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Properties

Label 4011.2.a.l
Level $4011$
Weight $2$
Character orbit 4011.a
Self dual yes
Analytic conductor $32.028$
Analytic rank $0$
Dimension $28$
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] = [4011,2,Mod(1,4011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4011, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4011.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4011 = 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4011.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: \(32.0279962507\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 6 q^{2} - 28 q^{3} + 34 q^{4} + 8 q^{5} + 6 q^{6} + 28 q^{7} - 15 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 6 q^{2} - 28 q^{3} + 34 q^{4} + 8 q^{5} + 6 q^{6} + 28 q^{7} - 15 q^{8} + 28 q^{9} + 4 q^{10} - 8 q^{11} - 34 q^{12} + 26 q^{13} - 6 q^{14} - 8 q^{15} + 62 q^{16} + 9 q^{17} - 6 q^{18} + 25 q^{19} + 20 q^{20} - 28 q^{21} + 3 q^{22} - 30 q^{23} + 15 q^{24} + 42 q^{25} + 25 q^{26} - 28 q^{27} + 34 q^{28} - 5 q^{29} - 4 q^{30} + 18 q^{31} - 26 q^{32} + 8 q^{33} + 30 q^{34} + 8 q^{35} + 34 q^{36} + 36 q^{37} - 2 q^{38} - 26 q^{39} + 28 q^{40} + 21 q^{41} + 6 q^{42} + 8 q^{43} - 20 q^{44} + 8 q^{45} + 24 q^{46} + 6 q^{47} - 62 q^{48} + 28 q^{49} - 48 q^{50} - 9 q^{51} + 54 q^{52} - 12 q^{53} + 6 q^{54} + 15 q^{55} - 15 q^{56} - 25 q^{57} + 19 q^{58} + 33 q^{59} - 20 q^{60} + 48 q^{61} + 28 q^{63} + 75 q^{64} + 21 q^{65} - 3 q^{66} + 27 q^{67} + 19 q^{68} + 30 q^{69} + 4 q^{70} - 45 q^{71} - 15 q^{72} + 61 q^{73} - 31 q^{74} - 42 q^{75} + 63 q^{76} - 8 q^{77} - 25 q^{78} + 35 q^{79} + 84 q^{80} + 28 q^{81} + 11 q^{82} + 43 q^{83} - 34 q^{84} + 43 q^{85} - q^{86} + 5 q^{87} - 27 q^{88} + 25 q^{89} + 4 q^{90} + 26 q^{91} - 102 q^{92} - 18 q^{93} + 55 q^{94} - 43 q^{95} + 26 q^{96} + 40 q^{97} - 6 q^{98} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.77095 −1.00000 5.67816 3.12132 2.77095 1.00000 −10.1920 1.00000 −8.64902
1.2 −2.75611 −1.00000 5.59616 −2.11611 2.75611 1.00000 −9.91144 1.00000 5.83224
1.3 −2.60224 −1.00000 4.77168 −2.58506 2.60224 1.00000 −7.21258 1.00000 6.72695
1.4 −2.55579 −1.00000 4.53209 4.15915 2.55579 1.00000 −6.47150 1.00000 −10.6299
1.5 −2.32219 −1.00000 3.39258 −2.33514 2.32219 1.00000 −3.23383 1.00000 5.42264
1.6 −2.23874 −1.00000 3.01194 1.19635 2.23874 1.00000 −2.26547 1.00000 −2.67832
1.7 −1.81456 −1.00000 1.29264 0.616523 1.81456 1.00000 1.28355 1.00000 −1.11872
1.8 −1.80294 −1.00000 1.25060 −3.82317 1.80294 1.00000 1.35113 1.00000 6.89294
1.9 −1.70611 −1.00000 0.910828 3.20896 1.70611 1.00000 1.85825 1.00000 −5.47486
1.10 −1.53559 −1.00000 0.358037 −0.986091 1.53559 1.00000 2.52138 1.00000 1.51423
1.11 −0.815618 −1.00000 −1.33477 3.08846 0.815618 1.00000 2.71990 1.00000 −2.51900
1.12 −0.786373 −1.00000 −1.38162 2.33863 0.786373 1.00000 2.65921 1.00000 −1.83903
1.13 −0.482188 −1.00000 −1.76749 −4.13435 0.482188 1.00000 1.81664 1.00000 1.99354
1.14 −0.386952 −1.00000 −1.85027 3.15570 0.386952 1.00000 1.48987 1.00000 −1.22110
1.15 −0.268740 −1.00000 −1.92778 −1.86903 0.268740 1.00000 1.05555 1.00000 0.502284
1.16 −0.0484532 −1.00000 −1.99765 1.78925 0.0484532 1.00000 0.193699 1.00000 −0.0866947
1.17 0.343679 −1.00000 −1.88188 −1.00728 −0.343679 1.00000 −1.33412 1.00000 −0.346179
1.18 0.544491 −1.00000 −1.70353 2.42913 −0.544491 1.00000 −2.01654 1.00000 1.32264
1.19 0.602953 −1.00000 −1.63645 −1.53490 −0.602953 1.00000 −2.19261 1.00000 −0.925472
1.20 1.03548 −1.00000 −0.927780 −1.23903 −1.03548 1.00000 −3.03166 1.00000 −1.28299
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(191\) \(1\)

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 4011.2.a.l ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4011.2.a.l ✓ 28 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 6 T_{2}^{27} - 27 T_{2}^{26} - 221 T_{2}^{25} + 212 T_{2}^{24} + 3487 T_{2}^{23} + \cdots + 96 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4011))\). Copy content Toggle raw display