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Properties

Label 975.2.v.b
Level $975$
Weight $2$
Character orbit 975.v
Analytic conductor $7.785$
Analytic rank $0$
Dimension $52$
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(196,975)]
 
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, 6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.196");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.v (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(13\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + q^{2} - 13 q^{3} - 17 q^{4} + 3 q^{5} + q^{6} - 20 q^{7} - 6 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + q^{2} - 13 q^{3} - 17 q^{4} + 3 q^{5} + q^{6} - 20 q^{7} - 6 q^{8} - 13 q^{9} + 4 q^{10} - 8 q^{11} - 7 q^{12} - 13 q^{13} - 10 q^{14} + 3 q^{15} - 9 q^{16} + q^{17} - 4 q^{18} + 20 q^{19} - 11 q^{20} + 5 q^{21} + 4 q^{22} - 15 q^{23} - 6 q^{24} + 23 q^{25} - 4 q^{26} - 13 q^{27} + 37 q^{28} - 5 q^{29} - 26 q^{30} + 26 q^{31} - 156 q^{32} + 12 q^{33} + 34 q^{34} - 12 q^{35} - 7 q^{36} + 32 q^{37} + 18 q^{38} - 13 q^{39} - 18 q^{40} + 49 q^{41} + 10 q^{42} - 76 q^{43} + 47 q^{44} - 2 q^{45} + 24 q^{46} + 22 q^{47} - 9 q^{48} - 52 q^{49} + 39 q^{50} + 6 q^{51} - 7 q^{52} + 25 q^{53} + q^{54} + 11 q^{55} + 45 q^{56} - 40 q^{57} - 13 q^{58} - 15 q^{59} + 19 q^{60} + 26 q^{61} + 62 q^{62} + 5 q^{63} + 6 q^{64} - 2 q^{65} - 11 q^{66} + 8 q^{67} + 62 q^{68} + 10 q^{69} + q^{70} + 9 q^{71} - 6 q^{72} + 46 q^{73} + 56 q^{74} + 18 q^{75} - 82 q^{76} + 49 q^{77} + q^{78} + 26 q^{79} + 60 q^{80} - 13 q^{81} - 126 q^{82} - 50 q^{83} - 8 q^{84} + 48 q^{85} + 24 q^{86} + 20 q^{87} - 62 q^{88} + 34 q^{89} - q^{90} + 5 q^{91} - 10 q^{92} - 104 q^{93} + q^{94} - 53 q^{95} + 24 q^{96} + 6 q^{97} - 47 q^{98} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
196.1 −0.852417 + 2.62347i −0.809017 − 0.587785i −4.53795 − 3.29701i 0.125937 − 2.23252i 2.23166 − 1.62139i 0.140369 8.05452 − 5.85195i 0.309017 + 0.951057i 5.74960 + 2.23343i
196.2 −0.729944 + 2.24654i −0.809017 − 0.587785i −2.89608 − 2.10412i 2.14685 + 0.625320i 1.91102 − 1.38844i 0.115049 3.01893 − 2.19338i 0.309017 + 0.951057i −2.97189 + 4.36654i
196.3 −0.535142 + 1.64700i −0.809017 − 0.587785i −0.808187 − 0.587182i −1.78604 − 1.34538i 1.40102 − 1.01790i −4.63434 −1.40246 + 1.01894i 0.309017 + 0.951057i 3.17162 − 2.22164i
196.4 −0.507569 + 1.56214i −0.809017 − 0.587785i −0.564611 − 0.410214i −0.597298 + 2.15482i 1.32883 − 0.965454i −1.83570 −1.73028 + 1.25712i 0.309017 + 0.951057i −3.06295 − 2.02678i
196.5 −0.360677 + 1.11005i −0.809017 − 0.587785i 0.515914 + 0.374834i −2.19168 − 0.443314i 0.944264 − 0.686048i 4.00658 −2.49069 + 1.80959i 0.309017 + 0.951057i 1.28259 − 2.27298i
196.6 −0.173592 + 0.534263i −0.809017 − 0.587785i 1.36273 + 0.990083i 2.16852 + 0.545458i 0.454471 − 0.330192i 2.51153 −1.67447 + 1.21657i 0.309017 + 0.951057i −0.667856 + 1.06387i
196.7 −0.0811464 + 0.249743i −0.809017 − 0.587785i 1.56225 + 1.13504i −1.80516 + 1.31961i 0.212444 − 0.154350i −1.74030 −0.835127 + 0.606755i 0.309017 + 0.951057i −0.183082 − 0.557909i
196.8 0.106420 − 0.327528i −0.809017 − 0.587785i 1.52208 + 1.10586i 1.59539 + 1.56676i −0.278612 + 0.202424i −2.33207 1.08141 − 0.785687i 0.309017 + 0.951057i 0.682940 − 0.355799i
196.9 0.129503 − 0.398568i −0.809017 − 0.587785i 1.47595 + 1.07234i 0.0328781 − 2.23583i −0.339042 + 0.246329i 2.54191 1.29662 − 0.942053i 0.309017 + 0.951057i −0.886872 − 0.302650i
196.10 0.548756 − 1.68890i −0.809017 − 0.587785i −0.933203 − 0.678011i −1.31321 + 1.80983i −1.43666 + 1.04380i −0.0522778 1.21613 − 0.883570i 0.309017 + 0.951057i 2.33598 + 3.21103i
196.11 0.650087 − 2.00076i −0.809017 − 0.587785i −1.96240 − 1.42577i 1.87342 − 1.22077i −1.70195 + 1.23654i 1.15752 −0.724458 + 0.526350i 0.309017 + 0.951057i −1.22459 − 4.54188i
196.12 0.743127 − 2.28711i −0.809017 − 0.587785i −3.06059 − 2.22365i −1.36759 − 1.76910i −1.94553 + 1.41351i −2.01853 −3.46908 + 2.52043i 0.309017 + 0.951057i −5.06241 + 1.81316i
196.13 0.753578 − 2.31928i −0.809017 − 0.587785i −3.19312 − 2.31994i 1.30897 + 1.81289i −1.97289 + 1.43339i −2.85975 −3.84106 + 2.79070i 0.309017 + 0.951057i 5.19102 − 1.66971i
391.1 −1.84780 − 1.34251i 0.309017 + 0.951057i 0.994019 + 3.05928i −1.49939 − 1.65886i 0.705799 − 2.17222i 3.36703 0.858754 − 2.64297i −0.809017 + 0.587785i 0.543537 + 5.07820i
391.2 −1.78383 − 1.29603i 0.309017 + 0.951057i 0.884332 + 2.72169i 2.16510 + 0.558886i 0.681364 − 2.09702i −1.55183 0.587172 − 1.80713i −0.809017 + 0.587785i −3.13784 − 3.80299i
391.3 −1.63846 − 1.19041i 0.309017 + 0.951057i 0.649445 + 1.99879i 1.18621 − 1.89549i 0.625838 − 1.92613i −2.67457 0.0636154 − 0.195788i −0.809017 + 0.587785i −4.19999 + 1.69361i
391.4 −0.981505 − 0.713105i 0.309017 + 0.951057i −0.163201 − 0.502281i −1.41325 + 1.73284i 0.374901 − 1.15383i 1.14267 −0.947800 + 2.91703i −0.809017 + 0.587785i 2.62281 − 0.692992i
391.5 −0.866129 − 0.629280i 0.309017 + 0.951057i −0.263847 − 0.812038i −1.59661 − 1.56552i 0.330832 − 1.01820i −1.75183 −0.944137 + 2.90576i −0.809017 + 0.587785i 0.397722 + 2.36065i
391.6 0.0369555 + 0.0268498i 0.309017 + 0.951057i −0.617389 − 1.90013i 1.33246 + 1.79570i −0.0141158 + 0.0434438i 3.80860 0.0564335 − 0.173685i −0.809017 + 0.587785i 0.00102777 + 0.102137i
391.7 0.266617 + 0.193709i 0.309017 + 0.951057i −0.584472 − 1.79882i 2.05645 − 0.878079i −0.101839 + 0.313428i −4.59361 0.396295 − 1.21967i −0.809017 + 0.587785i 0.718376 + 0.164241i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 196.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.v.b ✓ 52
25.d even 5 1 inner 975.2.v.b ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.v.b ✓ 52 1.a even 1 1 trivial
975.2.v.b ✓ 52 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{52} - T_{2}^{51} + 22 T_{2}^{50} - 21 T_{2}^{49} + 280 T_{2}^{48} - 202 T_{2}^{47} + 2697 T_{2}^{46} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\). Copy content Toggle raw display