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Properties

Label 6035.2.a.e
Level $6035$
Weight $2$
Character orbit 6035.a
Self dual yes
Analytic conductor $48.190$
Analytic rank $0$
Dimension $49$
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: \(49\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 49 q + q^{2} + 10 q^{3} + 55 q^{4} - 49 q^{5} + 2 q^{6} + 15 q^{7} + 3 q^{8} + 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 49 q + q^{2} + 10 q^{3} + 55 q^{4} - 49 q^{5} + 2 q^{6} + 15 q^{7} + 3 q^{8} + 43 q^{9} - q^{10} - 4 q^{11} + 32 q^{12} + 21 q^{13} + 8 q^{14} - 10 q^{15} + 63 q^{16} + 49 q^{17} + 12 q^{18} + 19 q^{19} - 55 q^{20} + 5 q^{21} + 10 q^{22} + 10 q^{23} + 7 q^{24} + 49 q^{25} - 20 q^{26} + 37 q^{27} + 48 q^{28} + 30 q^{29} - 2 q^{30} + 23 q^{31} + 2 q^{32} - q^{33} + q^{34} - 15 q^{35} + 39 q^{36} + 60 q^{37} + 13 q^{38} + 3 q^{39} - 3 q^{40} - 29 q^{41} + 32 q^{42} + 23 q^{43} + 11 q^{44} - 43 q^{45} + 6 q^{46} - 8 q^{47} + 96 q^{48} + 82 q^{49} + q^{50} + 10 q^{51} + 11 q^{52} + 15 q^{53} - 4 q^{54} + 4 q^{55} - 18 q^{56} + 44 q^{57} + 38 q^{58} - 28 q^{59} - 32 q^{60} + 107 q^{61} + 37 q^{62} + 54 q^{63} + 77 q^{64} - 21 q^{65} + 23 q^{66} + 11 q^{67} + 55 q^{68} + 27 q^{69} - 8 q^{70} - 49 q^{71} - 53 q^{72} + 97 q^{73} + 27 q^{74} + 10 q^{75} + 66 q^{76} + 19 q^{77} + 48 q^{78} + 31 q^{79} - 63 q^{80} - 7 q^{81} + 61 q^{82} + 6 q^{83} + 46 q^{84} - 49 q^{85} - 89 q^{86} + 36 q^{87} + 63 q^{88} - 20 q^{89} - 12 q^{90} + 35 q^{91} - 2 q^{92} + 47 q^{93} - 4 q^{94} - 19 q^{95} + 73 q^{96} + 92 q^{97} + 3 q^{98} + 5 q^{99}+O(q^{100}) \) 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 −2.76733 1.59965 5.65810 −1.00000 −4.42674 −1.31827 −10.1232 −0.441134 2.76733
1.2 −2.69755 3.18040 5.27679 −1.00000 −8.57929 2.84577 −8.83933 7.11493 2.69755
1.3 −2.65457 −1.67907 5.04675 −1.00000 4.45721 1.49488 −8.08781 −0.180722 2.65457
1.4 −2.46903 0.417352 4.09613 −1.00000 −1.03046 4.52546 −5.17542 −2.82582 2.46903
1.5 −2.45613 −2.72994 4.03255 −1.00000 6.70507 4.46164 −4.99220 4.45257 2.45613
1.6 −2.43490 0.528737 3.92876 −1.00000 −1.28742 −4.65981 −4.69634 −2.72044 2.43490
1.7 −2.18215 −0.779466 2.76177 −1.00000 1.70091 −1.35162 −1.66229 −2.39243 2.18215
1.8 −2.16678 2.61186 2.69492 −1.00000 −5.65931 0.691862 −1.50573 3.82180 2.16678
1.9 −1.93079 1.24985 1.72797 −1.00000 −2.41319 0.0149302 0.525242 −1.43789 1.93079
1.10 −1.89300 −1.14901 1.58346 −1.00000 2.17509 1.83563 0.788506 −1.67977 1.89300
1.11 −1.88687 −2.20455 1.56027 −1.00000 4.15970 −1.49671 0.829712 1.86005 1.88687
1.12 −1.68140 2.72713 0.827113 −1.00000 −4.58540 −4.54380 1.97209 4.43722 1.68140
1.13 −1.66746 0.542291 0.780412 −1.00000 −0.904247 −3.22049 2.03361 −2.70592 1.66746
1.14 −1.43198 2.63685 0.0505582 −1.00000 −3.77591 3.33757 2.79156 3.95299 1.43198
1.15 −1.39209 0.859708 −0.0620899 −1.00000 −1.19679 4.42516 2.87061 −2.26090 1.39209
1.16 −1.31251 −1.15146 −0.277313 −1.00000 1.51130 −2.40040 2.98900 −1.67415 1.31251
1.17 −1.23643 −2.99255 −0.471232 −1.00000 3.70009 −1.37052 3.05551 5.95534 1.23643
1.18 −0.658530 −1.42449 −1.56634 −1.00000 0.938072 −1.19176 2.34854 −0.970815 0.658530
1.19 −0.640985 2.98787 −1.58914 −1.00000 −1.91518 −0.781752 2.30058 5.92734 0.640985
1.20 −0.617520 −1.37173 −1.61867 −1.00000 0.847070 −0.736468 2.23460 −1.11836 0.617520
See all 49 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.49
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.e ✓ 49
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6035.2.a.e ✓ 49 1.a even 1 1 trivial