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Properties

Label 6035.2.a.h
Level $6035$
Weight $2$
Character orbit 6035.a
Self dual yes
Analytic conductor $48.190$
Analytic rank $0$
Dimension $59$
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] = [6035,2,Mod(1,6035)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6035, 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("6035.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6035 = 5 \cdot 17 \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6035.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: \(48.1897176198\)
Analytic rank: \(0\)
Dimension: \(59\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 59 q + 2 q^{2} + 6 q^{3} + 76 q^{4} + 59 q^{5} + 14 q^{6} + 9 q^{7} + 6 q^{8} + 97 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 59 q + 2 q^{2} + 6 q^{3} + 76 q^{4} + 59 q^{5} + 14 q^{6} + 9 q^{7} + 6 q^{8} + 97 q^{9} + 2 q^{10} + 6 q^{11} + 16 q^{12} + 25 q^{13} + 8 q^{14} + 6 q^{15} + 114 q^{16} - 59 q^{17} + 3 q^{18} + 35 q^{19} + 76 q^{20} + 41 q^{21} + 13 q^{22} - 2 q^{23} + 35 q^{24} + 59 q^{25} + 10 q^{26} + 27 q^{27} + 28 q^{28} + 10 q^{29} + 14 q^{30} + 15 q^{31} + 19 q^{32} - 3 q^{33} - 2 q^{34} + 9 q^{35} + 160 q^{36} + 56 q^{37} + 17 q^{38} + 25 q^{39} + 6 q^{40} + 47 q^{41} + 46 q^{42} + 27 q^{43} + 39 q^{44} + 97 q^{45} + 4 q^{46} - 8 q^{47} + 58 q^{48} + 174 q^{49} + 2 q^{50} - 6 q^{51} + 13 q^{52} + 17 q^{53} + 48 q^{54} + 6 q^{55} + 33 q^{56} + 6 q^{57} + 32 q^{58} + 30 q^{59} + 16 q^{60} + 85 q^{61} - 3 q^{62} + 14 q^{63} + 168 q^{64} + 25 q^{65} + 87 q^{66} + 21 q^{67} - 76 q^{68} + 111 q^{69} + 8 q^{70} - 59 q^{71} - 52 q^{72} + 41 q^{73} + 11 q^{74} + 6 q^{75} + 118 q^{76} + 55 q^{77} - 54 q^{78} + 43 q^{79} + 114 q^{80} + 207 q^{81} + 45 q^{82} + 10 q^{83} + 62 q^{84} - 59 q^{85} + 19 q^{86} + 8 q^{87} + 35 q^{88} + 86 q^{89} + 3 q^{90} + 47 q^{91} - 98 q^{92} + 41 q^{93} + 30 q^{94} + 35 q^{95} + 69 q^{96} + 72 q^{97} + 14 q^{98} + 25 q^{99}+O(q^{100}) \) 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 −2.78220 0.531208 5.74064 1.00000 −1.47793 5.08298 −10.4072 −2.71782 −2.78220
1.2 −2.75521 −3.32287 5.59117 1.00000 9.15519 0.0853844 −9.89442 8.04146 −2.75521
1.3 −2.69345 3.39811 5.25470 1.00000 −9.15266 −4.44179 −8.76637 8.54717 −2.69345
1.4 −2.65295 −2.09005 5.03815 1.00000 5.54480 2.26446 −8.06007 1.36830 −2.65295
1.5 −2.48619 −1.43684 4.18113 1.00000 3.57226 −2.10057 −5.42270 −0.935481 −2.48619
1.6 −2.46188 1.88140 4.06087 1.00000 −4.63179 −1.20679 −5.07361 0.539668 −2.46188
1.7 −2.44774 0.819540 3.99145 1.00000 −2.00602 2.11487 −4.87457 −2.32835 −2.44774
1.8 −2.42937 −1.27159 3.90183 1.00000 3.08915 −3.98798 −4.62024 −1.38307 −2.42937
1.9 −2.25292 1.80225 3.07566 1.00000 −4.06033 −3.69213 −2.42337 0.248105 −2.25292
1.10 −2.17980 2.67247 2.75152 1.00000 −5.82545 4.07077 −1.63817 4.14211 −2.17980
1.11 −2.11526 3.31197 2.47433 1.00000 −7.00568 4.53334 −1.00333 7.96913 −2.11526
1.12 −2.03514 −3.16809 2.14178 1.00000 6.44749 −3.29825 −0.288547 7.03677 −2.03514
1.13 −1.96419 −0.830203 1.85806 1.00000 1.63068 −0.855688 0.278801 −2.31076 −1.96419
1.14 −1.75849 −0.492394 1.09228 1.00000 0.865869 1.61123 1.59622 −2.75755 −1.75849
1.15 −1.71705 −3.23330 0.948270 1.00000 5.55174 4.30663 1.80588 7.45420 −1.71705
1.16 −1.47530 −1.44851 0.176504 1.00000 2.13699 2.79346 2.69020 −0.901812 −1.47530
1.17 −1.34008 −2.63171 −0.204177 1.00000 3.52671 −4.96340 2.95378 3.92590 −1.34008
1.18 −1.27257 0.337599 −0.380563 1.00000 −0.429619 −3.22826 3.02944 −2.88603 −1.27257
1.19 −1.25970 2.36319 −0.413145 1.00000 −2.97691 2.52906 3.03985 2.58464 −1.25970
1.20 −1.15770 0.151800 −0.659734 1.00000 −0.175739 −1.47605 3.07917 −2.97696 −1.15770
See all 59 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.59
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(17\) \(1\)
\(71\) \(1\)

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 6035.2.a.h ✓ 59
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6035.2.a.h ✓ 59 1.a even 1 1 trivial