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Properties

Label 975.2.bg.b
Level $975$
Weight $2$
Character orbit 975.bg
Analytic conductor $7.785$
Analytic rank $0$
Dimension $136$
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] = [975,2,Mod(79,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bg (of order \(10\), 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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(34\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q + 38 q^{4} + 2 q^{5} - 2 q^{6} + 30 q^{8} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q + 38 q^{4} + 2 q^{5} - 2 q^{6} + 30 q^{8} + 34 q^{9} - 4 q^{10} - 2 q^{11} + 24 q^{14} - 70 q^{16} - 10 q^{17} + 20 q^{19} + 58 q^{20} - 12 q^{21} - 20 q^{22} + 30 q^{23} - 24 q^{24} - 10 q^{25} + 8 q^{26} - 30 q^{28} + 10 q^{29} + 16 q^{30} - 26 q^{31} + 44 q^{34} - 22 q^{35} - 38 q^{36} - 30 q^{38} + 34 q^{39} - 36 q^{40} - 26 q^{41} + 10 q^{42} - 2 q^{44} - 2 q^{45} - 12 q^{46} - 40 q^{47} - 248 q^{49} + 30 q^{50} + 10 q^{53} + 2 q^{54} - 42 q^{55} - 130 q^{58} + 10 q^{60} - 24 q^{61} + 10 q^{63} + 92 q^{64} - 10 q^{65} + 4 q^{66} + 40 q^{67} + 194 q^{70} + 4 q^{71} + 30 q^{72} - 20 q^{73} - 36 q^{74} + 212 q^{76} + 20 q^{79} + 12 q^{80} - 34 q^{81} - 130 q^{83} + 8 q^{84} - 6 q^{85} + 36 q^{86} + 300 q^{88} + 72 q^{89} - 26 q^{90} - 12 q^{91} - 100 q^{92} + 66 q^{94} + 118 q^{95} - 4 q^{96} - 40 q^{97} - 90 q^{98} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1 −2.57944 − 0.838110i −0.587785 + 0.809017i 4.33304 + 3.14814i 1.68351 − 1.47166i 2.19420 − 1.59418i 4.86903i −5.34995 − 7.36358i −0.309017 − 0.951057i −5.57593 + 2.38509i
79.2 −2.49676 − 0.811245i 0.587785 − 0.809017i 3.95764 + 2.87539i −1.43951 + 1.71108i −2.12387 + 1.54308i − 3.43598i −4.46244 − 6.14203i −0.309017 − 0.951057i 4.98222 − 3.10436i
79.3 −2.45061 − 0.796250i 0.587785 − 0.809017i 3.75342 + 2.72702i −0.960674 − 2.01918i −2.08461 + 1.51456i − 0.520604i −3.99764 − 5.50228i −0.309017 − 0.951057i 0.746458 + 5.71316i
79.4 −2.22364 − 0.722504i −0.587785 + 0.809017i 2.80452 + 2.03760i −2.01577 − 0.967828i 1.89154 − 1.37428i − 1.27092i −2.01550 − 2.77409i −0.309017 − 0.951057i 3.78307 + 3.60850i
79.5 −2.07367 − 0.673777i −0.587785 + 0.809017i 2.22810 + 1.61881i 0.507883 + 2.17763i 1.76397 − 1.28160i 1.95104i −0.966438 − 1.33019i −0.309017 − 0.951057i 0.414051 − 4.85788i
79.6 −2.04297 − 0.663802i 0.587785 − 0.809017i 2.11507 + 1.53669i 2.09915 − 0.770431i −1.73786 + 1.26263i 0.800481i −0.775719 − 1.06769i −0.309017 − 0.951057i −4.79992 + 0.180548i
79.7 −1.76521 − 0.573551i −0.587785 + 0.809017i 1.16896 + 0.849302i 2.17974 + 0.498714i 1.50158 − 1.09096i − 0.449481i 0.605571 + 0.833497i −0.309017 − 0.951057i −3.56166 − 2.13053i
79.8 −1.75939 − 0.571660i 0.587785 − 0.809017i 1.15062 + 0.835977i 1.65967 + 1.49850i −1.49663 + 1.08736i − 4.74051i 0.628226 + 0.864679i −0.309017 − 0.951057i −2.06337 − 3.58521i
79.9 −1.44257 − 0.468718i 0.587785 − 0.809017i 0.243268 + 0.176745i −2.19062 + 0.448513i −1.22712 + 0.891555i − 0.980720i 1.51502 + 2.08525i −0.309017 − 0.951057i 3.37035 + 0.379776i
79.10 −1.29375 − 0.420365i 0.587785 − 0.809017i −0.120952 − 0.0878770i −0.0891634 − 2.23429i −1.10053 + 0.799581i − 0.00556712i 1.71870 + 2.36559i −0.309017 − 0.951057i −0.823861 + 2.92809i
79.11 −1.19803 − 0.389265i 0.587785 − 0.809017i −0.334274 − 0.242864i 0.321112 + 2.21289i −1.01911 + 0.740426i 3.87714i 1.78679 + 2.45930i −0.309017 − 0.951057i 0.476697 − 2.77612i
79.12 −1.18072 − 0.383639i −0.587785 + 0.809017i −0.371116 − 0.269631i −0.715875 + 2.11838i 1.00438 − 0.729724i 3.02751i 1.79419 + 2.46949i −0.309017 − 0.951057i 1.65794 − 2.22657i
79.13 −1.06441 − 0.345848i −0.587785 + 0.809017i −0.604675 − 0.439322i 0.768956 − 2.09969i 0.905442 − 0.657842i 0.854409i 1.80737 + 2.48763i −0.309017 − 0.951057i −1.54466 + 1.96899i
79.14 −0.682990 − 0.221917i −0.587785 + 0.809017i −1.20081 − 0.872437i −2.16913 − 0.543014i 0.580986 − 0.422111i 1.76028i 1.47075 + 2.02432i −0.309017 − 0.951057i 1.36099 + 0.852240i
79.15 −0.614207 − 0.199568i −0.587785 + 0.809017i −1.28061 − 0.930418i −0.982367 − 2.00872i 0.522476 − 0.379601i − 4.57857i 1.36008 + 1.87199i −0.309017 − 0.951057i 0.202500 + 1.42982i
79.16 −0.450262 − 0.146299i 0.587785 − 0.809017i −1.43670 − 1.04382i 2.23152 + 0.142507i −0.383016 + 0.278277i 2.26795i 1.05074 + 1.44622i −0.309017 − 0.951057i −0.983922 − 0.390635i
79.17 0.123282 + 0.0400566i 0.587785 − 0.809017i −1.60444 − 1.16569i −0.938461 − 2.02960i 0.104870 − 0.0761922i − 4.99962i −0.303489 − 0.417716i −0.309017 − 0.951057i −0.0343959 − 0.287804i
79.18 0.138199 + 0.0449037i −0.587785 + 0.809017i −1.60095 − 1.16316i 0.966599 + 2.01635i −0.117559 + 0.0854118i 1.75676i −0.339844 − 0.467755i −0.309017 − 0.951057i 0.0430416 + 0.322063i
79.19 0.177813 + 0.0577751i 0.587785 − 0.809017i −1.58975 − 1.15502i −1.90791 + 1.16613i 0.151257 − 0.109895i 2.49976i −0.435737 − 0.599741i −0.309017 − 0.951057i −0.406626 + 0.0971239i
79.20 0.383423 + 0.124582i −0.587785 + 0.809017i −1.48654 − 1.08004i −1.33309 + 1.79524i −0.326159 + 0.236968i − 3.47748i −0.909358 − 1.25162i −0.309017 − 0.951057i −0.734790 + 0.522256i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bg.b ✓ 136
25.e even 10 1 inner 975.2.bg.b ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bg.b ✓ 136 1.a even 1 1 trivial
975.2.bg.b ✓ 136 25.e even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{136} - 53 T_{2}^{134} - 10 T_{2}^{133} + 1547 T_{2}^{132} + 530 T_{2}^{131} - 32902 T_{2}^{130} + \cdots + 11885142361 \) acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\). Copy content Toggle raw display