Properties

Label 704.6.a.t
Level $704$
Weight $6$
Character orbit 704.a
Self dual yes
Analytic conductor $112.910$
Analytic rank $1$
Dimension $3$
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] = [704,6,Mod(1,704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("704.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 704.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: \(112.910209148\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 11)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 11) q^{3} + ( - \beta_{2} - 4 \beta_1 - 7) q^{5} + (5 \beta_{2} + 5 \beta_1 - 28) q^{7} + ( - \beta_{2} + 16 \beta_1 - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 11) q^{3} + ( - \beta_{2} - 4 \beta_1 - 7) q^{5} + (5 \beta_{2} + 5 \beta_1 - 28) q^{7} + ( - \beta_{2} + 16 \beta_1 - 8) q^{9} + 121 q^{11} + ( - 11 \beta_{2} + 37 \beta_1 - 178) q^{13} + ( - 14 \beta_{2} - 15 \beta_1 - 551) q^{15} + (9 \beta_{2} - 105 \beta_1 + 400) q^{17} + ( - 15 \beta_{2} + 15 \beta_1 + 450) q^{19} + (85 \beta_{2} - 63 \beta_1 + 352) q^{21} + ( - 60 \beta_{2} - 201 \beta_1 + 1069) q^{23} + (65 \beta_{2} - 70 \beta_1 + 26) q^{25} + ( - 34 \beta_{2} - 159 \beta_1 - 955) q^{27} + (40 \beta_{2} - 62 \beta_1 + 1176) q^{29} + (4 \beta_{2} - 89 \beta_1 + 1397) q^{31} + (121 \beta_1 + 1331) q^{33} + ( - 217 \beta_{2} + 167 \beta_1 - 8204) q^{35} + ( - 191 \beta_{2} + 364 \beta_1 - 6093) q^{37} + ( - 235 \beta_{2} + 139 \beta_1 + 2062) q^{39} + ( - 137 \beta_{2} + 967 \beta_1 + 1630) q^{41} + (10 \beta_{2} - 1160 \beta_1 - 8346) q^{43} + (6 \beta_{2} + 514 \beta_1 - 6322) q^{45} + (234 \beta_{2} + 102 \beta_1 + 5788) q^{47} + (320 \beta_{2} + 620 \beta_1 + 16077) q^{49} + (267 \beta_{2} - 233 \beta_1 - 7408) q^{51} + (262 \beta_{2} + 310 \beta_1 - 16878) q^{53} + ( - 121 \beta_{2} - 484 \beta_1 - 847) q^{55} + ( - 285 \beta_{2} + 705 \beta_1 + 6390) q^{57} + (486 \beta_{2} - 1683 \beta_1 - 523) q^{59} + (688 \beta_{2} + 598 \beta_1 - 6132) q^{61} + (378 \beta_{2} - 2198 \beta_1 + 5024) q^{63} + (573 \beta_{2} + 1983 \beta_1 - 3026) q^{65} + ( - 1162 \beta_{2} + 3089 \beta_1 - 17335) q^{67} + ( - 879 \beta_{2} + 784 \beta_1 - 12235) q^{69} + (826 \beta_{2} - 1457 \beta_1 - 12333) q^{71} + (231 \beta_{2} - 4677 \beta_1 + 6778) q^{73} + (1240 \beta_{2} - 1104 \beta_1 - 6524) q^{75} + (605 \beta_{2} + 605 \beta_1 - 3388) q^{77} + ( - 658 \beta_{2} + 1928 \beta_1 - 42578) q^{79} + ( - 210 \beta_{2} - 5230 \beta_1 - 27299) q^{81} + (1800 \beta_{2} + 2250 \beta_1 - 48126) q^{83} + ( - 499 \beta_{2} - 4807 \beta_1 + 36116) q^{85} + (782 \beta_{2} + 386 \beta_1 + 6588) q^{87} + ( - 1217 \beta_{2} + 1516 \beta_1 - 36519) q^{89} + ( - 462 \beta_{2} - 8226 \beta_1 - 33956) q^{91} + (161 \beta_{2} + 904 \beta_1 + 5293) q^{93} + (195 \beta_{2} - 1095 \beta_1 + 7830) q^{95} + (819 \beta_{2} + 2334 \beta_1 + 2723) q^{97} + ( - 121 \beta_{2} + 1936 \beta_1 - 968) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 34 q^{3} - 24 q^{5} - 84 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 34 q^{3} - 24 q^{5} - 84 q^{7} - 7 q^{9} + 363 q^{11} - 486 q^{13} - 1654 q^{15} + 1086 q^{17} + 1380 q^{19} + 908 q^{21} + 3066 q^{23} - 57 q^{25} - 2990 q^{27} + 3426 q^{29} + 4098 q^{31} + 4114 q^{33} - 24228 q^{35} - 17724 q^{37} + 6560 q^{39} + 5994 q^{41} - 26208 q^{43} - 18458 q^{45} + 17232 q^{47} + 48531 q^{49} - 22724 q^{51} - 50586 q^{53} - 2904 q^{55} + 20160 q^{57} - 3738 q^{59} - 18486 q^{61} + 12496 q^{63} - 7668 q^{65} - 47754 q^{67} - 35042 q^{69} - 39282 q^{71} + 15426 q^{73} - 21916 q^{75} - 10164 q^{77} - 125148 q^{79} - 86917 q^{81} - 143928 q^{83} + 104040 q^{85} + 19368 q^{87} - 106824 q^{89} - 109632 q^{91} + 16622 q^{93} + 22200 q^{95} + 9684 q^{97} - 847 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 52x - 38 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{2} + 6\nu + 34 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2\nu - 36 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 3\beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} - 3\beta _1 + 142 ) / 4 \) 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
−6.29828
8.04796
−0.749680
0 −3.48600 0 59.8722 0 −145.071 0 −230.848 0
1.2 0 16.8394 0 −75.2230 0 225.525 0 40.5643 0
1.3 0 20.6466 0 −8.64919 0 −164.454 0 183.283 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(11\) \( -1 \)

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 704.6.a.t 3
4.b odd 2 1 704.6.a.q 3
8.b even 2 1 176.6.a.i 3
8.d odd 2 1 11.6.a.b 3
24.f even 2 1 99.6.a.g 3
40.e odd 2 1 275.6.a.b 3
40.k even 4 2 275.6.b.b 6
56.e even 2 1 539.6.a.e 3
88.g even 2 1 121.6.a.d 3
264.p odd 2 1 1089.6.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.b 3 8.d odd 2 1
99.6.a.g 3 24.f even 2 1
121.6.a.d 3 88.g even 2 1
176.6.a.i 3 8.b even 2 1
275.6.a.b 3 40.e odd 2 1
275.6.b.b 6 40.k even 4 2
539.6.a.e 3 56.e even 2 1
704.6.a.q 3 4.b odd 2 1
704.6.a.t 3 1.a even 1 1 trivial
1089.6.a.r 3 264.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 34T_{3}^{2} + 217T_{3} + 1212 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(704))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 34 T^{2} + \cdots + 1212 \) Copy content Toggle raw display
$5$ \( T^{3} + 24 T^{2} + \cdots - 38954 \) Copy content Toggle raw display
$7$ \( T^{3} + 84 T^{2} + \cdots - 5380448 \) Copy content Toggle raw display
$11$ \( (T - 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 486 T^{2} + \cdots - 164136608 \) Copy content Toggle raw display
$17$ \( T^{3} - 1086 T^{2} + \cdots + 331752056 \) Copy content Toggle raw display
$19$ \( T^{3} - 1380 T^{2} + \cdots + 57024000 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 17004325928 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 4029189120 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 1094344400 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 541788167034 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 201929821568 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 2443875098544 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 70174939136 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 1850911309656 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 7759637437060 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 15233874751008 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 147288561330212 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 1290398551704 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 34539701265952 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 1279883216320 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 411597824719824 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 90320980174650 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 10221902527106 \) Copy content Toggle raw display
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