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Properties

Label 4013.2.a.b
Level $4013$
Weight $2$
Character orbit 4013.a
Self dual yes
Analytic conductor $32.044$
Analytic rank $1$
Dimension $157$
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] = [4013,2,Mod(1,4013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4013, 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("4013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4013 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4013.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: \(32.0439663311\)
Analytic rank: \(1\)
Dimension: \(157\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9} - 61 q^{10} - 27 q^{11} - 93 q^{12} - 97 q^{13} - 12 q^{14} - 36 q^{15} + 105 q^{16} - 45 q^{17} - 68 q^{18} - 128 q^{19} - 30 q^{20} - 26 q^{21} - 68 q^{22} - 41 q^{23} - 40 q^{24} + 102 q^{25} - 5 q^{26} - 189 q^{27} - 115 q^{28} - 26 q^{29} - 12 q^{30} - 88 q^{31} - 89 q^{32} - 52 q^{33} - 61 q^{34} - 87 q^{35} + 110 q^{36} - 62 q^{37} - 37 q^{38} - 20 q^{39} - 161 q^{40} - 34 q^{41} - 53 q^{42} - 254 q^{43} - 19 q^{44} - 46 q^{45} - 52 q^{46} - 76 q^{47} - 162 q^{48} + 96 q^{49} - 54 q^{50} - 76 q^{51} - 259 q^{52} - 48 q^{53} - 12 q^{54} - 194 q^{55} - 10 q^{56} - 30 q^{57} - 52 q^{58} - 64 q^{59} - 31 q^{60} - 107 q^{61} - 51 q^{62} - 106 q^{63} + 54 q^{64} - 17 q^{65} - 13 q^{66} - 193 q^{67} - 118 q^{68} - 55 q^{69} - 86 q^{70} - 11 q^{71} - 172 q^{72} - 173 q^{73} - 11 q^{74} - 209 q^{75} - 213 q^{76} - 84 q^{77} - 30 q^{78} - 111 q^{79} - 6 q^{80} + 157 q^{81} - 117 q^{82} - 154 q^{83} - 6 q^{84} - 91 q^{85} + 28 q^{86} - 165 q^{87} - 165 q^{88} - 32 q^{89} - 103 q^{90} - 200 q^{91} - 86 q^{92} - 39 q^{93} - 118 q^{94} - 22 q^{95} - 28 q^{96} - 151 q^{97} - 38 q^{98} - 91 q^{99}+O(q^{100}) \) 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 −2.81982 1.03159 5.95140 1.63220 −2.90891 3.00127 −11.1422 −1.93581 −4.60252
1.2 −2.78653 −0.812844 5.76476 −1.14246 2.26502 −4.07959 −10.4906 −2.33929 3.18350
1.3 −2.78090 −3.27207 5.73341 3.80445 9.09929 2.37189 −10.3822 7.70642 −10.5798
1.4 −2.68953 2.61392 5.23356 1.08630 −7.03022 −3.47806 −8.69675 3.83259 −2.92162
1.5 −2.65659 1.81764 5.05746 3.63888 −4.82872 −3.82226 −8.12240 0.303819 −9.66700
1.6 −2.65066 −3.08601 5.02601 −3.86295 8.17996 −0.380070 −8.02093 6.52343 10.2394
1.7 −2.61349 −2.30050 4.83033 −0.685238 6.01232 −4.02398 −7.39704 2.29228 1.79086
1.8 −2.60727 1.83175 4.79788 −1.15472 −4.77588 −1.01890 −7.29484 0.355310 3.01066
1.9 −2.54071 0.214630 4.45522 0.391559 −0.545314 2.42547 −6.23801 −2.95393 −0.994840
1.10 −2.53912 −1.94061 4.44715 3.91571 4.92744 −2.75166 −6.21361 0.765960 −9.94248
1.11 −2.52958 −1.28339 4.39880 −1.06771 3.24644 0.845401 −6.06796 −1.35291 2.70087
1.12 −2.51870 −2.55292 4.34385 0.976255 6.43005 1.70978 −5.90347 3.51742 −2.45890
1.13 −2.51500 −3.29902 4.32520 −0.150822 8.29703 −3.52495 −5.84787 7.88356 0.379318
1.14 −2.47196 3.17720 4.11056 1.49907 −7.85391 0.578951 −5.21722 7.09463 −3.70563
1.15 −2.42617 1.73131 3.88629 −0.987697 −4.20045 1.01261 −4.57644 −0.00256059 2.39632
1.16 −2.39096 −0.986021 3.71671 −2.27233 2.35754 −0.314319 −4.10458 −2.02776 5.43305
1.17 −2.35914 −1.36085 3.56554 2.69250 3.21043 1.19672 −3.69333 −1.14810 −6.35199
1.18 −2.34257 −1.48585 3.48762 0.440014 3.48071 1.63946 −3.48485 −0.792243 −1.03076
1.19 −2.28387 2.83481 3.21604 −3.24199 −6.47432 0.379335 −2.77728 5.03614 7.40427
1.20 −2.28352 −2.81571 3.21448 2.96925 6.42974 −4.34900 −2.77330 4.92822 −6.78035
See next 80 embeddings (of 157 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.157
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(4013\) \(1\)

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 4013.2.a.b ✓ 157
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4013.2.a.b ✓ 157 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{157} + 15 T_{2}^{156} - 113 T_{2}^{155} - 2786 T_{2}^{154} + 2130 T_{2}^{153} + 248978 T_{2}^{152} + \cdots + 608498 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4013))\). Copy content Toggle raw display