Properties

Label 1040.2.dh.a
Level $1040$
Weight $2$
Character orbit 1040.dh
Analytic conductor $8.304$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1040,2,Mod(289,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.dh (of order \(6\), degree \(2\), not 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: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 8x^{10} + 54x^{8} - 78x^{6} + 92x^{4} - 10x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} - \beta_1) q^{3} + (\beta_{7} + \beta_{6}) q^{5} + \beta_{2} q^{7} + ( - \beta_{7} + \beta_{5} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} - \beta_1) q^{3} + (\beta_{7} + \beta_{6}) q^{5} + \beta_{2} q^{7} + ( - \beta_{7} + \beta_{5} - \beta_{3} + 1) q^{9} + (\beta_{8} + \beta_{6} - \beta_{3} + 1) q^{11} + ( - \beta_{10} + \beta_{8} - \beta_{6} + \beta_{5} - \beta_{3} + 1) q^{13} + ( - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1) q^{15} + ( - \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} - 1) q^{17} + ( - \beta_{7} - 2 \beta_{5} - \beta_{3} - 2) q^{19} + ( - \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{4} - \beta_{2} + 1) q^{21} + ( - \beta_{10} + \beta_1) q^{23} + ( - \beta_{11} - \beta_{8} - \beta_{4} - \beta_{2} - 1) q^{25} + (\beta_{11} + \beta_{2}) q^{27} - 3 \beta_{5} q^{29} + ( - \beta_{11} + \beta_{10} + 2 \beta_{4} - \beta_{2} + 2) q^{31} + (\beta_{2} + 2 \beta_1) q^{33} + ( - \beta_{9} - 3 \beta_{5} + 2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{35} + ( - \beta_{11} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{3} - 1) q^{37} + (2 \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{39} + ( - \beta_{11} + \beta_{10} - 2 \beta_{9} - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + \beta_1) q^{41} + (\beta_{2} + 2 \beta_1) q^{43} + (\beta_{9} - 5 \beta_{5} + \beta_{3} + 2 \beta_{2} - \beta_1 - 5) q^{45} + ( - 2 \beta_{11} + 4 \beta_{10} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{2}) q^{47} + ( - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{3} - 1) q^{49} + (\beta_{8} + \beta_{7} + \beta_{6} - 1) q^{51} + (\beta_{11} + 3 \beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{2}) q^{53} + ( - \beta_{11} + \beta_{9} - \beta_{8} + \beta_{6} - 5 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{55} + (\beta_{11} + \beta_{2}) q^{57} + ( - 2 \beta_{9} + \beta_{7} + \beta_{3} - \beta_{2} + \beta_1) q^{59} + ( - 2 \beta_{7} + \beta_{5} - 2 \beta_{3} + 1) q^{61} + ( - \beta_{11} - 3 \beta_{10} + 2 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + \cdots + 2) q^{63}+ \cdots + ( - \beta_{11} + \beta_{10} + 2 \beta_{4} - \beta_{2} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{5} + 6 q^{9} + 4 q^{15} - 12 q^{19} - 8 q^{21} - 2 q^{25} + 18 q^{29} + 16 q^{31} - 10 q^{35} + 32 q^{39} + 14 q^{41} - 29 q^{45} + 6 q^{49} - 24 q^{51} + 26 q^{55} + 4 q^{59} + 6 q^{61} + 23 q^{65} - 24 q^{69} + 12 q^{71} - 2 q^{75} + 104 q^{79} + 14 q^{81} + 21 q^{85} + 20 q^{89} + 44 q^{91} - 20 q^{95} - 104 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 8x^{10} + 54x^{8} - 78x^{6} + 92x^{4} - 10x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 108\nu^{11} - 729\nu^{9} + 4768\nu^{7} - 1242\nu^{5} + 135\nu^{3} + 9934\nu ) / 1222 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 108\nu^{11} - 729\nu^{9} + 4768\nu^{7} - 1242\nu^{5} + 135\nu^{3} + 12378\nu ) / 1222 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 92 \nu^{11} + 293 \nu^{10} + 621 \nu^{9} - 2436 \nu^{8} - 4039 \nu^{7} + 16443 \nu^{6} + 1058 \nu^{5} - 26893 \nu^{4} - 115 \nu^{3} + 28014 \nu^{2} - 5181 \nu - 3045 ) / 2444 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 135 \nu^{11} - 16 \nu^{10} - 1064 \nu^{9} + 108 \nu^{8} + 7182 \nu^{7} - 729 \nu^{6} - 9801 \nu^{5} + 184 \nu^{4} + 12236 \nu^{3} - 20 \nu^{2} - 108 \nu - 3531 ) / 1222 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -135\nu^{10} + 1064\nu^{8} - 7182\nu^{6} + 9801\nu^{4} - 12236\nu^{2} + 108 ) / 1222 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 563 \nu^{11} - 655 \nu^{10} - 4564 \nu^{9} + 5185 \nu^{8} + 30807 \nu^{7} - 34846 \nu^{6} - 46495 \nu^{5} + 47553 \nu^{4} + 52486 \nu^{3} - 52601 \nu^{2} - 5705 \nu + 524 ) / 2444 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 92 \nu^{11} + 563 \nu^{10} - 621 \nu^{9} - 4564 \nu^{8} + 4039 \nu^{7} + 30807 \nu^{6} - 1058 \nu^{5} - 46495 \nu^{4} + 115 \nu^{3} + 52486 \nu^{2} + 5181 \nu - 5705 ) / 2444 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 655 \nu^{11} - 92 \nu^{10} + 5185 \nu^{9} + 621 \nu^{8} - 34846 \nu^{7} - 4039 \nu^{6} + 47553 \nu^{5} + 1058 \nu^{4} - 52601 \nu^{3} - 115 \nu^{2} + 524 \nu - 5181 ) / 2444 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 389\nu^{10} - 3084\nu^{8} + 20817\nu^{6} - 29219\nu^{4} + 35466\nu^{2} - 1222\nu - 3855 ) / 1222 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1195\nu^{11} - 9441\nu^{9} + 63574\nu^{7} - 86757\nu^{5} + 100323\nu^{3} - 956\nu ) / 1222 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1357\nu^{11} - 10840\nu^{9} + 73170\nu^{7} - 105117\nu^{5} + 124660\nu^{3} - 13550\nu ) / 1222 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{10} - 2\beta_{9} - 6\beta_{5} + 2\beta_{4} - 2\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} - 3\beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -14\beta_{9} + 2\beta_{7} - 34\beta_{5} + 2\beta_{3} - 7\beta_{2} + 7\beta _1 - 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23\beta_{11} - 37\beta_{10} - 16\beta_{8} + 16\beta_{6} - 16\beta_{5} + 16\beta_{3} + 37\beta _1 - 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23\beta_{11} - 23\beta_{10} + 8\beta_{8} + 8\beta_{7} + 8\beta_{6} - 46\beta_{4} + 23\beta_{2} - 99 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -108\beta_{7} - 108\beta_{5} + 108\beta_{3} - 145\beta_{2} + 237\beta _1 - 108 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 299 \beta_{11} - 299 \beta_{10} + 598 \beta_{9} + 108 \beta_{8} + 108 \beta_{6} + 1378 \beta_{5} - 598 \beta_{4} - 108 \beta_{3} + 598 \beta_{2} - 299 \beta _1 + 108 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -467\beta_{11} + 766\beta_{10} + 353\beta_{8} - 353\beta_{7} - 353\beta_{6} - 467\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3878\beta_{9} - 706\beta_{7} + 8918\beta_{5} - 706\beta_{3} + 1939\beta_{2} - 1939\beta _1 + 8918 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6045 \beta_{11} + 9923 \beta_{10} + 4584 \beta_{8} - 4584 \beta_{6} + 4584 \beta_{5} - 4584 \beta_{3} - 9923 \beta _1 + 4584 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(-1\) \(-1 - \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.286513 + 0.165418i
−2.20467 1.27287i
−1.02826 0.593667i
1.02826 + 0.593667i
2.20467 + 1.27287i
−0.286513 0.165418i
0.286513 0.165418i
−2.20467 + 1.27287i
−1.02826 + 0.593667i
1.02826 0.593667i
2.20467 1.27287i
−0.286513 + 0.165418i
0 −2.33117 + 1.34590i 0 −2.12291 + 0.702335i 0 2.90420 + 1.67674i 0 2.12291 3.67698i 0
289.2 0 −1.86449 + 1.07646i 0 −0.817544 2.08125i 0 −2.54486 1.46928i 0 0.817544 1.41603i 0
289.3 0 −0.298874 + 0.172555i 0 1.44045 + 1.71029i 0 −1.75765 1.01478i 0 −1.44045 + 2.49493i 0
289.4 0 0.298874 0.172555i 0 1.44045 1.71029i 0 1.75765 + 1.01478i 0 −1.44045 + 2.49493i 0
289.5 0 1.86449 1.07646i 0 −0.817544 + 2.08125i 0 2.54486 + 1.46928i 0 0.817544 1.41603i 0
289.6 0 2.33117 1.34590i 0 −2.12291 0.702335i 0 −2.90420 1.67674i 0 2.12291 3.67698i 0
529.1 0 −2.33117 1.34590i 0 −2.12291 0.702335i 0 2.90420 1.67674i 0 2.12291 + 3.67698i 0
529.2 0 −1.86449 1.07646i 0 −0.817544 + 2.08125i 0 −2.54486 + 1.46928i 0 0.817544 + 1.41603i 0
529.3 0 −0.298874 0.172555i 0 1.44045 1.71029i 0 −1.75765 + 1.01478i 0 −1.44045 2.49493i 0
529.4 0 0.298874 + 0.172555i 0 1.44045 + 1.71029i 0 1.75765 1.01478i 0 −1.44045 2.49493i 0
529.5 0 1.86449 + 1.07646i 0 −0.817544 2.08125i 0 2.54486 1.46928i 0 0.817544 + 1.41603i 0
529.6 0 2.33117 + 1.34590i 0 −2.12291 + 0.702335i 0 −2.90420 + 1.67674i 0 2.12291 + 3.67698i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.dh.a 12
4.b odd 2 1 65.2.n.a 12
5.b even 2 1 inner 1040.2.dh.a 12
12.b even 2 1 585.2.bs.a 12
13.c even 3 1 inner 1040.2.dh.a 12
20.d odd 2 1 65.2.n.a 12
20.e even 4 2 325.2.e.e 12
52.b odd 2 1 845.2.n.e 12
52.f even 4 2 845.2.l.f 24
52.i odd 6 1 845.2.b.e 6
52.i odd 6 1 845.2.n.e 12
52.j odd 6 1 65.2.n.a 12
52.j odd 6 1 845.2.b.d 6
52.l even 12 2 845.2.d.d 12
52.l even 12 2 845.2.l.f 24
60.h even 2 1 585.2.bs.a 12
65.n even 6 1 inner 1040.2.dh.a 12
156.p even 6 1 585.2.bs.a 12
260.g odd 2 1 845.2.n.e 12
260.u even 4 2 845.2.l.f 24
260.v odd 6 1 65.2.n.a 12
260.v odd 6 1 845.2.b.d 6
260.w odd 6 1 845.2.b.e 6
260.w odd 6 1 845.2.n.e 12
260.bc even 12 2 845.2.d.d 12
260.bc even 12 2 845.2.l.f 24
260.bg even 12 2 4225.2.a.bq 6
260.bj even 12 2 325.2.e.e 12
260.bj even 12 2 4225.2.a.br 6
780.br even 6 1 585.2.bs.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 4.b odd 2 1
65.2.n.a 12 20.d odd 2 1
65.2.n.a 12 52.j odd 6 1
65.2.n.a 12 260.v odd 6 1
325.2.e.e 12 20.e even 4 2
325.2.e.e 12 260.bj even 12 2
585.2.bs.a 12 12.b even 2 1
585.2.bs.a 12 60.h even 2 1
585.2.bs.a 12 156.p even 6 1
585.2.bs.a 12 780.br even 6 1
845.2.b.d 6 52.j odd 6 1
845.2.b.d 6 260.v odd 6 1
845.2.b.e 6 52.i odd 6 1
845.2.b.e 6 260.w odd 6 1
845.2.d.d 12 52.l even 12 2
845.2.d.d 12 260.bc even 12 2
845.2.l.f 24 52.f even 4 2
845.2.l.f 24 52.l even 12 2
845.2.l.f 24 260.u even 4 2
845.2.l.f 24 260.bc even 12 2
845.2.n.e 12 52.b odd 2 1
845.2.n.e 12 52.i odd 6 1
845.2.n.e 12 260.g odd 2 1
845.2.n.e 12 260.w odd 6 1
1040.2.dh.a 12 1.a even 1 1 trivial
1040.2.dh.a 12 5.b even 2 1 inner
1040.2.dh.a 12 13.c even 3 1 inner
1040.2.dh.a 12 65.n even 6 1 inner
4225.2.a.bq 6 260.bg even 12 2
4225.2.a.br 6 260.bj even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 12T_{3}^{10} + 109T_{3}^{8} - 412T_{3}^{6} + 1177T_{3}^{4} - 140T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 12 T^{10} + 109 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{6} + 3 T^{5} + 5 T^{4} + 10 T^{3} + \cdots + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 24 T^{10} + 397 T^{8} + \cdots + 160000 \) Copy content Toggle raw display
$11$ \( (T^{6} + 13 T^{4} + 16 T^{3} + 169 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 15 T^{10} + 39 T^{8} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 35 T^{10} + 1062 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$19$ \( (T^{6} + 6 T^{5} + 37 T^{4} + 14 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{10} + 109 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 40 T - 40)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} - 35 T^{10} + 1062 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$41$ \( (T^{6} - 7 T^{5} + 78 T^{4} + 213 T^{3} + \cdots + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 80 T^{10} + 6117 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$47$ \( (T^{6} + 236 T^{4} + 14640 T^{2} + \cdots + 270400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 171 T^{4} + 1040 T^{2} + \cdots + 400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 2 T^{5} + 59 T^{4} - 162 T^{3} + \cdots + 18496)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 3 T^{5} + 58 T^{4} - 83 T^{3} + \cdots + 13225)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 100 T^{10} + \cdots + 406586896 \) Copy content Toggle raw display
$71$ \( (T^{6} - 6 T^{5} + 37 T^{4} - 46 T^{3} + \cdots + 676)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 215 T^{4} + 13900 T^{2} + \cdots + 250000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 26 T^{2} + 180 T - 160)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 276 T^{4} + 23600 T^{2} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 10 T^{5} + 257 T^{4} + \cdots + 2515396)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 280 T^{10} + \cdots + 41740124416 \) Copy content Toggle raw display
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