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Properties

Label 4057.2.a.b
Level $4057$
Weight $2$
Character orbit 4057.a
Self dual yes
Analytic conductor $32.395$
Analytic rank $0$
Dimension $173$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4057,2,Mod(1,4057)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4057, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4057.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4057 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4057.a (trivial)

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: yes
Analytic conductor: \(32.3953081000\)
Analytic rank: \(0\)
Dimension: \(173\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 173 q + 21 q^{2} + 28 q^{3} + 179 q^{4} + 37 q^{5} + 16 q^{6} + 55 q^{7} + 60 q^{8} + 185 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 173 q + 21 q^{2} + 28 q^{3} + 179 q^{4} + 37 q^{5} + 16 q^{6} + 55 q^{7} + 60 q^{8} + 185 q^{9} + 27 q^{10} + 49 q^{11} + 57 q^{12} + 28 q^{13} + 48 q^{14} + 28 q^{15} + 183 q^{16} + 105 q^{17} + 42 q^{18} + 32 q^{19} + 100 q^{20} + 21 q^{21} + 11 q^{22} + 159 q^{23} + 35 q^{24} + 176 q^{25} + 38 q^{26} + 94 q^{27} + 94 q^{28} + 57 q^{29} + 14 q^{30} + 60 q^{31} + 136 q^{32} + 55 q^{33} + 14 q^{34} + 107 q^{35} + 185 q^{36} + 17 q^{37} + 99 q^{38} + 92 q^{39} + 68 q^{40} + 68 q^{41} + 27 q^{42} + 67 q^{43} + 85 q^{44} + 75 q^{45} + 5 q^{46} + 236 q^{47} + 78 q^{48} + 182 q^{49} + 65 q^{50} + 72 q^{51} + 53 q^{52} + 64 q^{53} + 23 q^{54} + 117 q^{55} + 116 q^{56} + 33 q^{57} - 11 q^{58} + 100 q^{59} - 15 q^{60} + 57 q^{61} + 171 q^{62} + 252 q^{63} + 160 q^{64} + 67 q^{65} + 30 q^{66} + 28 q^{67} + 291 q^{68} + 36 q^{69} - 29 q^{70} + 96 q^{71} + 96 q^{72} + 36 q^{73} + 69 q^{74} + 64 q^{75} + 38 q^{76} + 130 q^{77} - 37 q^{78} + 74 q^{79} + 205 q^{80} + 177 q^{81} + 50 q^{82} + 242 q^{83} - 7 q^{84} - 5 q^{85} + 15 q^{86} + 267 q^{87} - 19 q^{88} + 76 q^{89} + 52 q^{90} + 9 q^{91} + 245 q^{92} + 6 q^{93} + 12 q^{94} + 254 q^{95} + 21 q^{96} + 24 q^{97} + 131 q^{98} + 78 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.79357 −0.184512 5.80401 3.67653 0.515447 2.32712 −10.6268 −2.96596 −10.2706
1.2 −2.72648 −1.83490 5.43370 0.0923546 5.00282 −1.61753 −9.36192 0.366851 −0.251803
1.3 −2.66310 0.966338 5.09209 0.577617 −2.57345 −1.19050 −8.23455 −2.06619 −1.53825
1.4 −2.65496 0.953781 5.04881 −3.53515 −2.53225 −1.20517 −8.09447 −2.09030 9.38568
1.5 −2.64632 3.36435 5.00301 −0.608526 −8.90313 −1.27504 −7.94691 8.31882 1.61035
1.6 −2.59338 2.60010 4.72562 2.28989 −6.74304 3.32352 −7.06857 3.76050 −5.93856
1.7 −2.53776 −0.695721 4.44022 2.46054 1.76557 −0.104118 −6.19268 −2.51597 −6.24424
1.8 −2.49499 −3.13087 4.22495 −0.0652690 7.81148 4.64194 −5.55122 6.80236 0.162845
1.9 −2.47681 2.50900 4.13457 −1.89767 −6.21430 −3.71277 −5.28691 3.29508 4.70016
1.10 −2.46894 −1.76219 4.09566 −1.92223 4.35073 −2.19638 −5.17405 0.105301 4.74586
1.11 −2.46592 −1.93828 4.08077 3.20017 4.77964 −3.00541 −5.13101 0.756924 −7.89136
1.12 −2.42899 2.20546 3.89999 0.464788 −5.35705 3.96228 −4.61506 1.86407 −1.12896
1.13 −2.42722 0.952029 3.89140 3.86168 −2.31079 −1.94146 −4.59085 −2.09364 −9.37314
1.14 −2.41478 −1.34248 3.83116 −3.25239 3.24179 2.15859 −4.42184 −1.19775 7.85379
1.15 −2.39817 0.0414509 3.75122 −0.527139 −0.0994062 −0.611771 −4.19972 −2.99828 1.26417
1.16 −2.37634 −2.85087 3.64699 3.08937 6.77463 3.25728 −3.91381 5.12744 −7.34139
1.17 −2.36096 2.71902 3.57414 2.53477 −6.41951 −0.168639 −3.71650 4.39309 −5.98449
1.18 −2.31072 −0.516181 3.33941 −2.87941 1.19275 −4.60091 −3.09500 −2.73356 6.65350
1.19 −2.30318 −2.40321 3.30466 −1.13192 5.53503 −0.419576 −3.00486 2.77541 2.60701
1.20 −2.29531 1.57290 3.26844 −2.25740 −3.61030 1.04765 −2.91147 −0.525975 5.18142
See next 80 embeddings (of 173 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.173
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(4057\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4057.2.a.b ✓ 173
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4057.2.a.b ✓ 173 1.a even 1 1 trivial