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Properties

Label 946.2.f.f
Level $946$
Weight $2$
Character orbit 946.f
Analytic conductor $7.554$
Analytic rank $0$
Dimension $32$
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(345,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.345");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.f (of order \(5\), degree \(4\), 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: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{2} - 2 q^{3} - 8 q^{4} + 2 q^{5} + 13 q^{6} + 10 q^{7} - 8 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{2} - 2 q^{3} - 8 q^{4} + 2 q^{5} + 13 q^{6} + 10 q^{7} - 8 q^{8} - 14 q^{9} + 2 q^{10} + q^{11} - 22 q^{12} + 11 q^{13} + 10 q^{14} + 8 q^{15} - 8 q^{16} - 13 q^{17} + q^{18} + 39 q^{19} + 2 q^{20} - 2 q^{21} + 6 q^{22} - 26 q^{23} + 13 q^{24} - 28 q^{25} - 4 q^{26} + 28 q^{27} - 16 q^{29} + 8 q^{30} + 7 q^{31} + 32 q^{32} + 4 q^{33} + 12 q^{34} - 52 q^{35} + q^{36} + 20 q^{37} + 14 q^{38} + 6 q^{39} - 3 q^{40} + q^{41} + 8 q^{42} + 32 q^{43} + 11 q^{44} + 64 q^{45} + 14 q^{46} + 47 q^{47} - 2 q^{48} + 4 q^{49} - 3 q^{50} + 8 q^{51} - 4 q^{52} + 25 q^{53} - 82 q^{54} - 11 q^{55} - 20 q^{56} - 81 q^{57} - 16 q^{58} + 45 q^{59} - 2 q^{60} + 2 q^{62} + 36 q^{63} - 8 q^{64} - 106 q^{65} - q^{66} - 48 q^{67} - 13 q^{68} - 35 q^{69} + 48 q^{70} + 45 q^{71} - 14 q^{72} - 20 q^{73} + 20 q^{74} + 66 q^{75} - 106 q^{76} - 10 q^{77} - 34 q^{78} - 38 q^{79} - 3 q^{80} + 69 q^{81} + 36 q^{82} + 58 q^{83} - 7 q^{84} + 59 q^{85} - 8 q^{86} - 58 q^{87} + q^{88} - 12 q^{89} - 31 q^{90} - 9 q^{91} - q^{92} + 19 q^{93} - 18 q^{94} - 46 q^{95} - 2 q^{96} + 80 q^{97} + 44 q^{98} + 5 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
345.1 −0.809017 − 0.587785i −1.04487 + 3.21579i 0.309017 + 0.951057i 1.50570 − 1.09395i 2.73551 − 1.98747i −1.11651 − 3.43626i 0.309017 − 0.951057i −6.82247 − 4.95682i −1.86114
345.2 −0.809017 − 0.587785i −0.971791 + 2.99086i 0.309017 + 0.951057i −3.10146 + 2.25335i 2.54418 − 1.84846i 0.0661608 + 0.203622i 0.309017 − 0.951057i −5.57384 − 4.04963i 3.83362
345.3 −0.809017 − 0.587785i −0.668555 + 2.05760i 0.309017 + 0.951057i 0.639481 − 0.464610i 1.75030 − 1.27167i 0.706639 + 2.17481i 0.309017 − 0.951057i −1.35970 − 0.987879i −0.790441
345.4 −0.809017 − 0.587785i −0.456418 + 1.40471i 0.309017 + 0.951057i −2.17260 + 1.57849i 1.19492 − 0.868160i 1.16598 + 3.58851i 0.309017 − 0.951057i 0.662154 + 0.481083i 2.68548
345.5 −0.809017 − 0.587785i −0.367924 + 1.13235i 0.309017 + 0.951057i 3.00998 − 2.18688i 0.963238 − 0.699834i −0.576083 − 1.77300i 0.309017 − 0.951057i 1.28019 + 0.930114i −3.72054
345.6 −0.809017 − 0.587785i −0.0257000 + 0.0790964i 0.309017 + 0.951057i 0.508232 − 0.369252i 0.0672834 − 0.0488843i 0.741554 + 2.28227i 0.309017 − 0.951057i 2.42146 + 1.75929i −0.628210
345.7 −0.809017 − 0.587785i 0.192259 − 0.591712i 0.309017 + 0.951057i 1.83391 − 1.33241i −0.503340 + 0.365698i −1.10745 − 3.40837i 0.309017 − 0.951057i 2.11389 + 1.53583i −2.26684
345.8 −0.809017 − 0.587785i 0.606934 − 1.86795i 0.309017 + 0.951057i −1.72324 + 1.25201i −1.58897 + 1.15446i −0.734395 − 2.26024i 0.309017 − 0.951057i −0.693817 − 0.504087i 2.13004
603.1 0.309017 − 0.951057i −1.58668 + 1.15279i −0.809017 − 0.587785i −1.10927 − 3.41398i 0.606056 + 1.86525i 3.13171 + 2.27532i −0.809017 + 0.587785i 0.261571 − 0.805031i −3.58967
603.2 0.309017 − 0.951057i −1.40694 + 1.02220i −0.809017 − 0.587785i 1.22605 + 3.77340i 0.537403 + 1.65396i 3.18384 + 2.31320i −0.809017 + 0.587785i 0.00753266 − 0.0231831i 3.96759
603.3 0.309017 − 0.951057i −0.878674 + 0.638394i −0.809017 − 0.587785i −0.253759 − 0.780991i 0.335624 + 1.03294i −2.84064 − 2.06384i −0.809017 + 0.587785i −0.562530 + 1.73129i −0.821183
603.4 0.309017 − 0.951057i −0.807447 + 0.586645i −0.809017 − 0.587785i 0.417740 + 1.28567i 0.308417 + 0.949211i −0.896917 − 0.651649i −0.809017 + 0.587785i −0.619232 + 1.90580i 1.35184
603.5 0.309017 − 0.951057i 0.746572 − 0.542416i −0.809017 − 0.587785i −1.28481 − 3.95424i −0.285165 − 0.877648i −1.73594 − 1.26124i −0.809017 + 0.587785i −0.663897 + 2.04326i −4.15774
603.6 0.309017 − 0.951057i 1.33010 − 0.966372i −0.809017 − 0.587785i 1.04393 + 3.21289i −0.508052 − 1.56362i 0.756798 + 0.549846i −0.809017 + 0.587785i −0.0917685 + 0.282435i 3.37823
603.7 0.309017 − 0.951057i 1.93981 − 1.40935i −0.809017 − 0.587785i 0.315572 + 0.971229i −0.740941 − 2.28038i 3.96946 + 2.88398i −0.809017 + 0.587785i 0.849530 − 2.61458i 1.02121
603.8 0.309017 − 0.951057i 2.39933 − 1.74321i −0.809017 − 0.587785i 0.144543 + 0.444857i −0.916462 − 2.82058i 0.285788 + 0.207637i −0.809017 + 0.587785i 1.79093 − 5.51191i 0.467750
775.1 0.309017 + 0.951057i −1.58668 − 1.15279i −0.809017 + 0.587785i −1.10927 + 3.41398i 0.606056 − 1.86525i 3.13171 − 2.27532i −0.809017 − 0.587785i 0.261571 + 0.805031i −3.58967
775.2 0.309017 + 0.951057i −1.40694 − 1.02220i −0.809017 + 0.587785i 1.22605 − 3.77340i 0.537403 − 1.65396i 3.18384 − 2.31320i −0.809017 − 0.587785i 0.00753266 + 0.0231831i 3.96759
775.3 0.309017 + 0.951057i −0.878674 − 0.638394i −0.809017 + 0.587785i −0.253759 + 0.780991i 0.335624 − 1.03294i −2.84064 + 2.06384i −0.809017 − 0.587785i −0.562530 − 1.73129i −0.821183
775.4 0.309017 + 0.951057i −0.807447 − 0.586645i −0.809017 + 0.587785i 0.417740 − 1.28567i 0.308417 − 0.949211i −0.896917 + 0.651649i −0.809017 − 0.587785i −0.619232 − 1.90580i 1.35184
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 345.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.f.f ✓ 32
11.c even 5 1 inner 946.2.f.f ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.f.f ✓ 32 1.a even 1 1 trivial
946.2.f.f ✓ 32 11.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 2 T_{3}^{31} + 21 T_{3}^{30} + 4 T_{3}^{29} + 165 T_{3}^{28} + 98 T_{3}^{27} + 1931 T_{3}^{26} + \cdots + 26896 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display