950i −0.824157 − 1.92821i 1.51190 − 0.728091i
47.5 0.0448648 − 0.998993i −1.20426 + 1.82437i −0.995974 − 0.0896393i 1.14308 + 2.12419i 1.76851 + 1.28490i 0.102370 − 0.315062i −0.134233 + 0.990950i −0.699023 − 1.63545i 2.17334 − 1.04662i
47.6 0.0448648 − 0.998993i −0.982242 + 1.48803i −0.995974 − 0.0896393i 1.31404 + 2.44189i 1.44247 + 1.04801i 1.05473 − 3.24613i −0.134233 + 0.990950i −0.0703681 − 0.164634i 2.49838 − 1.20316i
47.7 0.0448648 − 0.998993i −0.922111 + 1.39694i −0.995974 − 0.0896393i −1.60946 − 2.99087i 1.35416 + 0.983856i −0.757463 + 2.33123i −0.134233 + 0.990950i 0.0779259 + 0.182317i −3.06007 + 1.47365i
47.8 0.0448648 − 0.998993i −0.426599 + 0.646269i −0.995974 − 0.0896393i 0.618420 + 1.14922i 0.626479 + 0.455164i −1.15810 + 3.56428i −0.134233 + 0.990950i 0.943397 + 2.20719i 1.17581 − 0.566238i
47.9 0.0448648 − 0.998993i −0.278525 + 0.421948i −0.995974 − 0.0896393i −0.393105 − 0.730511i 0.409027 + 0.297175i 0.306372 − 0.942916i −0.134233 + 0.990950i 1.07861 + 2.52354i −0.747412 + 0.359935i
47.10 0.0448648 − 0.998993i −0.260256 + 0.394271i −0.995974 − 0.0896393i −0.534106 − 0.992535i 0.382198 + 0.277683i 0.853890 − 2.62800i −0.134233 + 0.990950i 1.09136 + 2.55336i −1.01550 + 0.489038i
47.11 0.0448648 − 0.998993i −0.0378410 + 0.0573266i −0.995974 − 0.0896393i −1.06851 − 1.98563i 0.0555711 + 0.0403748i −0.109829 + 0.338020i −0.134233 + 0.990950i 1.17722 + 2.75424i −2.03157 + 0.978354i
47.12 0.0448648 − 0.998993i 0.224331 − 0.339847i −0.995974 − 0.0896393i 1.92887 + 3.58443i −0.329440 − 0.239352i 0.224469 − 0.690843i −0.134233 + 0.990950i 1.11390 + 2.60611i 3.66736 − 1.76611i
47.13 0.0448648 − 0.998993i 0.657152 − 0.995543i −0.995974 − 0.0896393i −0.247370 − 0.459690i −0.965058 − 0.701155i −1.08239 + 3.33125i −0.134233 + 0.990950i 0.619818 + 1.45014i −0.470326 + 0.226497i
47.14 0.0448648 − 0.998993i 0.719111 − 1.08941i −0.995974 − 0.0896393i 1.01933 + 1.89424i −1.05605 − 0.767263i −0.377666 + 1.16234i −0.134233 + 0.990950i 0.509390 + 1.19178i 1.93807 − 0.933324i
47.15 0.0448648 − 0.998993i 0.737039 − 1.11657i −0.995974 − 0.0896393i 0.300454 + 0.558337i −1.08237 − 0.786391i 1.48865 − 4.58159i −0.134233 + 0.990950i 0.475582 + 1.11268i 0.571255 − 0.275102i
47.16 0.0448648 − 0.998993i 0.753512 − 1.14152i −0.995974 − 0.0896393i −0.174196 − 0.323710i −1.10657 − 0.803968i −1.10084 + 3.38804i −0.134233 + 0.990950i 0.443783 + 1.03828i −0.331199 + 0.159497i
47.17 0.0448648 − 0.998993i 0.929838 − 1.40864i −0.995974 − 0.0896393i −1.88534 − 3.50354i −1.36551 − 0.992100i 0.422867 − 1.30145i −0.134233 + 0.990950i 0.0593952 + 0.138962i −3.58460 + 1.72625i
47.18 0.0448648 − 0.998993i 1.31516 − 1.99239i −0.995974 − 0.0896393i 1.26080 + 2.34297i −1.93138 − 1.40323i −0.104866 + 0.322745i −0.134233 + 0.990950i −1.06088 − 2.48204i 2.39717 − 1.15442i
47.19 0.0448648 − 0.998993i 1.33422 − 2.02125i −0.995974 − 0.0896393i −1.13037 − 2.10058i −1.95936 − 1.42356i −0.694040 + 2.13604i −0.134233 + 0.990950i −1.12625 − 2.63500i −2.14918 + 1.03499i
47.20 0.0448648 − 0.998993i 1.67084 − 2.53121i −0.995974 − 0.0896393i −0.775526 − 1.44117i −2.45370 − 1.78272i 0.728172 − 2.24108i −0.134233 + 0.990950i −2.43625 − 5.69990i −1.47451 + 0.710087i
See next 80 embeddings (of 504 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.21
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
43.e even 7 1 inner
473.v even 35 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.v.a ✓ 504
11.c even 5 1 inner 946.2.v.a ✓ 504
43.e even 7 1 inner 946.2.v.a ✓ 504
473.v even 35 1 inner 946.2.v.a ✓ 504
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.v.a ✓ 504 1.a even 1 1 trivial
946.2.v.a ✓ 504 11.c even 5 1 inner
946.2.v.a ✓ 504 43.e even 7 1 inner
946.2.v.a ✓ 504 473.v even 35 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{504} - 41 T_{3}^{502} - 2 T_{3}^{501} + 687 T_{3}^{500} + 222 T_{3}^{499} - 5573 T_{3}^{498} + \cdots + 25\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display