Properties

Label 8004.2.a.i
Level $8004$
Weight $2$
Character orbit 8004.a
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8004,2,Mod(1,8004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.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: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} + \cdots - 208 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_1 q^{5} - \beta_{12} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_1 q^{5} - \beta_{12} q^{7} + q^{9} - \beta_{4} q^{11} + \beta_{5} q^{13} - \beta_1 q^{15} - \beta_{13} q^{17} + ( - \beta_{9} + \beta_{8} - \beta_{5}) q^{19} + \beta_{12} q^{21} + q^{23} + (\beta_{14} - \beta_{13} - \beta_{10} - \beta_{8} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{25} - q^{27} + q^{29} + ( - \beta_{7} - \beta_{4}) q^{31} + \beta_{4} q^{33} + (\beta_{15} + \beta_{13} - 2 \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{3} - \beta_1 + 2) q^{35} + ( - \beta_{14} + 1) q^{37} - \beta_{5} q^{39} + (\beta_{8} - \beta_{3} - 1) q^{41} + ( - \beta_{8} - \beta_{4} + \beta_1 - 1) q^{43} + \beta_1 q^{45} + ( - \beta_{15} - \beta_{10} - \beta_{8} + \beta_{5} - \beta_{4} + \beta_1 - 1) q^{47} + ( - \beta_{15} - \beta_{13} + \beta_{9} + \beta_{7} + \beta_{4} + \beta_1 + 1) q^{49} + \beta_{13} q^{51} + (\beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{53} + ( - \beta_{11} + 2 \beta_{10} + \beta_{8} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 2) q^{55} + (\beta_{9} - \beta_{8} + \beta_{5}) q^{57} + ( - \beta_{15} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + \beta_1) q^{59} + (\beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{2} + 1) q^{61} - \beta_{12} q^{63} + (\beta_{13} - \beta_{12} - 2 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{65} + (\beta_{15} + \beta_{13} + \beta_{11} + \beta_{9} + \beta_{7} + \beta_{2} + \beta_1 - 1) q^{67} - q^{69} + (\beta_{15} - \beta_{14} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3}) q^{71} + (\beta_{13} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} + \beta_{2} + \beta_1 - 2) q^{73} + ( - \beta_{14} + \beta_{13} + \beta_{10} + \beta_{8} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 1) q^{75} + ( - \beta_{13} - \beta_{11} + \beta_{6} + \beta_{4} - \beta_1) q^{77} + (\beta_{15} + \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + \beta_1 - 1) q^{79} + q^{81} + ( - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{7} + \beta_{4} - \beta_{3}) q^{83} + ( - \beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{11} - 2 \beta_{8} - \beta_{7} - \beta_{4} + \beta_{3} + \cdots + 3) q^{85}+ \cdots - \beta_{4} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} + \cdots - 208 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 30\!\cdots\!14 \nu^{15} + \cdots + 52\!\cdots\!44 ) / 82\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!65 \nu^{15} + \cdots - 18\!\cdots\!00 ) / 13\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!49 \nu^{15} + \cdots + 39\!\cdots\!72 ) / 54\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15\!\cdots\!29 \nu^{15} + \cdots - 48\!\cdots\!00 ) / 82\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 43\!\cdots\!91 \nu^{15} + \cdots - 11\!\cdots\!60 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!65 \nu^{15} + \cdots + 43\!\cdots\!12 ) / 54\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 76\!\cdots\!15 \nu^{15} + \cdots - 27\!\cdots\!32 ) / 27\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!37 \nu^{15} + \cdots - 31\!\cdots\!68 ) / 54\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 48\!\cdots\!01 \nu^{15} + \cdots - 65\!\cdots\!16 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 19\!\cdots\!11 \nu^{15} + \cdots + 83\!\cdots\!60 ) / 54\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!97 \nu^{15} + \cdots + 44\!\cdots\!40 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 70\!\cdots\!27 \nu^{15} + \cdots - 31\!\cdots\!80 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 32\!\cdots\!95 \nu^{15} + \cdots - 12\!\cdots\!04 ) / 54\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 45\!\cdots\!52 \nu^{15} + \cdots + 39\!\cdots\!40 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{10} - \beta_{8} + \beta_{5} - \beta_{4} - \beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} + \beta_{10} + \beta_{9} - 2\beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} + 11\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{15} + 16 \beta_{14} - 20 \beta_{13} - 22 \beta_{10} + 3 \beta_{9} - 20 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} + 14 \beta_{5} - 18 \beta_{4} - \beta_{3} - 18 \beta_{2} + 20 \beta _1 + 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - 13 \beta_{12} + 27 \beta_{10} + 20 \beta_{9} + 12 \beta_{8} - 43 \beta_{7} - 24 \beta_{6} - 9 \beta_{5} + 28 \beta_{4} - \beta_{3} - 16 \beta_{2} + 143 \beta _1 + 65 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 96 \beta_{15} + 268 \beta_{14} - 381 \beta_{13} + 12 \beta_{12} - 22 \beta_{11} - 413 \beta_{10} + 68 \beta_{9} - 381 \beta_{8} + 70 \beta_{7} - 50 \beta_{6} + 214 \beta_{5} - 320 \beta_{4} - 34 \beta_{3} - 318 \beta_{2} + \cdots + 1014 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 94 \beta_{15} - 82 \beta_{14} - 40 \beta_{13} - 131 \beta_{12} + 33 \beta_{11} + 593 \beta_{10} + 384 \beta_{9} + 385 \beta_{8} - 763 \beta_{7} - 484 \beta_{6} - 290 \beta_{5} + 591 \beta_{4} - 23 \beta_{3} - 200 \beta_{2} + \cdots + 1084 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2185 \beta_{15} + 4636 \beta_{14} - 7081 \beta_{13} + 416 \beta_{12} - 736 \beta_{11} - 7451 \beta_{10} + 1312 \beta_{9} - 7090 \beta_{8} + 1421 \beta_{7} - 963 \beta_{6} + 3459 \beta_{5} - 5678 \beta_{4} + \cdots + 15516 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1625 \beta_{15} - 2295 \beta_{14} - 470 \beta_{13} - 1017 \beta_{12} + 1194 \beta_{11} + 12215 \beta_{10} + 7305 \beta_{9} + 9254 \beta_{8} - 12962 \beta_{7} - 9158 \beta_{6} - 6943 \beta_{5} + 11651 \beta_{4} + \cdots + 16430 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 43998 \beta_{15} + 81591 \beta_{14} - 129666 \beta_{13} + 10392 \beta_{12} - 17679 \beta_{11} - 133030 \beta_{10} + 24157 \beta_{9} - 129934 \beta_{8} + 28119 \beta_{7} - 16928 \beta_{6} + \cdots + 248780 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 24501 \beta_{15} - 54903 \beta_{14} - 95 \beta_{13} - 1533 \beta_{12} + 29931 \beta_{11} + 244048 \beta_{10} + 136881 \beta_{9} + 200061 \beta_{8} - 219178 \beta_{7} - 167791 \beta_{6} + \cdots + 235142 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 837746 \beta_{15} + 1448480 \beta_{14} - 2352986 \beta_{13} + 229352 \beta_{12} - 373284 \beta_{11} - 2372878 \beta_{10} + 436378 \beta_{9} - 2360164 \beta_{8} + 551556 \beta_{7} + \cdots + 4102327 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 331408 \beta_{15} - 1212971 \beta_{14} + 196253 \beta_{13} + 189349 \beta_{12} + 652250 \beta_{11} + 4785042 \beta_{10} + 2528884 \beta_{9} + 4105260 \beta_{8} - 3733385 \beta_{7} + \cdots + 3180682 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 15503403 \beta_{15} + 25829971 \beta_{14} - 42461733 \beta_{13} + 4762320 \beta_{12} - 7391012 \beta_{11} - 42417933 \beta_{10} + 7808976 \beta_{9} - 42674857 \beta_{8} + \cdots + 68876016 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3861844 \beta_{15} - 25604851 \beta_{14} + 7668573 \beta_{13} + 6343023 \beta_{12} + 13314661 \beta_{11} + 92598498 \beta_{10} + 46210791 \beta_{9} + 81763455 \beta_{8} + \cdots + 39683032 \) 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
−4.28632
−3.33956
−2.66915
−2.58465
−1.83485
−0.309856
−0.123882
0.144883
0.282383
1.27509
1.38601
2.30095
3.01816
3.56414
4.02229
4.15437
0 −1.00000 0 −4.28632 0 1.70119 0 1.00000 0
1.2 0 −1.00000 0 −3.33956 0 −2.20985 0 1.00000 0
1.3 0 −1.00000 0 −2.66915 0 −2.40707 0 1.00000 0
1.4 0 −1.00000 0 −2.58465 0 −3.26725 0 1.00000 0
1.5 0 −1.00000 0 −1.83485 0 1.71276 0 1.00000 0
1.6 0 −1.00000 0 −0.309856 0 −1.15608 0 1.00000 0
1.7 0 −1.00000 0 −0.123882 0 3.12124 0 1.00000 0
1.8 0 −1.00000 0 0.144883 0 −2.05535 0 1.00000 0
1.9 0 −1.00000 0 0.282383 0 5.02462 0 1.00000 0
1.10 0 −1.00000 0 1.27509 0 1.05879 0 1.00000 0
1.11 0 −1.00000 0 1.38601 0 −3.43605 0 1.00000 0
1.12 0 −1.00000 0 2.30095 0 −4.14667 0 1.00000 0
1.13 0 −1.00000 0 3.01816 0 3.11628 0 1.00000 0
1.14 0 −1.00000 0 3.56414 0 −0.955621 0 1.00000 0
1.15 0 −1.00000 0 4.02229 0 4.04331 0 1.00000 0
1.16 0 −1.00000 0 4.15437 0 −4.14427 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(-1\)
\(29\) \(-1\)

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 8004.2.a.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.i 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\( T_{5}^{16} - 5 T_{5}^{15} - 43 T_{5}^{14} + 234 T_{5}^{13} + 634 T_{5}^{12} - 4048 T_{5}^{11} - 3483 T_{5}^{10} + 32512 T_{5}^{9} - 137 T_{5}^{8} - 121665 T_{5}^{7} + 60168 T_{5}^{6} + 172218 T_{5}^{5} - 138024 T_{5}^{4} + \cdots - 208 \) Copy content Toggle raw display
\( T_{7}^{16} + 4 T_{7}^{15} - 63 T_{7}^{14} - 277 T_{7}^{13} + 1440 T_{7}^{12} + 7244 T_{7}^{11} - 14299 T_{7}^{10} - 91842 T_{7}^{9} + 48966 T_{7}^{8} + 599660 T_{7}^{7} + 124216 T_{7}^{6} - 1972024 T_{7}^{5} + \cdots - 1420576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T + 1)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 5 T^{15} - 43 T^{14} + 234 T^{13} + \cdots - 208 \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} - 63 T^{14} + \cdots - 1420576 \) Copy content Toggle raw display
$11$ \( T^{16} - 5 T^{15} - 90 T^{14} + \cdots - 211968 \) Copy content Toggle raw display
$13$ \( T^{16} - 8 T^{15} - 120 T^{14} + \cdots - 52214944 \) Copy content Toggle raw display
$17$ \( T^{16} - 7 T^{15} + \cdots + 1623072768 \) Copy content Toggle raw display
$19$ \( T^{16} - T^{15} - 205 T^{14} + \cdots - 1855731712 \) Copy content Toggle raw display
$23$ \( (T - 1)^{16} \) Copy content Toggle raw display
$29$ \( (T - 1)^{16} \) Copy content Toggle raw display
$31$ \( T^{16} + 2 T^{15} - 192 T^{14} + \cdots - 5833216 \) Copy content Toggle raw display
$37$ \( T^{16} - 14 T^{15} + \cdots - 258933952 \) Copy content Toggle raw display
$41$ \( T^{16} + T^{15} + \cdots + 254378586688 \) Copy content Toggle raw display
$43$ \( T^{16} + 13 T^{15} + \cdots + 19590801152 \) Copy content Toggle raw display
$47$ \( T^{16} + 4 T^{15} + \cdots + 241789371392 \) Copy content Toggle raw display
$53$ \( T^{16} - 19 T^{15} + \cdots - 4489400316672 \) Copy content Toggle raw display
$59$ \( T^{16} - 12 T^{15} + \cdots + 23868801024 \) Copy content Toggle raw display
$61$ \( T^{16} - 21 T^{15} + \cdots - 674407936 \) Copy content Toggle raw display
$67$ \( T^{16} + 11 T^{15} + \cdots + 3408931808 \) Copy content Toggle raw display
$71$ \( T^{16} - 7 T^{15} + \cdots - 75\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{16} + 13 T^{15} + \cdots - 75334284986368 \) Copy content Toggle raw display
$79$ \( T^{16} + 18 T^{15} + \cdots - 39769115858944 \) Copy content Toggle raw display
$83$ \( T^{16} - 25 T^{15} + \cdots + 918869999616 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 848564373554432 \) Copy content Toggle raw display
$97$ \( T^{16} - 5 T^{15} + \cdots - 1191522349056 \) Copy content Toggle raw display
show more
show less