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Properties

Label 6005.2.a.f
Level $6005$
Weight $2$
Character orbit 6005.a
Self dual yes
Analytic conductor $47.950$
Analytic rank $0$
Dimension $111$
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] = [6005,2,Mod(1,6005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6005, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6005 = 5 \cdot 1201 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6005.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: \(47.9501664138\)
Analytic rank: \(0\)
Dimension: \(111\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 111 q + 20 q^{2} + 40 q^{3} + 136 q^{4} + 111 q^{5} + 3 q^{6} + 39 q^{7} + 45 q^{8} + 139 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 111 q + 20 q^{2} + 40 q^{3} + 136 q^{4} + 111 q^{5} + 3 q^{6} + 39 q^{7} + 45 q^{8} + 139 q^{9} + 20 q^{10} + 36 q^{11} + 80 q^{12} + 36 q^{13} + 7 q^{14} + 40 q^{15} + 190 q^{16} + 38 q^{17} + 48 q^{18} + 77 q^{19} + 136 q^{20} + 11 q^{21} + 39 q^{22} + 82 q^{23} - 3 q^{24} + 111 q^{25} - 3 q^{26} + 130 q^{27} + 87 q^{28} + 20 q^{29} + 3 q^{30} + 41 q^{31} + 85 q^{32} + 33 q^{33} + 7 q^{34} + 39 q^{35} + 191 q^{36} + 80 q^{37} + 42 q^{38} + 21 q^{39} + 45 q^{40} + 16 q^{41} + 33 q^{42} + 164 q^{43} + 37 q^{44} + 139 q^{45} + 32 q^{46} + 148 q^{47} + 149 q^{48} + 160 q^{49} + 20 q^{50} + 51 q^{51} + 87 q^{52} + 83 q^{53} - 6 q^{54} + 36 q^{55} - 10 q^{56} + 28 q^{57} + 47 q^{58} + 14 q^{59} + 80 q^{60} + 20 q^{61} + 14 q^{62} + 120 q^{63} + 231 q^{64} + 36 q^{65} - 4 q^{66} + 253 q^{67} + 80 q^{68} + 6 q^{69} + 7 q^{70} + 5 q^{71} + 124 q^{72} + 64 q^{73} - 37 q^{74} + 40 q^{75} + 92 q^{76} + 63 q^{77} + 29 q^{78} + 91 q^{79} + 190 q^{80} + 187 q^{81} - 7 q^{82} + 63 q^{83} - 69 q^{84} + 38 q^{85} - 22 q^{86} + 57 q^{87} + 121 q^{88} - 6 q^{89} + 48 q^{90} + 119 q^{91} + 104 q^{92} + 14 q^{93} - q^{94} + 77 q^{95} - 38 q^{96} + 96 q^{97} + 81 q^{98} + 106 q^{99}+O(q^{100}) \) 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 −2.74668 1.63661 5.54423 1.00000 −4.49523 −2.87647 −9.73484 −0.321512 −2.74668
1.2 −2.72268 3.38210 5.41301 1.00000 −9.20838 2.13000 −9.29255 8.43858 −2.72268
1.3 −2.71589 0.174668 5.37604 1.00000 −0.474377 2.58754 −9.16893 −2.96949 −2.71589
1.4 −2.65700 −0.218117 5.05966 1.00000 0.579538 4.50653 −8.12951 −2.95242 −2.65700
1.5 −2.63642 3.34656 4.95072 1.00000 −8.82296 −3.64864 −7.77935 8.19949 −2.63642
1.6 −2.62342 −2.19431 4.88231 1.00000 5.75659 1.89929 −7.56149 1.81500 −2.62342
1.7 −2.61069 −1.33217 4.81572 1.00000 3.47788 −0.680814 −7.35098 −1.22533 −2.61069
1.8 −2.56895 1.63205 4.59952 1.00000 −4.19265 3.45435 −6.67805 −0.336428 −2.56895
1.9 −2.47000 −2.23996 4.10090 1.00000 5.53271 3.40548 −5.18922 2.01743 −2.47000
1.10 −2.41916 1.95905 3.85233 1.00000 −4.73925 −1.88529 −4.48108 0.837875 −2.41916
1.11 −2.34362 −0.853258 3.49256 1.00000 1.99971 −1.48080 −3.49800 −2.27195 −2.34362
1.12 −2.32829 0.726641 3.42095 1.00000 −1.69183 −3.13935 −3.30840 −2.47199 −2.32829
1.13 −2.32774 1.75930 3.41838 1.00000 −4.09520 3.92990 −3.30162 0.0951392 −2.32774
1.14 −2.32622 −1.23055 3.41132 1.00000 2.86255 −0.720355 −3.28304 −1.48574 −2.32622
1.15 −2.23901 3.10819 3.01316 1.00000 −6.95927 3.97064 −2.26846 6.66087 −2.23901
1.16 −2.17647 1.64482 2.73702 1.00000 −3.57990 −1.13164 −1.60410 −0.294573 −2.17647
1.17 −2.03891 2.79275 2.15716 1.00000 −5.69418 −0.0959578 −0.320429 4.79947 −2.03891
1.18 −1.98616 −2.45604 1.94482 1.00000 4.87807 −0.764099 0.109603 3.03211 −1.98616
1.19 −1.98040 0.765234 1.92199 1.00000 −1.51547 −3.35858 0.154490 −2.41442 −1.98040
1.20 −1.91654 −2.98629 1.67314 1.00000 5.72336 0.938196 0.626446 5.91794 −1.91654
See next 80 embeddings (of 111 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.111
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1201\) \(1\)

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 6005.2.a.f ✓ 111
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6005.2.a.f ✓ 111 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{111} - 20 T_{2}^{110} + 21 T_{2}^{109} + 2205 T_{2}^{108} - 12753 T_{2}^{107} - 99984 T_{2}^{106} + \cdots - 1711692 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\). Copy content Toggle raw display