Properties

Label 6037.2.a.b
Level $6037$
Weight $2$
Character orbit 6037.a
Self dual yes
Analytic conductor $48.206$
Analytic rank $0$
Dimension $259$
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] = [6037,2,Mod(1,6037)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6037, 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("6037.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6037 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6037.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: \(48.2056877002\)
Analytic rank: \(0\)
Dimension: \(259\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 259 q + 47 q^{2} + 29 q^{3} + 273 q^{4} + 38 q^{5} + 24 q^{6} + 42 q^{7} + 141 q^{8} + 286 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 259 q + 47 q^{2} + 29 q^{3} + 273 q^{4} + 38 q^{5} + 24 q^{6} + 42 q^{7} + 141 q^{8} + 286 q^{9} + 18 q^{10} + 108 q^{11} + 46 q^{12} + 33 q^{13} + 35 q^{14} + 40 q^{15} + 301 q^{16} + 67 q^{17} + 117 q^{18} + 69 q^{19} + 103 q^{20} + 24 q^{21} + 42 q^{22} + 162 q^{23} + 45 q^{24} + 291 q^{25} + 41 q^{26} + 101 q^{27} + 87 q^{28} + 78 q^{29} + 48 q^{30} + 25 q^{31} + 314 q^{32} + 67 q^{33} + 9 q^{34} + 252 q^{35} + 337 q^{36} + 49 q^{37} + 59 q^{38} + 93 q^{39} + 44 q^{40} + 60 q^{41} + 38 q^{42} + 178 q^{43} + 171 q^{44} + 67 q^{45} + 43 q^{46} + 185 q^{47} + 67 q^{48} + 273 q^{49} + 204 q^{50} + 145 q^{51} + 83 q^{52} + 112 q^{53} + 60 q^{54} + 57 q^{55} + 93 q^{56} + 109 q^{57} + 63 q^{58} + 228 q^{59} + 53 q^{60} + 20 q^{61} + 126 q^{62} + 153 q^{63} + 345 q^{64} + 113 q^{65} + 5 q^{66} + 208 q^{67} + 166 q^{68} + 10 q^{69} + 69 q^{70} + 150 q^{71} + 331 q^{72} + 75 q^{73} + 84 q^{74} + 72 q^{75} + 102 q^{76} + 166 q^{77} + 69 q^{78} + 52 q^{79} + 180 q^{80} + 327 q^{81} + 43 q^{82} + 434 q^{83} + 75 q^{85} + 133 q^{86} + 144 q^{87} + 111 q^{88} + 78 q^{89} - 8 q^{90} + 35 q^{91} + 372 q^{92} + 160 q^{93} + 36 q^{94} + 154 q^{95} + 60 q^{96} + 35 q^{97} + 254 q^{98} + 234 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.75717 −1.09794 5.60201 2.84974 3.02722 3.79647 −9.93137 −1.79452 −7.85722
1.2 −2.66503 2.65532 5.10237 −0.476820 −7.07650 −1.66384 −8.26789 4.05073 1.27074
1.3 −2.65823 −1.47107 5.06620 −0.123247 3.91044 −2.04180 −8.15068 −0.835960 0.327620
1.4 −2.65707 2.92115 5.06000 0.928477 −7.76170 3.41739 −8.13063 5.53314 −2.46702
1.5 −2.64523 1.84208 4.99727 3.41959 −4.87275 2.32041 −7.92847 0.393274 −9.04562
1.6 −2.61164 1.28286 4.82065 −1.11903 −3.35038 −4.20141 −7.36653 −1.35426 2.92250
1.7 −2.60200 −2.50808 4.77039 0.421972 6.52603 −1.06326 −7.20855 3.29049 −1.09797
1.8 −2.60071 −0.749599 4.76372 −3.60076 1.94949 −2.40322 −7.18764 −2.43810 9.36455
1.9 −2.58809 −2.19108 4.69820 0.221659 5.67071 0.704833 −6.98319 1.80083 −0.573673
1.10 −2.58711 −3.26786 4.69312 2.66521 8.45431 2.69955 −6.96738 7.67894 −6.89517
1.11 −2.58398 0.626068 4.67694 −1.39938 −1.61774 0.699914 −6.91715 −2.60804 3.61596
1.12 −2.56682 2.69132 4.58858 1.90137 −6.90815 −1.37100 −6.64441 4.24322 −4.88048
1.13 −2.47379 −1.46896 4.11964 2.18568 3.63389 −2.37075 −5.24354 −0.842162 −5.40693
1.14 −2.46629 1.74062 4.08257 −1.65001 −4.29287 1.73775 −5.13620 0.0297629 4.06940
1.15 −2.46237 −0.199276 4.06325 −0.688498 0.490692 0.363231 −5.08049 −2.96029 1.69534
1.16 −2.44806 0.638947 3.99301 2.71697 −1.56418 −2.58742 −4.87901 −2.59175 −6.65132
1.17 −2.42988 0.381455 3.90430 −2.17078 −0.926887 −2.70764 −4.62721 −2.85449 5.27472
1.18 −2.42952 −2.16225 3.90258 −1.13309 5.25323 1.20579 −4.62237 1.67531 2.75288
1.19 −2.38315 −2.49497 3.67942 −2.67299 5.94590 −2.23204 −4.00231 3.22488 6.37013
1.20 −2.38263 −0.344809 3.67692 0.273793 0.821552 0.727572 −3.99549 −2.88111 −0.652346
See next 80 embeddings (of 259 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.259
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(6037\) \(-1\)

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 6037.2.a.b 259
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6037.2.a.b 259 1.a even 1 1 trivial