Properties

Label 1340.2.i.b
Level $1340$
Weight $2$
Character orbit 1340.i
Analytic conductor $10.700$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1340,2,Mod(841,1340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1340, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1340.841");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1340.i (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(10.6999538709\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.5808268944.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - x^{5} + 24x^{4} - 7x^{3} + 29x^{2} + 12x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - q^{5} + ( - \beta_{6} + \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - q^{5} + ( - \beta_{6} + \beta_1) q^{7} + \beta_{2} q^{9} + ( - \beta_{6} + 4 \beta_{4} - 4) q^{11} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{13} + \beta_{3} q^{15} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{17} + (2 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{2}) q^{19} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_1 + 2) q^{21} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{23} + q^{25} + (\beta_{7} + 2 \beta_{3} + 1) q^{27} + ( - 2 \beta_{6} + \beta_{4} + 2 \beta_1 - 1) q^{29} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2) q^{31} + ( - \beta_{5} + \beta_{4} - 5 \beta_1 - 1) q^{33} + (\beta_{6} - \beta_1) q^{35} + (\beta_{7} + 5 \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{3} - 5 \beta_{2} - 3 \beta_1) q^{37} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{39} + ( - \beta_{5} - 4 \beta_{4} + \beta_1 + 4) q^{41} + (\beta_{7} + 2 \beta_{3} + 5) q^{43} - \beta_{2} q^{45} + (3 \beta_{6} + \beta_{5} - 5 \beta_{4} - 2 \beta_1 + 5) q^{47} + (\beta_{7} - \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_1) q^{49} + ( - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{51} + (3 \beta_{7} - \beta_{3} + 1) q^{53} + (\beta_{6} - 4 \beta_{4} + 4) q^{55} + (\beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{57} + (3 \beta_{7} - 3 \beta_{3} - 2 \beta_{2} + 1) q^{59} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{61} + (\beta_{6} + 2 \beta_{4} - 2) q^{63} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{65} + (2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 3 \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{67} + (3 \beta_{6} - 5 \beta_{4} - 3 \beta_{2}) q^{69} + (5 \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 - 1) q^{71} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{73} - \beta_{3} q^{75} + (3 \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} - 4 \beta_1) q^{77} + (\beta_{6} - 5 \beta_{5} - \beta_{4} - 3 \beta_1 + 1) q^{79} + (\beta_{7} - \beta_{3} - 4 \beta_{2} - 6) q^{81} + ( - \beta_{6} - \beta_{3} + \beta_{2} - \beta_1) q^{83} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{85} + (2 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} - 3 \beta_1 + 4) q^{87} + (6 \beta_{3} - 2 \beta_{2}) q^{89} + (\beta_{7} + \beta_{3} + \beta_{2} - 1) q^{91} + ( - 2 \beta_{6} - 3 \beta_{5} + \beta_{4} - 3 \beta_1 - 1) q^{93} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{2}) q^{95} + (4 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} - 7 \beta_{2}) q^{97} + ( - 3 \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 8 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 8 q^{5} + 2 q^{7} - 2 q^{9} - 15 q^{11} + 3 q^{13} - 2 q^{15} - 3 q^{17} - q^{19} + 7 q^{21} - 6 q^{23} + 8 q^{25} + 2 q^{27} - 5 q^{31} - 8 q^{33} - 2 q^{35} + 3 q^{37} + 14 q^{39} + 18 q^{41} + 34 q^{43} + 2 q^{45} + 14 q^{47} + 12 q^{49} + 4 q^{51} + 4 q^{53} + 15 q^{55} - 8 q^{57} + 12 q^{59} - q^{61} - 9 q^{63} - 3 q^{65} - 13 q^{67} - 17 q^{69} - 9 q^{71} - q^{73} + 2 q^{75} - q^{77} + 5 q^{79} - 40 q^{81} + 3 q^{85} + 13 q^{87} - 8 q^{89} - 14 q^{91} - 2 q^{93} + q^{95} - 13 q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - x^{5} + 24x^{4} - 7x^{3} + 29x^{2} + 12x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -110\nu^{7} + 161\nu^{6} - 506\nu^{5} - 374\nu^{4} - 1748\nu^{3} - 902\nu^{2} - 616\nu - 8939 ) / 3593 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 115\nu^{7} - 5\nu^{6} + 529\nu^{5} + 391\nu^{4} + 3134\nu^{3} + 943\nu^{2} + 644\nu + 1996 ) / 3593 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 499\nu^{7} - 959\nu^{6} + 3014\nu^{5} - 2615\nu^{4} + 10412\nu^{3} - 16029\nu^{2} + 10699\nu + 3412 ) / 14372 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 703\nu^{7} - 343\nu^{6} + 1078\nu^{5} + 953\nu^{4} + 3724\nu^{3} - 5733\nu^{2} - 34149\nu - 3920 ) / 14372 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1057\nu^{7} - 2233\nu^{6} + 7018\nu^{5} - 9341\nu^{4} + 24244\nu^{3} - 37323\nu^{2} + 29633\nu - 25520 ) / 14372 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 460\nu^{7} - 20\nu^{6} + 2116\nu^{5} + 1564\nu^{4} + 8943\nu^{3} + 3772\nu^{2} + 2576\nu + 4391 ) / 3593 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 3\beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + 4\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{6} - \beta_{5} + 12\beta_{4} - \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{7} - 2\beta_{6} - 6\beta_{5} + 8\beta_{4} - 17\beta_{3} + 2\beta_{2} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{7} - 10\beta_{3} + 23\beta_{2} + 53 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 18\beta_{6} + 31\beta_{5} - 53\beta_{4} + 76\beta _1 + 53 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1340\mathbb{Z}\right)^\times\).

\(n\) \(537\) \(671\) \(1141\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
841.1
−0.945610 + 1.63784i
−0.352312 + 0.610223i
0.658715 1.14093i
1.13921 1.97316i
−0.945610 1.63784i
−0.352312 0.610223i
0.658715 + 1.14093i
1.13921 + 1.97316i
0 −1.89122 0 −1.00000 0 −1.23397 + 2.13729i 0 0.576713 0
841.2 0 −0.704624 0 −1.00000 0 0.899440 1.55788i 0 −2.50350 0
841.3 0 1.31743 0 −1.00000 0 1.29090 2.23591i 0 −1.26438 0
841.4 0 2.27841 0 −1.00000 0 0.0436225 0.0755564i 0 2.19117 0
1101.1 0 −1.89122 0 −1.00000 0 −1.23397 2.13729i 0 0.576713 0
1101.2 0 −0.704624 0 −1.00000 0 0.899440 + 1.55788i 0 −2.50350 0
1101.3 0 1.31743 0 −1.00000 0 1.29090 + 2.23591i 0 −1.26438 0
1101.4 0 2.27841 0 −1.00000 0 0.0436225 + 0.0755564i 0 2.19117 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 841.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1340.2.i.b 8
67.c even 3 1 inner 1340.2.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1340.2.i.b 8 1.a even 1 1 trivial
1340.2.i.b 8 67.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - T_{3}^{3} - 5T_{3}^{2} + 3T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1340, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} - 5 T^{2} + 3 T + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + 10 T^{6} - 12 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 15 T^{7} + 147 T^{6} + \cdots + 13456 \) Copy content Toggle raw display
$13$ \( T^{8} - 3 T^{7} + 35 T^{6} + 134 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} + 17 T^{6} + 8 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{8} + T^{7} + 39 T^{6} - 54 T^{5} + \cdots + 1936 \) Copy content Toggle raw display
$23$ \( T^{8} + 6 T^{7} + 44 T^{6} + \cdots + 3481 \) Copy content Toggle raw display
$29$ \( T^{8} + 30 T^{6} - 80 T^{5} + \cdots + 2809 \) Copy content Toggle raw display
$31$ \( T^{8} + 5 T^{7} + 73 T^{6} + \cdots + 85264 \) Copy content Toggle raw display
$37$ \( T^{8} - 3 T^{7} + 131 T^{6} + \cdots + 3655744 \) Copy content Toggle raw display
$41$ \( T^{8} - 18 T^{7} + 220 T^{6} + \cdots + 32041 \) Copy content Toggle raw display
$43$ \( (T^{4} - 17 T^{3} + 73 T^{2} + 21 T - 236)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 14 T^{7} + 204 T^{6} + \cdots + 32041 \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} - 92 T^{2} + 252 T + 568)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 6 T^{3} - 84 T^{2} + 292 T + 2048)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + T^{7} + 79 T^{6} + \cdots + 2238016 \) Copy content Toggle raw display
$67$ \( T^{8} + 13 T^{7} + 283 T^{6} + \cdots + 20151121 \) Copy content Toggle raw display
$71$ \( T^{8} + 9 T^{7} + 269 T^{6} + \cdots + 135424 \) Copy content Toggle raw display
$73$ \( T^{8} + T^{7} + 43 T^{6} - 130 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$79$ \( T^{8} - 5 T^{7} + 267 T^{6} + \cdots + 113379904 \) Copy content Toggle raw display
$83$ \( T^{8} + 16 T^{6} + 36 T^{5} + 255 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} - 264 T^{2} + 320 T + 9136)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 13 T^{7} + \cdots + 562069264 \) Copy content Toggle raw display
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