Properties

Label 869.2.l.a
Level $869$
Weight $2$
Character orbit 869.l
Analytic conductor $6.939$
Analytic rank $0$
Dimension $8$
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] = [869,2,Mod(315,869)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("869.315");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 869.l (of order \(10\), degree \(4\), 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: \(6.93899993565\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.484000000.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{2} + (\beta_{7} + \beta_{6} - \beta_{4} - \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + (3 \beta_{3} + 2 \beta_{2} + 3) q^{5} + (2 \beta_{7} + \beta_{6}) q^{6} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{7} + (2 \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{8} + (5 \beta_{5} - \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{2} + (\beta_{7} + \beta_{6} - \beta_{4} - \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + (3 \beta_{3} + 2 \beta_{2} + 3) q^{5} + (2 \beta_{7} + \beta_{6}) q^{6} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{7} + (2 \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{8} + (5 \beta_{5} - \beta_{3} - \beta_{2}) q^{9} + ( - \beta_{5} + 6 \beta_{3} + 5 \beta_{2} + 3) q^{10} + ( - 2 \beta_{5} - 2 \beta_{3} + \beta_{2}) q^{11} + (\beta_{7} + 2 \beta_{6} - \beta_1) q^{12} + (2 \beta_{5} - 4 \beta_{3} - 4) q^{13} + ( - \beta_{7} - \beta_{6} + \beta_{4} + \beta_1) q^{14} + (6 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 3 \beta_1) q^{15} + ( - \beta_{3} - 4 \beta_{2} - 1) q^{16} - 3 \beta_{6} q^{17} + (4 \beta_{5} + 4 \beta_{3} - \beta_{2} + 1) q^{18} + ( - 4 \beta_{5} + 4 \beta_{3} + 8 \beta_{2} + 4) q^{19} + (3 \beta_{5} + 2 \beta_{3} + 2 \beta_{2}) q^{20} + ( - 5 \beta_{5} + 4 \beta_{3} - \beta_{2} + 2) q^{21} + ( - \beta_{5} - 4 \beta_{3} + \beta_{2} + 2) q^{22} + ( - 5 \beta_{5} + 5 \beta_{2} + 1) q^{23} + (\beta_{7} + \beta_{6} - 3 \beta_{4} - 4 \beta_1) q^{24} + (3 \beta_{5} + 5 \beta_{3} - 3) q^{25} + (6 \beta_{5} - 6 \beta_{3} - 4 \beta_{2} - 4) q^{26} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 4 \beta_1) q^{27} - \beta_{6} q^{28} + ( - 2 \beta_{6} + 2 \beta_{4} + \beta_1) q^{29} + (8 \beta_{7} + \beta_{6} + 8 \beta_{4} - \beta_1) q^{30} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2}) q^{31} + (\beta_{5} - 6 \beta_{3} - 5 \beta_{2} - 3) q^{32} + ( - 4 \beta_{7} + \beta_{6} - \beta_{4} + 4 \beta_1) q^{33} + ( - 3 \beta_{7} - 6 \beta_{6} + 3 \beta_1) q^{34} + ( - 3 \beta_{7} - 3 \beta_{6} + \beta_{4} + 4 \beta_1) q^{35} + (4 \beta_{5} - \beta_{3} - 4) q^{36} + (2 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - \beta_1) q^{37} + (4 \beta_{3} + 12 \beta_{2} + 4) q^{38} + ( - 6 \beta_{7} + 2 \beta_{6}) q^{39} + (\beta_{5} + \beta_{3} - 4 \beta_{2} + 4) q^{40} + ( - 2 \beta_{7} - 5 \beta_{6} - 2 \beta_{4} + 5 \beta_1) q^{41} + ( - 8 \beta_{5} + \beta_{3} + \beta_{2}) q^{42} + ( - \beta_{7} + 6 \beta_{4} + \beta_1) q^{43} + ( - 4 \beta_{5} + \beta_{3} + 3 \beta_{2} + 4) q^{44} + (13 \beta_{5} - 13 \beta_{2} - 5) q^{45} + ( - \beta_{5} - 4 \beta_{3} + 6 \beta_{2} + 2) q^{46} + (3 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} - 4 \beta_1) q^{47} + ( - 2 \beta_{7} - 4 \beta_{6} - 4 \beta_{4} + \beta_1) q^{48} + ( - 2 \beta_{3} - 4 \beta_{2} - 2) q^{49} + (6 \beta_{5} + 5 \beta_{3} - 3 \beta_{2} - 11) q^{50} + ( - 3 \beta_{5} - 3 \beta_{3} - 9 \beta_{2} + 9) q^{51} + ( - 2 \beta_{5} - 2) q^{52} + (3 \beta_{7} - \beta_{4} - 3 \beta_1) q^{54} + ( - \beta_{5} - 3 \beta_{3} + 10 \beta_{2} + 5) q^{55} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 3 \beta_1) q^{56} + (4 \beta_{7} + 4 \beta_{6} + 8 \beta_{4} + 4 \beta_1) q^{57} + ( - 2 \beta_{7} - 3 \beta_{6} + 2 \beta_{4} + \beta_1) q^{58} + (2 \beta_{7} + \beta_{6} + \beta_{4} - \beta_1) q^{59} + (5 \beta_{7} + 3 \beta_{6} + 4 \beta_{4} - 6 \beta_1) q^{60} + (2 \beta_{7} + 3 \beta_{6}) q^{61} + ( - 3 \beta_{5} - 3 \beta_{3} - \beta_{2} + 1) q^{62} + (\beta_{7} - 5 \beta_{6} + \beta_{4} + 5 \beta_1) q^{63} + (5 \beta_{5} - 6 \beta_{3} - 6 \beta_{2}) q^{64} + (10 \beta_{5} - 16 \beta_{3} - 6 \beta_{2} - 8) q^{65} + ( - 3 \beta_{7} + 6 \beta_{6} - 5 \beta_{4}) q^{66} + (8 \beta_{5} - 8 \beta_{2} - 7) q^{67} + ( - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{4}) q^{68} + ( - 4 \beta_{7} + \beta_{6} + 4 \beta_{4} + 9 \beta_1) q^{69} + ( - 6 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 3 \beta_1) q^{70} + ( - 2 \beta_{7} + 2 \beta_{6} - 8 \beta_{4} - 4 \beta_1) q^{71} + (12 \beta_{5} - 11 \beta_{3} - 6 \beta_{2} - 1) q^{72} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{73} + (5 \beta_{7} + 5 \beta_{6} + 5 \beta_{4} - 5 \beta_1) q^{74} + (5 \beta_{7} - 5 \beta_{6} + 8 \beta_{4} - 3 \beta_1) q^{75} + (8 \beta_{3} + 8 \beta_{2} + 4) q^{76} + (2 \beta_{7} + 2 \beta_{6} - \beta_{4} - 5 \beta_1) q^{77} + ( - 4 \beta_{7} + 4 \beta_{6} - 6 \beta_{4} - 8 \beta_1) q^{78} + ( - \beta_{7} - \beta_{6} - 4 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 8) q^{79} + ( - 11 \beta_{5} + 11) q^{80} + (2 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 4) q^{81} + ( - 7 \beta_{7} - 5 \beta_{6} - 4 \beta_{4} + 10 \beta_1) q^{82} + (10 \beta_{5} - 4 \beta_{3} - 5 \beta_{2} - 6) q^{83} + ( - \beta_{5} - \beta_{3} - 3 \beta_{2} + 3) q^{84} + ( - 6 \beta_{7} - 9 \beta_{6} - 6 \beta_{4} + 9 \beta_1) q^{85} + ( - \beta_{7} + \beta_{6} + 5 \beta_{4} - 6 \beta_1) q^{86} + ( - 8 \beta_{5} + 2 \beta_{3} - 6 \beta_{2} + 1) q^{87} + ( - 3 \beta_{5} - \beta_{3} + 3 \beta_{2} - 5) q^{88} + (2 \beta_{5} - 2 \beta_{2} - 1) q^{89} + (5 \beta_{5} + 8 \beta_{3} - 18 \beta_{2} - 10) q^{90} + (4 \beta_{7} + 2 \beta_{6} - 4 \beta_{4} - 6 \beta_1) q^{91} + ( - 5 \beta_{5} + 5 \beta_{3} + 6 \beta_{2} + 6) q^{92} + ( - 3 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 4 \beta_1) q^{93} + 5 \beta_{7} q^{94} + (8 \beta_{5} + 8 \beta_{3} + 12 \beta_{2} - 12) q^{95} + ( - 8 \beta_{7} - \beta_{6} - 8 \beta_{4} + \beta_1) q^{96} + 2 \beta_{5} q^{97} + ( - 2 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} - 2) q^{98} + ( - 11 \beta_{5} + 8 \beta_{3} + 17 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{2} + 6 q^{4} + 14 q^{5} + 10 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{2} + 6 q^{4} + 14 q^{5} + 10 q^{8} + 14 q^{9} - 2 q^{11} - 20 q^{13} + 2 q^{16} + 10 q^{18} - 2 q^{20} + 20 q^{22} - 12 q^{23} - 28 q^{25} - 22 q^{36} + 40 q^{40} - 20 q^{42} + 16 q^{44} + 12 q^{45} + 10 q^{46} - 4 q^{49} - 80 q^{50} + 90 q^{51} - 20 q^{52} + 24 q^{55} + 10 q^{62} + 34 q^{64} - 24 q^{67} + 50 q^{72} + 10 q^{73} + 40 q^{79} + 66 q^{80} + 32 q^{81} - 10 q^{83} + 30 q^{84} - 50 q^{88} - 50 q^{90} + 16 q^{92} - 120 q^{95} + 4 q^{97} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{6} - 37\nu^{4} + 629\nu^{2} - 363 ) / 1991 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -28\nu^{6} + 148\nu^{4} - 525\nu^{2} - 539 ) / 1991 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -28\nu^{7} + 148\nu^{5} - 525\nu^{3} - 539\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\nu^{6} + 73\nu^{4} + 750\nu^{2} + 2761 ) / 1991 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -61\nu^{7} + 38\nu^{5} - 646\nu^{3} - 1672\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 68\nu^{7} - 75\nu^{5} + 1275\nu^{3} + 3300\nu ) / 1991 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} + 4\beta_{6} + \beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 10\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{7} + 17\beta_{4} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{5} - 37\beta_{3} - 75\beta_{2} - 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -38\beta_{7} - 75\beta_{6} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/869\mathbb{Z}\right)^\times\).

\(n\) \(475\) \(793\)
\(\chi(n)\) \(1 + \beta_{2} + \beta_{3} - \beta_{5}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
315.1
0.476925 + 1.46782i
−0.476925 1.46782i
−1.73855 1.26313i
1.73855 + 1.26313i
−1.73855 + 1.26313i
1.73855 1.26313i
0.476925 1.46782i
−0.476925 + 1.46782i
0.690983 0.951057i −2.79197 0.907165i 0.190983 + 0.587785i 2.30902 1.67760i −2.79197 + 2.02848i 0.771681 + 2.37499i 2.92705 + 0.951057i 4.54508 + 3.30220i 3.35520i
315.2 0.690983 0.951057i 2.79197 + 0.907165i 0.190983 + 0.587785i 2.30902 1.67760i 2.79197 2.02848i −0.771681 2.37499i 2.92705 + 0.951057i 4.54508 + 3.30220i 3.35520i
552.1 1.80902 + 0.587785i −1.48490 2.04378i 1.30902 + 0.951057i 1.19098 + 3.66547i −1.48490 4.57004i 1.07448 + 0.780656i −0.427051 0.587785i −1.04508 + 3.21644i 7.33094i
552.2 1.80902 + 0.587785i 1.48490 + 2.04378i 1.30902 + 0.951057i 1.19098 + 3.66547i 1.48490 + 4.57004i −1.07448 0.780656i −0.427051 0.587785i −1.04508 + 3.21644i 7.33094i
710.1 1.80902 0.587785i −1.48490 + 2.04378i 1.30902 0.951057i 1.19098 3.66547i −1.48490 + 4.57004i 1.07448 0.780656i −0.427051 + 0.587785i −1.04508 3.21644i 7.33094i
710.2 1.80902 0.587785i 1.48490 2.04378i 1.30902 0.951057i 1.19098 3.66547i 1.48490 4.57004i −1.07448 + 0.780656i −0.427051 + 0.587785i −1.04508 3.21644i 7.33094i
789.1 0.690983 + 0.951057i −2.79197 + 0.907165i 0.190983 0.587785i 2.30902 + 1.67760i −2.79197 2.02848i 0.771681 2.37499i 2.92705 0.951057i 4.54508 3.30220i 3.35520i
789.2 0.690983 + 0.951057i 2.79197 0.907165i 0.190983 0.587785i 2.30902 + 1.67760i 2.79197 + 2.02848i −0.771681 + 2.37499i 2.92705 0.951057i 4.54508 3.30220i 3.35520i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 315.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
79.b odd 2 1 inner
869.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 869.2.l.a 8
11.d odd 10 1 inner 869.2.l.a 8
79.b odd 2 1 inner 869.2.l.a 8
869.l even 10 1 inner 869.2.l.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.2.l.a 8 1.a even 1 1 trivial
869.2.l.a 8 11.d odd 10 1 inner
869.2.l.a 8 79.b odd 2 1 inner
869.2.l.a 8 869.l even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5T_{2}^{3} + 10T_{2}^{2} - 10T_{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(869, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 5 T^{3} + 10 T^{2} - 10 T + 5)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 10 T^{6} + 60 T^{4} + \cdots + 3025 \) Copy content Toggle raw display
$5$ \( (T^{4} - 7 T^{3} + 34 T^{2} - 88 T + 121)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 9 T^{6} + 31 T^{4} - 11 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} - 9 T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 10 T^{3} + 60 T^{2} + 120 T + 80)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 54 T^{6} + 1296 T^{4} + \cdots + 793881 \) Copy content Toggle raw display
$19$ \( (T^{4} + 320 T + 1280)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 3 T - 29)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + 5 T^{6} + 400 T^{4} + \cdots + 75625 \) Copy content Toggle raw display
$31$ \( (T^{4} + 10 T^{2} + 25 T + 25)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 25 T^{6} + 5875 T^{4} + \cdots + 1890625 \) Copy content Toggle raw display
$41$ \( T^{8} + 69 T^{6} + \cdots + 111746041 \) Copy content Toggle raw display
$43$ \( (T^{4} - 272 T^{2} + 18491)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 175 T^{6} + 11500 T^{4} + \cdots + 1890625 \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} + 5 T^{6} + 235 T^{4} + \cdots + 3025 \) Copy content Toggle raw display
$61$ \( T^{8} + 134 T^{6} + 6856 T^{4} + \cdots + 121 \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T - 71)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} - 100 T^{6} + \cdots + 484000000 \) Copy content Toggle raw display
$73$ \( (T^{4} - 5 T^{3} + 5 T^{2} + 5 T + 5)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 40 T^{7} + 799 T^{6} + \cdots + 38950081 \) Copy content Toggle raw display
$83$ \( (T^{4} + 5 T^{3} - 15 T^{2} + 435 T + 4205)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \) Copy content Toggle raw display
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