Properties

Label 8.14.a.b
Level $8$
Weight $14$
Character orbit 8.a
Self dual yes
Analytic conductor $8.578$
Analytic rank $0$
Dimension $2$
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] = [8,14,Mod(1,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 8.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: \(8.57847431615\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{781}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 195 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7} \)
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 \(\beta = 64\sqrt{781}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 436) q^{3} + ( - 12 \beta + 9238) q^{5} + (222 \beta + 55464) q^{7} + ( - 872 \beta + 1794749) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 436) q^{3} + ( - 12 \beta + 9238) q^{5} + (222 \beta + 55464) q^{7} + ( - 872 \beta + 1794749) q^{9} + ( - 531 \beta + 8237020) q^{11} + (10452 \beta + 9372286) q^{13} + ( - 14470 \beta + 42415480) q^{15} + ( - 38184 \beta - 76896814) q^{17} + (93819 \beta - 59373820) q^{19} + (41328 \beta - 685990368) q^{21} + ( - 111462 \beta + 359134456) q^{23} + ( - 221712 \beta - 674709937) q^{25} + ( - 580618 \beta + 2876892808) q^{27} + (2374788 \beta + 154670670) q^{29} + (1482936 \beta + 2883752096) q^{31} + ( - 8468536 \beta + 5289996976) q^{33} + (1385268 \beta - 8009695632) q^{35} + (10246116 \beta - 5810776650) q^{37} + ( - 4815214 \beta - 29349380456) q^{39} + (8079216 \beta + 655584138) q^{41} + ( - 9339771 \beta - 14797810052) q^{43} + ( - 29592524 \beta + 50053976126) q^{45} + (16143540 \beta + 6156808944) q^{47} + (24626016 \beta + 63845578073) q^{49} + (60248590 \beta + 88622688680) q^{51} + ( - 50827740 \beta - 19003003514) q^{53} + ( - 103749618 \beta + 96477465832) q^{55} + (100278904 \beta - 326011714864) q^{57} + ( - 136967391 \beta + 126672955852) q^{59} + (14150004 \beta - 323622192146) q^{61} + (350069670 \beta - 519726611448) q^{63} + ( - 15911856 \beta - 314647187756) q^{65} + ( - 36810993 \beta + 809996903156) q^{67} + ( - 407731888 \beta + 513146885728) q^{69} + ( - 352709874 \beta - 520135071256) q^{71} + ( - 86109192 \beta + 2002641954346) q^{73} + (578043505 \beta + 415077834380) q^{75} + (1799167056 \beta + 79756388448) q^{77} + ( - 525520788 \beta - 1260888786032) q^{79} + ( - 1739792600 \beta + 250298701529) q^{81} + ( - 996493749 \beta - 145243115452) q^{83} + (570017976 \beta + 755423627276) q^{85} + (880736898 \beta - 7529453404968) q^{87} + ( - 639382536 \beta - 4361877828870) q^{89} + (2660357220 \beta + 7942549238448) q^{91} + ( - 2237192000 \beta - 3486560759680) q^{93} + (1579185762 \beta - 4149992101288) q^{95} + (4881770040 \beta + 4800856149986) q^{97} + ( - 8135693159 \beta + 16264611663212) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 872 q^{3} + 18476 q^{5} + 110928 q^{7} + 3589498 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 872 q^{3} + 18476 q^{5} + 110928 q^{7} + 3589498 q^{9} + 16474040 q^{11} + 18744572 q^{13} + 84830960 q^{15} - 153793628 q^{17} - 118747640 q^{19} - 1371980736 q^{21} + 718268912 q^{23} - 1349419874 q^{25} + 5753785616 q^{27} + 309341340 q^{29} + 5767504192 q^{31} + 10579993952 q^{33} - 16019391264 q^{35} - 11621553300 q^{37} - 58698760912 q^{39} + 1311168276 q^{41} - 29595620104 q^{43} + 100107952252 q^{45} + 12313617888 q^{47} + 127691156146 q^{49} + 177245377360 q^{51} - 38006007028 q^{53} + 192954931664 q^{55} - 652023429728 q^{57} + 253345911704 q^{59} - 647244384292 q^{61} - 1039453222896 q^{63} - 629294375512 q^{65} + 1619993806312 q^{67} + 1026293771456 q^{69} - 1040270142512 q^{71} + 4005283908692 q^{73} + 830155668760 q^{75} + 159512776896 q^{77} - 2521777572064 q^{79} + 500597403058 q^{81} - 290486230904 q^{83} + 1510847254552 q^{85} - 15058906809936 q^{87} - 8723755657740 q^{89} + 15885098476896 q^{91} - 6973121519360 q^{93} - 8299984202576 q^{95} + 9601712299972 q^{97} + 32529223326424 q^{99}+O(q^{100}) \) 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
14.4732
−13.4732
0 −1352.57 0 −12224.8 0 452526. 0 235118. 0
1.2 0 2224.57 0 30700.8 0 −341598. 0 3.35438e6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 8.14.a.b 2
3.b odd 2 1 72.14.a.c 2
4.b odd 2 1 16.14.a.e 2
5.b even 2 1 200.14.a.b 2
5.c odd 4 2 200.14.c.b 4
8.b even 2 1 64.14.a.j 2
8.d odd 2 1 64.14.a.l 2
12.b even 2 1 144.14.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.14.a.b 2 1.a even 1 1 trivial
16.14.a.e 2 4.b odd 2 1
64.14.a.j 2 8.b even 2 1
64.14.a.l 2 8.d odd 2 1
72.14.a.c 2 3.b odd 2 1
144.14.a.n 2 12.b even 2 1
200.14.a.b 2 5.b even 2 1
200.14.c.b 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 872T_{3} - 3008880 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(8))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 872 T - 3008880 \) Copy content Toggle raw display
$5$ \( T^{2} - 18476 T - 375311900 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 154582077888 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 66946512008464 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 261630161766908 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 12\!\cdots\!40 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 24\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 89\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 18\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 12\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 30\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 20\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 60\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 79\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 79\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 43\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 65\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 12\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 39\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 70\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 31\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 53\!\cdots\!04 \) Copy content Toggle raw display
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