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Properties

Label 1040.2.k.e
Level $1040$
Weight $2$
Character orbit 1040.k
Analytic conductor $8.304$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1040,2,Mod(961,1040)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.k (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 36x^{4} - 52x^{3} + 50x^{2} + 140x + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 520)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{2} q^{5} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{7} + (\beta_{7} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{2} q^{5} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{7} + (\beta_{7} + 2) q^{9} - \beta_{4} q^{11} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{13} + (\beta_{5} + \beta_{2}) q^{15} + ( - \beta_{7} - \beta_{3} + \beta_1 - 1) q^{17} + (\beta_{6} - \beta_{5}) q^{19} + (\beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{21} + (\beta_{3} + 2 \beta_1 + 2) q^{23} - q^{25} + ( - \beta_{7} + 3 \beta_{3} - \beta_1 + 1) q^{27} + ( - \beta_{7} - 2 \beta_{3} + 1) q^{29} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{2}) q^{31} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{33} + ( - \beta_{3} + \beta_1 - 2) q^{35} + (3 \beta_{5} - \beta_{4} + 3 \beta_{2}) q^{37} + (\beta_{7} + 2 \beta_{5} - \beta_{4} + 5) q^{39} + ( - \beta_{6} + \beta_{5} + \beta_{4}) q^{41} + (\beta_{7} + \beta_1 + 3) q^{43} + (\beta_{6} - 2 \beta_{2}) q^{45} + (2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2}) q^{47} + ( - 2 \beta_{3} + 4 \beta_1 - 5) q^{49} + ( - 4 \beta_{3} - 4) q^{51} + (\beta_{7} - \beta_{3} - \beta_1 - 1) q^{53} + \beta_1 q^{55} + ( - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} + 7 \beta_{2}) q^{57} + ( - \beta_{6} - \beta_{5} - 2 \beta_{2}) q^{59} + ( - \beta_{7} + 2 \beta_{3} + 2 \beta_1 + 5) q^{61} + (5 \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{63} + (\beta_{5} + \beta_{2} - \beta_1 - 1) q^{65} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2}) q^{67} + (\beta_{7} + 4 \beta_{3} - 2 \beta_1 + 9) q^{69} + ( - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{71} + (\beta_{5} - 3 \beta_{4} + \beta_{2}) q^{73} - \beta_{3} q^{75} + (\beta_{7} + \beta_{3} + 3 \beta_1 - 5) q^{77} + (2 \beta_{3} + 2 \beta_1 + 4) q^{79} + (\beta_{7} - 4 \beta_{3} + 2 \beta_1 + 6) q^{81} + (3 \beta_{5} - \beta_{4} + 9 \beta_{2}) q^{83} + ( - \beta_{6} - \beta_{5} + \beta_{4}) q^{85} + ( - \beta_{7} - 3 \beta_{3} + \beta_1 - 11) q^{87} + ( - 2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2}) q^{89} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{91} + ( - 7 \beta_{5} - \beta_{4} - 5 \beta_{2}) q^{93} + ( - \beta_{7} + \beta_{3} + 1) q^{95} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{2}) q^{97} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 16 q^{9} - 4 q^{13} - 4 q^{17} + 12 q^{23} - 8 q^{25} - 4 q^{27} + 16 q^{29} - 12 q^{35} + 40 q^{39} + 24 q^{43} - 32 q^{49} - 16 q^{51} - 4 q^{53} + 32 q^{61} - 8 q^{65} + 56 q^{69} + 4 q^{75} - 44 q^{77} + 24 q^{79} + 64 q^{81} - 76 q^{87} + 32 q^{91} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 36x^{4} - 52x^{3} + 50x^{2} + 140x + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -295\nu^{7} + 581\nu^{6} - 1975\nu^{5} + 12197\nu^{4} - 28834\nu^{3} + 14811\nu^{2} + 52682\nu + 217006 ) / 103451 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4733 \nu^{7} + 18790 \nu^{6} - 22920 \nu^{5} - 15070 \nu^{4} - 128766 \nu^{3} + 794160 \nu^{2} - 577130 \nu - 329546 ) / 1448314 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 371\nu^{7} - 380\nu^{6} - 1023\nu^{5} + 15170\nu^{4} + 10312\nu^{3} - 9509\nu^{2} - 26978\nu + 283268 ) / 103451 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4804 \nu^{7} - 30853 \nu^{6} + 39176 \nu^{5} + 9329 \nu^{4} - 16490 \nu^{3} - 693923 \nu^{2} + 263566 \nu + 195258 ) / 724157 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23665 \nu^{7} - 93950 \nu^{6} + 114600 \nu^{5} + 75350 \nu^{4} + 643830 \nu^{3} - 2522486 \nu^{2} + 2885650 \nu + 1647730 ) / 1448314 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37489 \nu^{7} - 35610 \nu^{6} + 179096 \nu^{5} + 120284 \nu^{4} + 1042570 \nu^{3} + 42688 \nu^{2} + 4620886 \nu + 2631678 ) / 1448314 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2968\nu^{7} - 3040\nu^{6} - 8184\nu^{5} + 17909\nu^{4} + 82496\nu^{3} - 76072\nu^{2} - 215824\nu + 93673 ) / 103451 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 5\beta_{5} - 5\beta_{4} + 5\beta_{3} + 8\beta_{2} - 5\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + 8\beta_{3} - 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{7} + 5\beta_{6} - 16\beta_{5} + 13\beta_{4} + 16\beta_{3} - 29\beta_{2} - 13\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{6} - 57\beta_{5} + \beta_{4} - 180\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 41\beta_{7} + 41\beta_{6} - 110\beta_{5} + 70\beta_{4} - 110\beta_{3} - 211\beta_{2} + 70\beta _1 + 101 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
1.90427 + 1.90427i
1.90427 − 1.90427i
−1.61013 − 1.61013i
−1.61013 + 1.61013i
1.47812 + 1.47812i
1.47812 − 1.47812i
−0.772270 − 0.772270i
−0.772270 + 0.772270i
0 −3.25249 0 − 1.00000i 0 1.80854i 0 7.57872 0
961.2 0 −3.25249 0 1.00000i 0 − 1.80854i 0 7.57872 0
961.3 0 −1.18501 0 − 1.00000i 0 − 5.22025i 0 −1.59576 0
961.4 0 −1.18501 0 1.00000i 0 5.22025i 0 −1.59576 0
961.5 0 −0.369700 0 − 1.00000i 0 0.956248i 0 −2.86332 0
961.6 0 −0.369700 0 1.00000i 0 − 0.956248i 0 −2.86332 0
961.7 0 2.80720 0 − 1.00000i 0 − 3.54454i 0 4.88037 0
961.8 0 2.80720 0 1.00000i 0 3.54454i 0 4.88037 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.k.e 8
4.b odd 2 1 520.2.k.b ✓ 8
12.b even 2 1 4680.2.g.k 8
13.b even 2 1 inner 1040.2.k.e 8
20.d odd 2 1 2600.2.k.c 8
20.e even 4 1 2600.2.f.e 8
20.e even 4 1 2600.2.f.f 8
52.b odd 2 1 520.2.k.b ✓ 8
52.f even 4 1 6760.2.a.bc 4
52.f even 4 1 6760.2.a.bd 4
156.h even 2 1 4680.2.g.k 8
260.g odd 2 1 2600.2.k.c 8
260.p even 4 1 2600.2.f.e 8
260.p even 4 1 2600.2.f.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.k.b ✓ 8 4.b odd 2 1
520.2.k.b ✓ 8 52.b odd 2 1
1040.2.k.e 8 1.a even 1 1 trivial
1040.2.k.e 8 13.b even 2 1 inner
2600.2.f.e 8 20.e even 4 1
2600.2.f.e 8 260.p even 4 1
2600.2.f.f 8 20.e even 4 1
2600.2.f.f 8 260.p even 4 1
2600.2.k.c 8 20.d odd 2 1
2600.2.k.c 8 260.g odd 2 1
4680.2.g.k 8 12.b even 2 1
4680.2.g.k 8 156.h even 2 1
6760.2.a.bc 4 52.f even 4 1
6760.2.a.bd 4 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2T_{3}^{3} - 8T_{3}^{2} - 14T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} - 8 T^{2} - 14 T - 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 44 T^{6} + 512 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( T^{8} + 28 T^{6} + 180 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + 20 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} + 2 T^{3} - 40 T^{2} - 128 T - 64)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 104 T^{6} + 3004 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
$23$ \( (T^{4} - 6 T^{3} - 64 T^{2} + 390 T + 56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} - 44 T^{2} + 452 T - 664)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 216 T^{6} + 16612 T^{4} + \cdots + 6130576 \) Copy content Toggle raw display
$37$ \( T^{8} + 172 T^{6} + 7728 T^{4} + \cdots + 53824 \) Copy content Toggle raw display
$41$ \( T^{8} + 124 T^{6} + 4144 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$43$ \( (T^{4} - 12 T^{3} - 8 T^{2} + 338 T - 236)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 284 T^{6} + 24384 T^{4} + \cdots + 7311616 \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{3} - 60 T^{2} - 72 T + 128)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 96 T^{6} + 3052 T^{4} + \cdots + 50176 \) Copy content Toggle raw display
$61$ \( (T^{4} - 16 T^{3} - 48 T^{2} + 876 T + 2264)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 388 T^{6} + \cdots + 63744256 \) Copy content Toggle raw display
$71$ \( T^{8} + 348 T^{6} + 35212 T^{4} + \cdots + 1430416 \) Copy content Toggle raw display
$73$ \( T^{8} + 236 T^{6} + 14896 T^{4} + \cdots + 222784 \) Copy content Toggle raw display
$79$ \( (T^{4} - 12 T^{3} - 64 T^{2} + 672 T + 1024)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 388 T^{6} + 35088 T^{4} + \cdots + 891136 \) Copy content Toggle raw display
$89$ \( T^{8} + 336 T^{6} + 27136 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$97$ \( T^{8} + 336 T^{6} + 30176 T^{4} + \cdots + 215296 \) Copy content Toggle raw display
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