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Properties

Label 8619.2.a.bk
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
Analytic rank $0$
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] = [8619,2,Mod(1,8619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8619, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.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: \(68.8230615021\)
Analytic rank: \(0\)
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: \( x^{11} - 17x^{9} + 100x^{7} - 7x^{6} - 241x^{5} + 43x^{4} + 215x^{3} - 48x^{2} - 54x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 663)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_{4} q^{5} - \beta_1 q^{6} + ( - \beta_{7} - \beta_1) q^{7} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_{4} q^{5} - \beta_1 q^{6} + ( - \beta_{7} - \beta_1) q^{7} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{8}+ \cdots + ( - \beta_{8} + \beta_{5} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 11 q^{3} + 12 q^{4} - 3 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 11 q^{3} + 12 q^{4} - 3 q^{7} + 11 q^{9} - 9 q^{10} - 2 q^{11} + 12 q^{12} + 21 q^{14} + 18 q^{16} - 11 q^{17} + 21 q^{20} - 3 q^{21} + 3 q^{22} + 18 q^{23} + 7 q^{25} + 11 q^{27} - 19 q^{28} + 15 q^{29} - 9 q^{30} + 13 q^{31} - 35 q^{32} - 2 q^{33} + 15 q^{35} + 12 q^{36} + 18 q^{37} + 5 q^{38} + 12 q^{40} + 21 q^{42} - 11 q^{43} - 3 q^{44} + 29 q^{46} - 11 q^{47} + 18 q^{48} + 28 q^{49} + 2 q^{50} - 11 q^{51} + 19 q^{53} + 13 q^{55} + 43 q^{56} - 68 q^{58} + 19 q^{59} + 21 q^{60} + 18 q^{61} + 6 q^{62} - 3 q^{63} + 20 q^{64} + 3 q^{66} - 2 q^{67} - 12 q^{68} + 18 q^{69} + 58 q^{70} - 11 q^{71} - 30 q^{73} + 35 q^{74} + 7 q^{75} + 10 q^{76} + 20 q^{77} - 14 q^{79} + 43 q^{80} + 11 q^{81} - 35 q^{82} - 5 q^{83} - 19 q^{84} + 11 q^{86} + 15 q^{87} - 5 q^{88} - 3 q^{89} - 9 q^{90} + 54 q^{92} + 13 q^{93} + 55 q^{94} + 26 q^{95} - 35 q^{96} - 31 q^{97} + 54 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 17x^{9} + 100x^{7} - 7x^{6} - 241x^{5} + 43x^{4} + 215x^{3} - 48x^{2} - 54x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13 \nu^{10} + 56 \nu^{9} + 137 \nu^{8} - 680 \nu^{7} - 426 \nu^{6} + 2555 \nu^{5} + 359 \nu^{4} + \cdots - 396 ) / 146 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19 \nu^{10} + 8 \nu^{9} - 335 \nu^{8} - 118 \nu^{7} + 2004 \nu^{6} + 511 \nu^{5} - 4725 \nu^{4} + \cdots - 432 ) / 146 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19 \nu^{10} - 8 \nu^{9} + 335 \nu^{8} + 118 \nu^{7} - 2004 \nu^{6} - 511 \nu^{5} + 4725 \nu^{4} + \cdots + 870 ) / 146 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8 \nu^{10} + 12 \nu^{9} + 118 \nu^{8} - 177 \nu^{7} - 571 \nu^{6} + 949 \nu^{5} + 979 \nu^{4} + \cdots + 9 ) / 73 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12 \nu^{10} - 18 \nu^{9} - 177 \nu^{8} + 229 \nu^{7} + 893 \nu^{6} - 949 \nu^{5} - 1797 \nu^{4} + \cdots - 196 ) / 73 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 35 \nu^{10} + 16 \nu^{9} + 571 \nu^{8} - 236 \nu^{7} - 3146 \nu^{6} + 1387 \nu^{5} + 6537 \nu^{4} + \cdots + 12 ) / 146 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33 \nu^{10} + 60 \nu^{9} - 505 \nu^{8} - 958 \nu^{7} + 2474 \nu^{6} + 4745 \nu^{5} - 4449 \nu^{4} + \cdots - 466 ) / 146 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28 \nu^{10} + 31 \nu^{9} - 486 \nu^{8} - 439 \nu^{7} + 2838 \nu^{6} + 1825 \nu^{5} - 6529 \nu^{4} + \cdots - 652 ) / 73 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} + \beta_{6} + \beta_{5} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{8} + 2\beta_{7} + 2\beta_{6} + 9\beta_{5} + 7\beta_{4} + \beta_{3} + 9\beta_{2} + 31\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{10} - \beta_{9} - 9 \beta_{8} + 2 \beta_{7} + 11 \beta_{6} + 12 \beta_{5} - 2 \beta_{4} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{10} - \beta_{9} - 13 \beta_{8} + 26 \beta_{7} + 25 \beta_{6} + 72 \beta_{5} + 41 \beta_{4} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 26 \beta_{10} - 12 \beta_{9} - 70 \beta_{8} + 33 \beta_{7} + 97 \beta_{6} + 116 \beta_{5} - 26 \beta_{4} + \cdots + 530 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 33 \beta_{10} - 15 \beta_{9} - 127 \beta_{8} + 246 \beta_{7} + 238 \beta_{6} + 560 \beta_{5} + \cdots + 357 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 246 \beta_{10} - 106 \beta_{9} - 534 \beta_{8} + 375 \beta_{7} + 800 \beta_{6} + 1033 \beta_{5} + \cdots + 3442 \) Copy content Toggle raw display

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

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
2.79773
2.20613
1.51799
1.11755
0.668709
0.221438
−0.627071
−1.03895
−2.05699
−2.38166
−2.42488
−2.79773 1.00000 5.82732 2.12968 −2.79773 −4.32320 −10.7078 1.00000 −5.95827
1.2 −2.20613 1.00000 2.86703 −1.69398 −2.20613 0.125669 −1.91278 1.00000 3.73715
1.3 −1.51799 1.00000 0.304301 3.77954 −1.51799 0.501477 2.57406 1.00000 −5.73732
1.4 −1.11755 1.00000 −0.751078 0.988234 −1.11755 −3.28484 3.07447 1.00000 −1.10440
1.5 −0.668709 1.00000 −1.55283 −2.85507 −0.668709 −2.03046 2.37581 1.00000 1.90921
1.6 −0.221438 1.00000 −1.95096 1.56526 −0.221438 2.74727 0.874896 1.00000 −0.346608
1.7 0.627071 1.00000 −1.60678 −2.72481 0.627071 −0.709008 −2.26171 1.00000 −1.70865
1.8 1.03895 1.00000 −0.920578 −1.38648 1.03895 3.80411 −3.03434 1.00000 −1.44049
1.9 2.05699 1.00000 2.23122 −3.23889 2.05699 −4.94134 0.475611 1.00000 −6.66237
1.10 2.38166 1.00000 3.67232 0.495625 2.38166 4.91459 3.98291 1.00000 1.18041
1.11 2.42488 1.00000 3.88005 2.94090 2.42488 0.195727 4.55889 1.00000 7.13133
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(1\)
\(17\) \(1\)

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 8619.2.a.bk 11
13.b even 2 1 8619.2.a.bl 11
13.e even 6 2 663.2.i.h ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.i.h ✓ 22 13.e even 6 2
8619.2.a.bk 11 1.a even 1 1 trivial
8619.2.a.bl 11 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{11} - 17T_{2}^{9} + 100T_{2}^{7} + 7T_{2}^{6} - 241T_{2}^{5} - 43T_{2}^{4} + 215T_{2}^{3} + 48T_{2}^{2} - 54T_{2} - 12 \) Copy content Toggle raw display
\( T_{5}^{11} - 31 T_{5}^{9} - 3 T_{5}^{8} + 343 T_{5}^{7} + 27 T_{5}^{6} - 1666 T_{5}^{5} + 61 T_{5}^{4} + \cdots + 1074 \) Copy content Toggle raw display
\( T_{7}^{11} + 3 T_{7}^{10} - 48 T_{7}^{9} - 136 T_{7}^{8} + 714 T_{7}^{7} + 1848 T_{7}^{6} - 3429 T_{7}^{5} + \cdots + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} - 17 T^{9} + \cdots - 12 \) Copy content Toggle raw display
$3$ \( (T - 1)^{11} \) Copy content Toggle raw display
$5$ \( T^{11} - 31 T^{9} + \cdots + 1074 \) Copy content Toggle raw display
$7$ \( T^{11} + 3 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{11} + 2 T^{10} + \cdots - 21984 \) Copy content Toggle raw display
$13$ \( T^{11} \) Copy content Toggle raw display
$17$ \( (T + 1)^{11} \) Copy content Toggle raw display
$19$ \( T^{11} - 99 T^{9} + \cdots - 199984 \) Copy content Toggle raw display
$23$ \( T^{11} - 18 T^{10} + \cdots + 4168200 \) Copy content Toggle raw display
$29$ \( T^{11} - 15 T^{10} + \cdots + 7413879 \) Copy content Toggle raw display
$31$ \( T^{11} - 13 T^{10} + \cdots + 5883392 \) Copy content Toggle raw display
$37$ \( T^{11} - 18 T^{10} + \cdots - 9996544 \) Copy content Toggle raw display
$41$ \( T^{11} - 242 T^{9} + \cdots - 26397972 \) Copy content Toggle raw display
$43$ \( T^{11} + 11 T^{10} + \cdots - 89588 \) Copy content Toggle raw display
$47$ \( T^{11} + 11 T^{10} + \cdots - 589824 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots - 253668864 \) Copy content Toggle raw display
$59$ \( T^{11} - 19 T^{10} + \cdots - 63729216 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 13417558528 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots - 410555493 \) Copy content Toggle raw display
$71$ \( T^{11} + 11 T^{10} + \cdots + 39667200 \) Copy content Toggle raw display
$73$ \( T^{11} + 30 T^{10} + \cdots - 41263072 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots + 28684706816 \) Copy content Toggle raw display
$83$ \( T^{11} + 5 T^{10} + \cdots - 1161936 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots + 247352688 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 345966464 \) Copy content Toggle raw display
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