Properties

Label 8035.2.a.d
Level $8035$
Weight $2$
Character orbit 8035.a
Self dual yes
Analytic conductor $64.160$
Analytic rank $1$
Dimension $140$
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] = [8035,2,Mod(1,8035)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8035, 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("8035.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8035 = 5 \cdot 1607 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8035.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: \(64.1597980241\)
Analytic rank: \(1\)
Dimension: \(140\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 140 q - 20 q^{2} - 12 q^{3} + 144 q^{4} - 140 q^{5} - 15 q^{7} - 63 q^{8} + 134 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 140 q - 20 q^{2} - 12 q^{3} + 144 q^{4} - 140 q^{5} - 15 q^{7} - 63 q^{8} + 134 q^{9} + 20 q^{10} - 26 q^{11} - 31 q^{12} - 32 q^{13} - 37 q^{14} + 12 q^{15} + 152 q^{16} - 69 q^{17} - 64 q^{18} + 37 q^{19} - 144 q^{20} - 43 q^{21} - 25 q^{22} - 63 q^{23} - 5 q^{24} + 140 q^{25} - 16 q^{26} - 48 q^{27} - 52 q^{28} - 136 q^{29} + 25 q^{31} - 151 q^{32} - 48 q^{33} + 29 q^{34} + 15 q^{35} + 120 q^{36} - 82 q^{37} - 69 q^{38} - 26 q^{39} + 63 q^{40} - 11 q^{41} - 35 q^{42} - 54 q^{43} - 83 q^{44} - 134 q^{45} + 25 q^{46} - 39 q^{47} - 83 q^{48} + 215 q^{49} - 20 q^{50} - 75 q^{51} - 56 q^{52} - 196 q^{53} - 29 q^{54} + 26 q^{55} - 132 q^{56} - 110 q^{57} - 29 q^{58} - 31 q^{59} + 31 q^{60} - 18 q^{61} - 107 q^{62} - 67 q^{63} + 165 q^{64} + 32 q^{65} - 16 q^{66} - 50 q^{67} - 201 q^{68} - 46 q^{69} + 37 q^{70} - 84 q^{71} - 200 q^{72} - 70 q^{73} - 101 q^{74} - 12 q^{75} + 118 q^{76} - 166 q^{77} - 106 q^{78} - 35 q^{79} - 152 q^{80} + 116 q^{81} - 72 q^{82} - 66 q^{83} - 60 q^{84} + 69 q^{85} - 66 q^{86} - 75 q^{87} - 101 q^{88} + 8 q^{89} + 64 q^{90} + 2 q^{91} - 197 q^{92} - 134 q^{93} + 65 q^{94} - 37 q^{95} + 6 q^{96} - 73 q^{97} - 151 q^{98} - 51 q^{99}+O(q^{100}) \) 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 −2.82578 1.42747 5.98503 −1.00000 −4.03370 3.78558 −11.2608 −0.962340 2.82578
1.2 −2.79540 −1.02268 5.81427 −1.00000 2.85879 3.00451 −10.6624 −1.95413 2.79540
1.3 −2.76648 3.20166 5.65342 −1.00000 −8.85734 −4.04586 −10.1071 7.25064 2.76648
1.4 −2.72603 −2.40516 5.43123 −1.00000 6.55652 −0.808497 −9.35364 2.78477 2.72603
1.5 −2.70698 2.06859 5.32774 −1.00000 −5.59964 3.95188 −9.00814 1.27907 2.70698
1.6 −2.70543 0.0377636 5.31934 −1.00000 −0.102167 −3.03479 −8.98024 −2.99857 2.70543
1.7 −2.70407 −1.94382 5.31198 −1.00000 5.25623 −4.62711 −8.95580 0.778451 2.70407
1.8 −2.69188 −1.94222 5.24620 −1.00000 5.22822 1.57451 −8.73838 0.772219 2.69188
1.9 −2.67655 −2.96805 5.16395 −1.00000 7.94414 4.47636 −8.46847 5.80930 2.67655
1.10 −2.66914 −3.32233 5.12432 −1.00000 8.86776 −1.95943 −8.33924 8.03786 2.66914
1.11 −2.56068 −0.755855 4.55711 −1.00000 1.93551 −1.57635 −6.54794 −2.42868 2.56068
1.12 −2.54398 1.20144 4.47181 −1.00000 −3.05645 −0.416723 −6.28824 −1.55653 2.54398
1.13 −2.54024 0.460298 4.45282 −1.00000 −1.16927 −2.08486 −6.23076 −2.78813 2.54024
1.14 −2.53622 2.11151 4.43241 −1.00000 −5.35524 2.12387 −6.16912 1.45846 2.53622
1.15 −2.44538 −0.0835630 3.97986 −1.00000 0.204343 5.18435 −4.84151 −2.99302 2.44538
1.16 −2.42652 0.577409 3.88800 −1.00000 −1.40110 3.20436 −4.58127 −2.66660 2.42652
1.17 −2.41030 −2.44694 3.80955 −1.00000 5.89785 −4.60731 −4.36157 2.98749 2.41030
1.18 −2.40990 2.64302 3.80764 −1.00000 −6.36944 −3.14385 −4.35625 3.98558 2.40990
1.19 −2.36612 −3.07539 3.59854 −1.00000 7.27675 3.28740 −3.78235 6.45802 2.36612
1.20 −2.31791 3.17051 3.37269 −1.00000 −7.34893 3.65453 −3.18176 7.05211 2.31791
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.140
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(1607\) \(1\)

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 8035.2.a.d 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8035.2.a.d 140 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{140} + 20 T_{2}^{139} - 12 T_{2}^{138} - 2859 T_{2}^{137} - 12842 T_{2}^{136} + 182399 T_{2}^{135} + 1453069 T_{2}^{134} - 6322791 T_{2}^{133} - 86478855 T_{2}^{132} + 87774355 T_{2}^{131} + 3474623906 T_{2}^{130} + \cdots - 846834921 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8035))\). Copy content Toggle raw display