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Properties

Label 975.2.v.d
Level $975$
Weight $2$
Character orbit 975.v
Analytic conductor $7.785$
Analytic rank $0$
Dimension $68$
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: \(68\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + q^{2} - 17 q^{3} - 19 q^{4} + q^{5} + q^{6} + 12 q^{7} - 6 q^{8} - 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + q^{2} - 17 q^{3} - 19 q^{4} + q^{5} + q^{6} + 12 q^{7} - 6 q^{8} - 17 q^{9} + 10 q^{10} - 16 q^{11} - 19 q^{12} + 17 q^{13} + 26 q^{14} + q^{15} - 3 q^{16} - 7 q^{17} - 4 q^{18} + 4 q^{19} + 31 q^{20} - 3 q^{21} - 14 q^{22} - 15 q^{23} - 6 q^{24} + 3 q^{25} + 4 q^{26} - 17 q^{27} + 17 q^{28} + 3 q^{29} - 20 q^{30} + 12 q^{31} + 84 q^{32} + 4 q^{33} - 22 q^{34} + 60 q^{35} - 19 q^{36} + 18 q^{38} + 17 q^{39} - 62 q^{40} - 37 q^{41} - 34 q^{42} + 36 q^{43} - 45 q^{44} - 4 q^{45} - 52 q^{46} - 10 q^{47} - 3 q^{48} + 76 q^{49} - 41 q^{50} - 2 q^{51} + 19 q^{52} + q^{53} + q^{54} - 5 q^{55} - 55 q^{56} + 24 q^{57} + 3 q^{58} - 35 q^{59} - 29 q^{60} - 54 q^{61} - 26 q^{62} - 3 q^{63} + 24 q^{64} + 4 q^{65} + 11 q^{66} - 68 q^{67} + 30 q^{68} + 10 q^{69} + 97 q^{70} - 51 q^{71} - 6 q^{72} - 26 q^{73} + 48 q^{74} - 2 q^{75} + 70 q^{76} + 9 q^{77} - q^{78} + 18 q^{79} - 104 q^{80} - 17 q^{81} + 254 q^{82} + 18 q^{83} - 28 q^{84} + 30 q^{85} - 24 q^{86} - 12 q^{87} - 28 q^{88} - 10 q^{89} - 5 q^{90} + 3 q^{91} - 50 q^{92} - 28 q^{93} - q^{94} + 83 q^{95} - 36 q^{96} - 42 q^{97} + 25 q^{98} + 24 q^{99}+O(q^{100}) \) 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 −0.830155 + 2.55496i −0.809017 − 0.587785i −4.22060 − 3.06645i −0.489608 + 2.18181i 2.17337 − 1.57905i −4.83531 6.99165 − 5.07973i 0.309017 + 0.951057i −5.16797 − 3.06217i
196.2 −0.736759 + 2.26751i −0.809017 − 0.587785i −2.98076 − 2.16565i −1.43714 − 1.71307i 1.92886 − 1.40140i 0.178618 3.24901 − 2.36055i 0.309017 + 0.951057i 4.94324 − 1.99662i
196.3 −0.695635 + 2.14094i −0.809017 − 0.587785i −2.48170 − 1.80306i −2.14557 + 0.629717i 1.82120 − 1.32318i 1.57504 1.94421 − 1.41255i 0.309017 + 0.951057i 0.144341 − 5.03159i
196.4 −0.608017 + 1.87128i −0.809017 − 0.587785i −1.51398 − 1.09997i 2.03157 + 0.934190i 1.59181 − 1.15652i −2.72251 −0.204726 + 0.148742i 0.309017 + 0.951057i −2.98337 + 3.23365i
196.5 −0.596991 + 1.83735i −0.809017 − 0.587785i −1.40142 − 1.01819i 1.40134 − 1.74249i 1.56294 − 1.13554i 5.14586 −0.418482 + 0.304045i 0.309017 + 0.951057i 2.36497 + 3.61499i
196.6 −0.305313 + 0.939657i −0.809017 − 0.587785i 0.828295 + 0.601791i 2.05988 − 0.869987i 0.799320 − 0.580740i −2.30384 −2.41701 + 1.75606i 0.309017 + 0.951057i 0.188580 + 2.20120i
196.7 −0.209599 + 0.645078i −0.809017 − 0.587785i 1.24584 + 0.905155i −1.50384 − 1.65483i 0.548736 − 0.398680i −0.482639 −1.94250 + 1.41131i 0.309017 + 0.951057i 1.38270 − 0.623248i
196.8 −0.165931 + 0.510684i −0.809017 − 0.587785i 1.38477 + 1.00609i 0.333153 − 2.21111i 0.434414 − 0.315620i −1.37474 −1.61240 + 1.17148i 0.309017 + 0.951057i 1.07390 + 0.537028i
196.9 −0.0764222 + 0.235203i −0.809017 − 0.587785i 1.56855 + 1.13962i −0.610244 + 2.15119i 0.200076 − 0.145364i 3.56621 −0.788067 + 0.572564i 0.309017 + 0.951057i −0.459330 − 0.307930i
196.10 0.106105 − 0.326556i −0.809017 − 0.587785i 1.52265 + 1.10627i −2.22540 + 0.218144i −0.277786 + 0.201823i −4.00008 1.07839 − 0.783498i 0.309017 + 0.951057i −0.164889 + 0.749865i
196.11 0.218515 − 0.672521i −0.809017 − 0.587785i 1.21350 + 0.881659i 2.22941 − 0.172418i −0.572080 + 0.415641i 3.28057 2.00226 − 1.45473i 0.309017 + 0.951057i 0.371205 − 1.53700i
196.12 0.346147 − 1.06533i −0.809017 − 0.587785i 0.602921 + 0.438048i −1.89299 − 1.19020i −0.906225 + 0.658411i 2.45466 2.48782 − 1.80750i 0.309017 + 0.951057i −1.92321 + 1.60468i
196.13 0.368795 − 1.13503i −0.809017 − 0.587785i 0.465741 + 0.338380i −0.911086 + 2.04204i −0.965518 + 0.701490i −1.17603 2.48687 − 1.80682i 0.309017 + 0.951057i 1.98178 + 1.78721i
196.14 0.512569 − 1.57753i −0.809017 − 0.587785i −0.607825 − 0.441611i 2.19347 + 0.434377i −1.34192 + 0.974964i 1.05950 1.67564 − 1.21742i 0.309017 + 0.951057i 1.80955 − 3.23761i
196.15 0.715069 − 2.20076i −0.809017 − 0.587785i −2.71397 − 1.97182i 0.229002 − 2.22431i −1.87208 + 1.36014i 0.291330 −2.53602 + 1.84252i 0.309017 + 0.951057i −4.73142 − 2.09451i
196.16 0.805170 − 2.47806i −0.809017 − 0.587785i −3.87444 − 2.81495i 1.38259 + 1.75740i −2.10796 + 1.53152i 4.82208 −5.87927 + 4.27154i 0.309017 + 0.951057i 5.46816 − 2.01113i
196.17 0.843435 − 2.59582i −0.809017 − 0.587785i −4.40889 − 3.20325i −2.07158 + 0.841764i −2.20814 + 1.60431i −2.47872 −7.61739 + 5.53436i 0.309017 + 0.951057i 0.437832 + 6.08742i
391.1 −2.21153 − 1.60677i 0.309017 + 0.951057i 1.69112 + 5.20475i 0.315069 − 2.21376i 0.844730 − 2.59981i −0.242268 2.93340 − 9.02809i −0.809017 + 0.587785i −4.25379 + 4.38956i
391.2 −1.84951 − 1.34375i 0.309017 + 0.951057i 0.996997 + 3.06844i −2.10731 + 0.747832i 0.706450 − 2.17423i 2.17111 0.866356 − 2.66637i −0.809017 + 0.587785i 4.90239 + 1.44857i
391.3 −1.67590 − 1.21761i 0.309017 + 0.951057i 0.708030 + 2.17909i 2.22875 − 0.180701i 0.640138 − 1.97014i 4.08964 0.186430 − 0.573773i −0.809017 + 0.587785i −3.95520 − 2.41093i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 196.17
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.d ✓ 68
25.d even 5 1 inner 975.2.v.d ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.v.d ✓ 68 1.a even 1 1 trivial
975.2.v.d ✓ 68 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{68} - T_{2}^{67} + 27 T_{2}^{66} - 26 T_{2}^{65} + 427 T_{2}^{64} - 392 T_{2}^{63} + \cdots + 126025 \) acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\). Copy content Toggle raw display