Properties

Label 8464.2.a.ce
Level $8464$
Weight $2$
Character orbit 8464.a
Self dual yes
Analytic conductor $67.585$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8464,2,Mod(1,8464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8464, 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("8464.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.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: \(67.5853802708\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 19x^{8} + 15x^{7} + 129x^{6} - 62x^{5} - 387x^{4} + 47x^{3} + 447x^{2} + 106x - 23 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 92)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1) q^{3} + ( - \beta_{6} - \beta_{3} + \beta_{2} + 1) q^{5} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{2}) q^{7} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1) q^{3} + ( - \beta_{6} - \beta_{3} + \beta_{2} + 1) q^{5} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{2}) q^{7} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_{3}) q^{9} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{11} + ( - \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{2}) q^{13} + ( - \beta_{8} - \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_1) q^{15} + (\beta_{5} + \beta_{4} - \beta_1 + 2) q^{17} + ( - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 2) q^{19} + ( - \beta_{8} - \beta_{7} - 3 \beta_{6} + 2 \beta_{4} - \beta_{3} - \beta_{2}) q^{21} + (\beta_{9} + \beta_{8} + \beta_{5} - 3 \beta_{4} - \beta_{3} + 2 \beta_{2} + 2) q^{25} + (\beta_{9} - 2 \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1 - 2) q^{27} + (\beta_{8} - 2 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{29} + ( - \beta_{9} + 3 \beta_{8} + \beta_{6} - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 3) q^{31} + ( - \beta_{9} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 4) q^{33} + (2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{35} + (3 \beta_{8} + 4 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{37} + ( - \beta_{8} - \beta_{7} + 3 \beta_{6} + 3 \beta_{3} + \beta_{2} + 2) q^{39} + (4 \beta_{8} + \beta_{7} - 2 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{41} + ( - 2 \beta_{9} - 2 \beta_{8} - \beta_{7} - 5 \beta_{6} - \beta_{5} - 4 \beta_{3} - \beta_{2} + \cdots - 3) q^{43}+ \cdots + ( - \beta_{9} + 5 \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + 7 \beta_{3} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{3} + 12 q^{5} - q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{3} + 12 q^{5} - q^{7} + 9 q^{9} - 10 q^{11} + 3 q^{13} + 3 q^{15} + 23 q^{17} - 11 q^{19} + 15 q^{21} + 14 q^{25} - 7 q^{27} - 6 q^{29} + 12 q^{31} + 25 q^{33} + 15 q^{35} + 6 q^{37} + 7 q^{39} - 8 q^{41} - 11 q^{43} + 39 q^{45} - 13 q^{47} - 3 q^{49} - 31 q^{51} + 29 q^{53} - 24 q^{55} + 18 q^{57} + 5 q^{59} + 27 q^{61} - 14 q^{63} + 39 q^{65} - 20 q^{71} + 49 q^{73} + 43 q^{75} - 40 q^{77} - 8 q^{79} - 38 q^{81} - 26 q^{83} + 37 q^{85} + 53 q^{87} + 60 q^{89} - 13 q^{91} - 7 q^{93} + 24 q^{95} + 27 q^{97} - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 19x^{8} + 15x^{7} + 129x^{6} - 62x^{5} - 387x^{4} + 47x^{3} + 447x^{2} + 106x - 23 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 73 \nu^{9} + 543 \nu^{8} - 1646 \nu^{7} - 8609 \nu^{6} + 11317 \nu^{5} + 42735 \nu^{4} - 19782 \nu^{3} - 71499 \nu^{2} - 12722 \nu + 4765 ) / 439 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 115 \nu^{9} - 203 \nu^{8} - 1709 \nu^{7} + 2735 \nu^{6} + 7665 \nu^{5} - 9617 \nu^{4} - 12100 \nu^{3} + 8709 \nu^{2} + 3009 \nu - 179 ) / 439 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 152 \nu^{9} + 24 \nu^{8} + 2423 \nu^{7} + 107 \nu^{6} - 12078 \nu^{5} - 4490 \nu^{4} + 20154 \nu^{3} + 14264 \nu^{2} - 1450 \nu - 859 ) / 439 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 131 \nu^{9} + 788 \nu^{8} - 2611 \nu^{7} - 12803 \nu^{6} + 16099 \nu^{5} + 65786 \nu^{4} - 23995 \nu^{3} - 112755 \nu^{2} - 26390 \nu + 6843 ) / 439 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 234 \nu^{9} - 171 \nu^{8} + 3863 \nu^{7} + 3463 \nu^{6} - 19853 \nu^{5} - 23213 \nu^{4} + 30576 \nu^{3} + 48946 \nu^{2} + 10002 \nu - 2495 ) / 439 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 243 \nu^{9} - 262 \nu^{8} + 4096 \nu^{7} + 4795 \nu^{6} - 21579 \nu^{5} - 29053 \nu^{4} + 33508 \nu^{3} + 57785 \nu^{2} + 12244 \nu - 3756 ) / 439 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 276 \nu^{9} + 136 \nu^{8} + 4365 \nu^{7} - 1296 \nu^{6} - 21469 \nu^{5} - 713 \nu^{4} + 35186 \nu^{3} + 13955 \nu^{2} - 2217 \nu - 624 ) / 439 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 608 \nu^{9} - 96 \nu^{8} - 9692 \nu^{7} - 428 \nu^{6} + 47873 \nu^{5} + 17960 \nu^{4} - 75348 \nu^{3} - 57056 \nu^{2} - 7370 \nu + 1241 ) / 439 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 3\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{9} - 12\beta_{8} + 7\beta_{7} - 11\beta_{6} + 9\beta_{4} - 8\beta_{3} - \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{9} + 36\beta_{6} + 12\beta_{5} + 8\beta_{4} + 12\beta_{3} + 30\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -36\beta_{9} - 114\beta_{8} + 50\beta_{7} - 98\beta_{6} + 77\beta_{4} - 67\beta_{3} - 13\beta_{2} - 5\beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 98 \beta_{9} + 2 \beta_{8} - 8 \beta_{7} + 342 \beta_{6} + 114 \beta_{5} + 56 \beta_{4} + 113 \beta_{3} - 5 \beta_{2} + 205 \beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 342 \beta_{9} - 996 \beta_{8} + 369 \beta_{7} - 830 \beta_{6} - 2 \beta_{5} + 634 \beta_{4} - 568 \beta_{3} - 131 \beta_{2} - 82 \beta _1 + 453 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 830 \beta_{9} + 55 \beta_{8} - 147 \beta_{7} + 3020 \beta_{6} + 996 \beta_{5} + 369 \beta_{4} + 986 \beta_{3} - 80 \beta_{2} + 1521 \beta _1 + 105 \) 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
−1.47688
−0.438180
−2.89832
−1.76281
−1.81997
2.35717
2.78660
2.04744
0.137466
2.06749
0 −2.78660 0 1.56806 0 −0.164812 0 4.76513 0
1.2 0 −2.35717 0 −0.253546 0 −4.09726 0 2.55623 0
1.3 0 −2.06749 0 1.32049 0 −2.84410 0 1.27450 0
1.4 0 −2.04744 0 3.83712 0 1.88184 0 1.19203 0
1.5 0 −0.137466 0 −2.71167 0 0.713269 0 −2.98110 0
1.6 0 0.438180 0 1.88076 0 4.92809 0 −2.80800 0
1.7 0 1.47688 0 −0.965939 0 −1.75417 0 −0.818837 0
1.8 0 1.76281 0 −0.748966 0 −0.199329 0 0.107515 0
1.9 0 1.81997 0 4.16547 0 −2.02299 0 0.312301 0
1.10 0 2.89832 0 3.90821 0 2.55947 0 5.40024 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(1\)

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 8464.2.a.ce 10
4.b odd 2 1 2116.2.a.j 10
23.b odd 2 1 8464.2.a.cd 10
23.c even 11 2 368.2.m.d 20
92.b even 2 1 2116.2.a.i 10
92.g odd 22 2 92.2.e.a 20
276.o even 22 2 828.2.q.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.2.e.a 20 92.g odd 22 2
368.2.m.d 20 23.c even 11 2
828.2.q.a 20 276.o even 22 2
2116.2.a.i 10 92.b even 2 1
2116.2.a.j 10 4.b odd 2 1
8464.2.a.cd 10 23.b odd 2 1
8464.2.a.ce 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8464))\):

\( T_{3}^{10} + T_{3}^{9} - 19T_{3}^{8} - 15T_{3}^{7} + 129T_{3}^{6} + 62T_{3}^{5} - 387T_{3}^{4} - 47T_{3}^{3} + 447T_{3}^{2} - 106T_{3} - 23 \) Copy content Toggle raw display
\( T_{5}^{10} - 12 T_{5}^{9} + 40 T_{5}^{8} + 36 T_{5}^{7} - 415 T_{5}^{6} + 505 T_{5}^{5} + 517 T_{5}^{4} - 968 T_{5}^{3} - 242 T_{5}^{2} + 484 T_{5} + 121 \) Copy content Toggle raw display
\( T_{7}^{10} + T_{7}^{9} - 33 T_{7}^{8} - 46 T_{7}^{7} + 277 T_{7}^{6} + 389 T_{7}^{5} - 742 T_{7}^{4} - 909 T_{7}^{3} + 451 T_{7}^{2} + 235 T_{7} + 23 \) Copy content Toggle raw display
\( T_{13}^{10} - 3 T_{13}^{9} - 77 T_{13}^{8} + 142 T_{13}^{7} + 2219 T_{13}^{6} - 1731 T_{13}^{5} - 27614 T_{13}^{4} - 157 T_{13}^{3} + 132231 T_{13}^{2} + 64499 T_{13} - 122429 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} - 19 T^{8} - 15 T^{7} + \cdots - 23 \) Copy content Toggle raw display
$5$ \( T^{10} - 12 T^{9} + 40 T^{8} + \cdots + 121 \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} - 33 T^{8} - 46 T^{7} + \cdots + 23 \) Copy content Toggle raw display
$11$ \( T^{10} + 10 T^{9} - 4 T^{8} + \cdots - 5071 \) Copy content Toggle raw display
$13$ \( T^{10} - 3 T^{9} - 77 T^{8} + \cdots - 122429 \) Copy content Toggle raw display
$17$ \( T^{10} - 23 T^{9} + 195 T^{8} + \cdots - 373 \) Copy content Toggle raw display
$19$ \( T^{10} + 11 T^{9} - 42 T^{8} - 869 T^{7} + \cdots + 32 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} + 6 T^{9} - 149 T^{8} + \cdots - 3287197 \) Copy content Toggle raw display
$31$ \( T^{10} - 12 T^{9} - 57 T^{8} + \cdots - 1442167 \) Copy content Toggle raw display
$37$ \( T^{10} - 6 T^{9} - 130 T^{8} + \cdots - 1366177 \) Copy content Toggle raw display
$41$ \( T^{10} + 8 T^{9} - 182 T^{8} + \cdots + 8112103 \) Copy content Toggle raw display
$43$ \( T^{10} + 11 T^{9} - 221 T^{8} + \cdots - 12256861 \) Copy content Toggle raw display
$47$ \( T^{10} + 13 T^{9} - 139 T^{8} + \cdots - 3761152 \) Copy content Toggle raw display
$53$ \( T^{10} - 29 T^{9} + 139 T^{8} + \cdots + 11440683 \) Copy content Toggle raw display
$59$ \( T^{10} - 5 T^{9} - 254 T^{8} + \cdots - 2504953 \) Copy content Toggle raw display
$61$ \( T^{10} - 27 T^{9} + 57 T^{8} + \cdots - 28126141 \) Copy content Toggle raw display
$67$ \( T^{10} - 232 T^{8} + 33 T^{7} + \cdots - 3854731 \) Copy content Toggle raw display
$71$ \( T^{10} + 20 T^{9} - 33 T^{8} + \cdots - 243869 \) Copy content Toggle raw display
$73$ \( T^{10} - 49 T^{9} + \cdots + 111243241 \) Copy content Toggle raw display
$79$ \( T^{10} + 8 T^{9} - 363 T^{8} + \cdots + 1102091 \) Copy content Toggle raw display
$83$ \( T^{10} + 26 T^{9} - 22 T^{8} + \cdots - 4377889 \) Copy content Toggle raw display
$89$ \( T^{10} - 60 T^{9} + 1308 T^{8} + \cdots + 12390653 \) Copy content Toggle raw display
$97$ \( T^{10} - 27 T^{9} - 67 T^{8} + \cdots - 38572369 \) Copy content Toggle raw display
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