Properties

Label 4001.2.a.b
Level $4001$
Weight $2$
Character orbit 4001.a
Self dual yes
Analytic conductor $31.948$
Analytic rank $0$
Dimension $184$
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] = [4001,2,Mod(1,4001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4001, 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("4001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4001.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: \(31.9481458487\)
Analytic rank: \(0\)
Dimension: \(184\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18} + 86 q^{19} + 26 q^{20} + 22 q^{21} + 54 q^{22} + 55 q^{23} + 72 q^{24} + 241 q^{25} + 32 q^{26} + 97 q^{27} + 75 q^{28} + 27 q^{29} - 10 q^{30} + 276 q^{31} + 20 q^{33} + 122 q^{34} + 30 q^{35} + 278 q^{36} + 42 q^{37} + 14 q^{38} + 113 q^{39} + 115 q^{40} + 39 q^{41} + 15 q^{42} + 65 q^{43} + 32 q^{44} + 54 q^{45} + 65 q^{46} + 82 q^{47} + 117 q^{48} + 297 q^{49} + 4 q^{50} + 45 q^{51} + 136 q^{52} + 21 q^{53} + 93 q^{54} + 252 q^{55} + 74 q^{56} + 14 q^{57} + 54 q^{58} + 95 q^{59} + 58 q^{60} + 131 q^{61} + 14 q^{62} + 88 q^{63} + 368 q^{64} - 9 q^{65} + 52 q^{66} + 90 q^{67} + 27 q^{68} + 101 q^{69} + 18 q^{70} + 117 q^{71} - 15 q^{72} + 72 q^{73} + 7 q^{74} + 150 q^{75} + 148 q^{76} + 7 q^{77} + 22 q^{78} + 287 q^{79} + 43 q^{80} + 244 q^{81} + 86 q^{82} + 25 q^{83} + 14 q^{84} + 41 q^{85} + 25 q^{86} + 82 q^{87} + 115 q^{88} + 48 q^{89} + 78 q^{90} + 272 q^{91} + 69 q^{92} + 44 q^{93} + 161 q^{94} + 37 q^{95} + 129 q^{96} + 106 q^{97} - 46 q^{98} + 53 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.80114 −3.24231 5.84639 −0.362239 9.08218 0.683027 −10.7743 7.51260 1.01468
1.2 −2.79357 0.695080 5.80404 −1.43429 −1.94176 3.16464 −10.6269 −2.51686 4.00680
1.3 −2.77961 −0.970175 5.72623 0.973772 2.69671 0.392591 −10.3575 −2.05876 −2.70671
1.4 −2.76351 −0.782711 5.63697 4.06144 2.16303 1.66325 −10.0508 −2.38736 −11.2238
1.5 −2.73134 2.60660 5.46019 −2.78013 −7.11951 −4.77010 −9.45095 3.79438 7.59346
1.6 −2.71583 2.95270 5.37572 2.81232 −8.01903 −1.71576 −9.16786 5.71846 −7.63778
1.7 −2.69288 −1.22667 5.25159 2.75099 3.30328 −4.02180 −8.75615 −1.49527 −7.40807
1.8 −2.68105 −2.59933 5.18802 −3.48579 6.96892 −5.16960 −8.54725 3.75650 9.34557
1.9 −2.66682 0.260886 5.11190 −1.12952 −0.695735 −1.67133 −8.29887 −2.93194 3.01221
1.10 −2.65266 3.05290 5.03661 −1.94746 −8.09830 2.48037 −8.05510 6.32018 5.16596
1.11 −2.64594 −1.99415 5.00101 −3.20075 5.27640 3.40947 −7.94051 0.976617 8.46900
1.12 −2.63285 1.55445 4.93191 −4.32053 −4.09264 −0.850828 −7.71930 −0.583679 11.3753
1.13 −2.61467 2.84813 4.83651 0.542328 −7.44692 4.00529 −7.41654 5.11183 −1.41801
1.14 −2.61059 −0.259768 4.81518 −4.09316 0.678147 2.08780 −7.34928 −2.93252 10.6856
1.15 −2.55462 2.40673 4.52611 3.51168 −6.14829 3.52515 −6.45325 2.79234 −8.97103
1.16 −2.43211 0.645473 3.91518 −1.15456 −1.56986 1.52754 −4.65795 −2.58336 2.80801
1.17 −2.42514 −1.95794 3.88128 1.35530 4.74827 4.91337 −4.56237 0.833523 −3.28679
1.18 −2.40164 −1.91938 3.76787 −2.41584 4.60966 −1.95333 −4.24578 0.684021 5.80197
1.19 −2.38954 1.02374 3.70989 −0.691971 −2.44626 1.59189 −4.08584 −1.95196 1.65349
1.20 −2.38517 0.919938 3.68905 3.44498 −2.19421 −3.87561 −4.02868 −2.15371 −8.21687
See next 80 embeddings (of 184 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.184
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(4001\) \( -1 \)

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 4001.2.a.b 184
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4001.2.a.b 184 1.a even 1 1 trivial