Properties

Label 869.2.a.e
Level $869$
Weight $2$
Character orbit 869.a
Self dual yes
Analytic conductor $6.939$
Analytic rank $1$
Dimension $11$
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] = [869,2,Mod(1,869)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("869.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 869.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: \(6.93899993565\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 3x^{10} - 10x^{9} + 32x^{8} + 30x^{7} - 111x^{6} - 23x^{5} + 151x^{4} - 32x^{3} - 69x^{2} + 38x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{8} + \beta_{7} + \beta_1) q^{4} + ( - \beta_{10} - \beta_{5} + \beta_{4}) q^{5} + ( - \beta_{8} - \beta_{7} - \beta_{4} - 1) q^{6} + (\beta_{10} - \beta_{9} - \beta_{8} - \beta_{3} - 1) q^{7} + ( - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{3} + \beta_{2} - \beta_1) q^{8} + (2 \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{8} + \beta_{7} + \beta_1) q^{4} + ( - \beta_{10} - \beta_{5} + \beta_{4}) q^{5} + ( - \beta_{8} - \beta_{7} - \beta_{4} - 1) q^{6} + (\beta_{10} - \beta_{9} - \beta_{8} - \beta_{3} - 1) q^{7} + ( - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{3} + \beta_{2} - \beta_1) q^{8} + (2 \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4}) q^{9} + ( - \beta_{6} + \beta_{5} + \beta_1 - 1) q^{10} + q^{11} + (2 \beta_{10} + \beta_{8} - \beta_{2} + 2 \beta_1 - 1) q^{12} + (\beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{10} - \beta_{9} + \beta_{6} + \beta_{4} - 2 \beta_{2} + \beta_1) q^{14} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{15} + (\beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{16} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{7} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1 + 2) q^{18} + ( - \beta_{10} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 4) q^{19} + (2 \beta_{10} + \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{20} + ( - 2 \beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 1) q^{21} - \beta_1 q^{22} + (\beta_{10} + \beta_{8} + \beta_{5} - 3 \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{23} + ( - 2 \beta_{9} - 3 \beta_{8} - 2 \beta_{7} - \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + \cdots - 3 \beta_1) q^{24}+ \cdots + (2 \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 3 q^{2} - 3 q^{3} + 7 q^{4} - 8 q^{5} - 11 q^{6} - 7 q^{7} - 9 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q - 3 q^{2} - 3 q^{3} + 7 q^{4} - 8 q^{5} - 11 q^{6} - 7 q^{7} - 9 q^{8} + 10 q^{9} - 6 q^{10} + 11 q^{11} + 6 q^{12} - 13 q^{13} + q^{15} + 7 q^{16} - 3 q^{17} + 17 q^{18} - 45 q^{19} - 16 q^{21} - 3 q^{22} + 7 q^{23} - 24 q^{24} + 9 q^{25} + 10 q^{26} - 36 q^{27} - 13 q^{28} - 25 q^{29} - 22 q^{30} - 21 q^{31} - 26 q^{32} - 3 q^{33} - 14 q^{34} - 16 q^{35} - 16 q^{36} + 2 q^{37} + 2 q^{38} + 3 q^{39} + 9 q^{40} - 21 q^{41} + 26 q^{42} - 4 q^{43} + 7 q^{44} - 31 q^{45} - 16 q^{46} - 17 q^{47} + 35 q^{48} + 10 q^{50} - 33 q^{51} - 38 q^{52} + 13 q^{53} + 15 q^{54} - 8 q^{55} + 7 q^{56} - 29 q^{58} - 13 q^{59} - q^{60} - 23 q^{61} + 29 q^{62} - 12 q^{63} + 11 q^{64} + 6 q^{65} - 11 q^{66} - 26 q^{67} + 14 q^{68} + 25 q^{69} + 14 q^{70} + 31 q^{71} - 24 q^{72} - 18 q^{73} + 12 q^{74} + 6 q^{75} - 30 q^{76} - 7 q^{77} + 6 q^{78} + 11 q^{79} + 18 q^{80} + 31 q^{81} + 2 q^{82} - 11 q^{83} + 3 q^{84} - 14 q^{85} + 59 q^{86} + 3 q^{87} - 9 q^{88} - 26 q^{89} + 25 q^{90} - 25 q^{91} + 46 q^{92} + 11 q^{93} - 27 q^{94} + 29 q^{95} - 25 q^{96} - 10 q^{97} + 15 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 3x^{10} - 10x^{9} + 32x^{8} + 30x^{7} - 111x^{6} - 23x^{5} + 151x^{4} - 32x^{3} - 69x^{2} + 38x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2 \nu^{10} + 5 \nu^{9} + 19 \nu^{8} - 44 \nu^{7} - 54 \nu^{6} + 97 \nu^{5} + 56 \nu^{4} - 15 \nu^{3} - 24 \nu^{2} - 56 \nu + 29 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5 \nu^{10} - 9 \nu^{9} - 65 \nu^{8} + 96 \nu^{7} + 296 \nu^{6} - 323 \nu^{5} - 539 \nu^{4} + 398 \nu^{3} + 256 \nu^{2} - 203 \nu + 43 ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{9} - 4\nu^{8} - 6\nu^{7} + 37\nu^{6} - 5\nu^{5} - 96\nu^{4} + 56\nu^{3} + 66\nu^{2} - 59\nu + 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11 \nu^{10} + 24 \nu^{9} + 129 \nu^{8} - 242 \nu^{7} - 528 \nu^{6} + 747 \nu^{5} + 903 \nu^{4} - 814 \nu^{3} - 447 \nu^{2} + 336 \nu - 26 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{10} - 6\nu^{9} - 19\nu^{8} + 60\nu^{7} + 54\nu^{6} - 185\nu^{5} - 51\nu^{4} + 206\nu^{3} - 8\nu^{2} - 71\nu + 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13 \nu^{10} + 29 \nu^{9} + 155 \nu^{8} - 307 \nu^{7} - 645 \nu^{6} + 1047 \nu^{5} + 1106 \nu^{4} - 1396 \nu^{3} - 499 \nu^{2} + 721 \nu - 144 ) / 7 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13 \nu^{10} - 29 \nu^{9} - 155 \nu^{8} + 307 \nu^{7} + 645 \nu^{6} - 1047 \nu^{5} - 1106 \nu^{4} + 1396 \nu^{3} + 506 \nu^{2} - 728 \nu + 130 ) / 7 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12 \nu^{10} + 23 \nu^{9} + 156 \nu^{8} - 257 \nu^{7} - 709 \nu^{6} + 946 \nu^{5} + 1288 \nu^{4} - 1378 \nu^{3} - 620 \nu^{2} + 791 \nu - 148 ) / 7 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18 \nu^{10} + 45 \nu^{9} + 199 \nu^{8} - 466 \nu^{7} - 745 \nu^{6} + 1531 \nu^{5} + 1134 \nu^{4} - 1913 \nu^{3} - 419 \nu^{2} + 861 \nu - 159 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{3} - \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + 8\beta_{8} + 7\beta_{7} + \beta_{5} + 2\beta_{3} + 8\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15 \beta_{10} - 6 \beta_{9} + 10 \beta_{8} + 2 \beta_{7} + 8 \beta_{6} + \beta_{5} + \beta_{4} - 5 \beta_{3} - 9 \beta_{2} + 29 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{10} + 10 \beta_{9} + 57 \beta_{8} + 45 \beta_{7} + 2 \beta_{6} + 10 \beta_{5} + 2 \beta_{4} + 18 \beta_{3} - 5 \beta_{2} + 56 \beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 97 \beta_{10} - 29 \beta_{9} + 86 \beta_{8} + 28 \beta_{7} + 54 \beta_{6} + 15 \beta_{5} + 12 \beta_{4} - 17 \beta_{3} - 69 \beta_{2} + 183 \beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 45 \beta_{10} + 75 \beta_{9} + 398 \beta_{8} + 289 \beta_{7} + 29 \beta_{6} + 84 \beta_{5} + 25 \beta_{4} + 133 \beta_{3} - 73 \beta_{2} + 390 \beta _1 + 220 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 614 \beta_{10} - 122 \beta_{9} + 695 \beta_{8} + 275 \beta_{7} + 350 \beta_{6} + 157 \beta_{5} + 104 \beta_{4} - 13 \beta_{3} - 510 \beta_{2} + 1212 \beta _1 + 149 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 460 \beta_{10} + 520 \beta_{9} + 2777 \beta_{8} + 1883 \beta_{7} + 289 \beta_{6} + 667 \beta_{5} + 228 \beta_{4} + 940 \beta_{3} - 748 \beta_{2} + 2750 \beta _1 + 1276 \) 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
2.69357
2.31683
1.97721
0.899857
0.705076
0.530334
0.218892
−1.20556
−1.28044
−1.53728
−2.31850
−2.69357 1.90179 5.25534 −0.450672 −5.12261 −3.86504 −8.76851 0.616807 1.21392
1.2 −2.31683 1.04519 3.36771 1.41367 −2.42153 1.87521 −3.16874 −1.90758 −3.27524
1.3 −1.97721 −1.38294 1.90936 −1.77953 2.73437 1.41235 0.179222 −1.08746 3.51851
1.4 −0.899857 0.683852 −1.19026 −2.15524 −0.615369 0.912096 2.87078 −2.53235 1.93941
1.5 −0.705076 −0.198112 −1.50287 4.30175 0.139684 −3.77767 2.46979 −2.96075 −3.03306
1.6 −0.530334 −3.19833 −1.71875 −3.55323 1.69618 1.07063 1.97218 7.22931 1.88440
1.7 −0.218892 2.03311 −1.95209 −0.181934 −0.445032 −2.70151 0.865079 1.13355 0.0398238
1.8 1.20556 −3.41025 −0.546636 1.06206 −4.11125 −0.0877359 −3.07011 8.62983 1.28037
1.9 1.28044 2.31346 −0.360481 −4.21606 2.96224 −2.21465 −3.02245 2.35209 −5.39840
1.10 1.53728 −0.827868 0.363239 −1.90635 −1.27267 3.93115 −2.51617 −2.31463 −2.93060
1.11 2.31850 −1.95990 3.37543 −0.534453 −4.54401 −3.55483 3.18893 0.841191 −1.23913
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(79\) \(-1\)

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 869.2.a.e 11
3.b odd 2 1 7821.2.a.i 11
11.b odd 2 1 9559.2.a.j 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.2.a.e 11 1.a even 1 1 trivial
7821.2.a.i 11 3.b odd 2 1
9559.2.a.j 11 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(869))\):

\( T_{2}^{11} + 3 T_{2}^{10} - 10 T_{2}^{9} - 32 T_{2}^{8} + 30 T_{2}^{7} + 111 T_{2}^{6} - 23 T_{2}^{5} - 151 T_{2}^{4} - 32 T_{2}^{3} + 69 T_{2}^{2} + 38 T_{2} + 5 \) Copy content Toggle raw display
\( T_{3}^{11} + 3 T_{3}^{10} - 17 T_{3}^{9} - 42 T_{3}^{8} + 112 T_{3}^{7} + 194 T_{3}^{6} - 325 T_{3}^{5} - 349 T_{3}^{4} + 376 T_{3}^{3} + 214 T_{3}^{2} - 131 T_{3} - 31 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} + 3 T^{10} - 10 T^{9} - 32 T^{8} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( T^{11} + 3 T^{10} - 17 T^{9} - 42 T^{8} + \cdots - 31 \) Copy content Toggle raw display
$5$ \( T^{11} + 8 T^{10} - 151 T^{8} - 401 T^{7} + \cdots - 31 \) Copy content Toggle raw display
$7$ \( T^{11} + 7 T^{10} - 14 T^{9} - 174 T^{8} + \cdots - 277 \) Copy content Toggle raw display
$11$ \( (T - 1)^{11} \) Copy content Toggle raw display
$13$ \( T^{11} + 13 T^{10} + 2 T^{9} + \cdots + 9409 \) Copy content Toggle raw display
$17$ \( T^{11} + 3 T^{10} - 96 T^{9} + \cdots + 216379 \) Copy content Toggle raw display
$19$ \( T^{11} + 45 T^{10} + 863 T^{9} + \cdots + 1856061 \) Copy content Toggle raw display
$23$ \( T^{11} - 7 T^{10} - 119 T^{9} + 763 T^{8} + \cdots - 65 \) Copy content Toggle raw display
$29$ \( T^{11} + 25 T^{10} + 150 T^{9} + \cdots + 1260825 \) Copy content Toggle raw display
$31$ \( T^{11} + 21 T^{10} + 70 T^{9} + \cdots + 51571 \) Copy content Toggle raw display
$37$ \( T^{11} - 2 T^{10} - 174 T^{9} + \cdots - 140409 \) Copy content Toggle raw display
$41$ \( T^{11} + 21 T^{10} + 13 T^{9} + \cdots - 3406745 \) Copy content Toggle raw display
$43$ \( T^{11} + 4 T^{10} - 209 T^{9} + \cdots - 6715983 \) Copy content Toggle raw display
$47$ \( T^{11} + 17 T^{10} + \cdots - 212684653 \) Copy content Toggle raw display
$53$ \( T^{11} - 13 T^{10} - 307 T^{9} + \cdots - 22915995 \) Copy content Toggle raw display
$59$ \( T^{11} + 13 T^{10} + \cdots - 1153634569 \) Copy content Toggle raw display
$61$ \( T^{11} + 23 T^{10} + \cdots + 681315441 \) Copy content Toggle raw display
$67$ \( T^{11} + 26 T^{10} + 103 T^{9} + \cdots + 233779 \) Copy content Toggle raw display
$71$ \( T^{11} - 31 T^{10} + \cdots - 2455102431 \) Copy content Toggle raw display
$73$ \( T^{11} + 18 T^{10} + \cdots - 5014749653 \) Copy content Toggle raw display
$79$ \( (T - 1)^{11} \) Copy content Toggle raw display
$83$ \( T^{11} + 11 T^{10} + \cdots + 2678096451 \) Copy content Toggle raw display
$89$ \( T^{11} + 26 T^{10} + \cdots + 9424367157 \) Copy content Toggle raw display
$97$ \( T^{11} + 10 T^{10} + \cdots - 5360436977 \) Copy content Toggle raw display
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