Properties

Label 1040.6.a.c
Level $1040$
Weight $6$
Character orbit 1040.a
Self dual yes
Analytic conductor $166.799$
Analytic rank $0$
Dimension $2$
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] = [1040,6,Mod(1,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1040.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: \(166.799172605\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{145}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 130)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{145}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 3) q^{3} - 25 q^{5} + (\beta - 65) q^{7} + (6 \beta - 89) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 3) q^{3} - 25 q^{5} + (\beta - 65) q^{7} + (6 \beta - 89) q^{9} + (30 \beta - 102) q^{11} - 169 q^{13} + (25 \beta + 75) q^{15} + ( - 172 \beta - 202) q^{17} + ( - 56 \beta - 56) q^{19} + (62 \beta + 50) q^{21} + ( - 153 \beta - 131) q^{23} + 625 q^{25} + (314 \beta + 126) q^{27} + (204 \beta - 3406) q^{29} + ( - 32 \beta + 160) q^{31} + (12 \beta - 4044) q^{33} + ( - 25 \beta + 1625) q^{35} + (472 \beta - 6638) q^{37} + (169 \beta + 507) q^{39} + (14 \beta - 17540) q^{41} + (943 \beta - 1267) q^{43} + ( - 150 \beta + 2225) q^{45} + ( - 411 \beta + 11019) q^{47} + ( - 130 \beta - 12437) q^{49} + (718 \beta + 25546) q^{51} + (488 \beta - 22054) q^{53} + ( - 750 \beta + 2550) q^{55} + (224 \beta + 8288) q^{57} + ( - 936 \beta + 10944) q^{59} + ( - 2174 \beta - 20636) q^{61} + ( - 479 \beta + 6655) q^{63} + 4225 q^{65} + (667 \beta - 51615) q^{67} + (590 \beta + 22578) q^{69} + ( - 4940 \beta + 7740) q^{71} + ( - 3350 \beta + 27044) q^{73} + ( - 625 \beta - 1875) q^{75} + ( - 2052 \beta + 10980) q^{77} + ( - 5204 \beta - 13604) q^{79} + ( - 2526 \beta - 24281) q^{81} + ( - 731 \beta - 21345) q^{83} + (4300 \beta + 5050) q^{85} + (2794 \beta - 19362) q^{87} + ( - 2716 \beta - 34066) q^{89} + ( - 169 \beta + 10985) q^{91} + ( - 64 \beta + 4160) q^{93} + (1400 \beta + 1400) q^{95} + (3878 \beta + 104728) q^{97} + ( - 3282 \beta + 35178) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 50 q^{5} - 130 q^{7} - 178 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 50 q^{5} - 130 q^{7} - 178 q^{9} - 204 q^{11} - 338 q^{13} + 150 q^{15} - 404 q^{17} - 112 q^{19} + 100 q^{21} - 262 q^{23} + 1250 q^{25} + 252 q^{27} - 6812 q^{29} + 320 q^{31} - 8088 q^{33} + 3250 q^{35} - 13276 q^{37} + 1014 q^{39} - 35080 q^{41} - 2534 q^{43} + 4450 q^{45} + 22038 q^{47} - 24874 q^{49} + 51092 q^{51} - 44108 q^{53} + 5100 q^{55} + 16576 q^{57} + 21888 q^{59} - 41272 q^{61} + 13310 q^{63} + 8450 q^{65} - 103230 q^{67} + 45156 q^{69} + 15480 q^{71} + 54088 q^{73} - 3750 q^{75} + 21960 q^{77} - 27208 q^{79} - 48562 q^{81} - 42690 q^{83} + 10100 q^{85} - 38724 q^{87} - 68132 q^{89} + 21970 q^{91} + 8320 q^{93} + 2800 q^{95} + 209456 q^{97} + 70356 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
6.52080
−5.52080
0 −15.0416 0 −25.0000 0 −52.9584 0 −16.7504 0
1.2 0 9.04159 0 −25.0000 0 −77.0416 0 −161.250 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(13\) \( +1 \)

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 1040.6.a.c 2
4.b odd 2 1 130.6.a.c 2
20.d odd 2 1 650.6.a.f 2
20.e even 4 2 650.6.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.6.a.c 2 4.b odd 2 1
650.6.a.f 2 20.d odd 2 1
650.6.b.f 4 20.e even 4 2
1040.6.a.c 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 6T_{3} - 136 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1040))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 6T - 136 \) Copy content Toggle raw display
$5$ \( (T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 130T + 4080 \) Copy content Toggle raw display
$11$ \( T^{2} + 204T - 120096 \) Copy content Toggle raw display
$13$ \( (T + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 404 T - 4248876 \) Copy content Toggle raw display
$19$ \( T^{2} + 112T - 451584 \) Copy content Toggle raw display
$23$ \( T^{2} + 262 T - 3377144 \) Copy content Toggle raw display
$29$ \( T^{2} + 6812 T + 5566516 \) Copy content Toggle raw display
$31$ \( T^{2} - 320T - 122880 \) Copy content Toggle raw display
$37$ \( T^{2} + 13276 T + 11759364 \) Copy content Toggle raw display
$41$ \( T^{2} + 35080 T + 307623180 \) Copy content Toggle raw display
$43$ \( T^{2} + 2534 T - 127335816 \) Copy content Toggle raw display
$47$ \( T^{2} - 22038 T + 96924816 \) Copy content Toggle raw display
$53$ \( T^{2} + 44108 T + 451848036 \) Copy content Toggle raw display
$59$ \( T^{2} - 21888 T - 7262784 \) Copy content Toggle raw display
$61$ \( T^{2} + 41272 T - 259465524 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 2599599320 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 3478614400 \) Copy content Toggle raw display
$73$ \( T^{2} - 54088 T - 895884564 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 3741765504 \) Copy content Toggle raw display
$83$ \( T^{2} + 42690 T + 378126680 \) Copy content Toggle raw display
$89$ \( T^{2} + 68132 T + 90877236 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 8787315804 \) Copy content Toggle raw display
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