Properties

Label 8.12.a.b
Level $8$
Weight $12$
Character orbit 8.a
Self dual yes
Analytic conductor $6.147$
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] = [8,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 12 \)
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: \(6.14674544448\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{109}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 27 \) 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 \(\beta = 64\sqrt{109}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 28) q^{3} + ( - 12 \beta + 3934) q^{5} + (42 \beta + 45528) q^{7} + (56 \beta + 270101) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 28) q^{3} + ( - 12 \beta + 3934) q^{5} + (42 \beta + 45528) q^{7} + (56 \beta + 270101) q^{9} + ( - 693 \beta + 79540) q^{11} + (948 \beta + 525238) q^{13} + (3598 \beta - 5247416) q^{15} + ( - 10824 \beta + 715442) q^{17} + ( - 4587 \beta - 10933300) q^{19} + (46704 \beta + 20026272) q^{21} + ( - 24738 \beta + 17903368) q^{23} + ( - 94416 \beta + 30939047) q^{25} + (94522 \beta + 27604696) q^{27} + (59556 \beta - 114413850) q^{29} + ( - 28632 \beta + 32361056) q^{31} + (60136 \beta - 307172432) q^{33} + ( - 381108 \beta - 45910704) q^{35} + ( - 81564 \beta + 37779390) q^{37} + (551782 \beta + 437954536) q^{39} + (647088 \beta + 600607098) q^{41} + (324051 \beta - 22759916) q^{43} + ( - 3020908 \beta + 762553526) q^{45} + ( - 816420 \beta - 614539632) q^{47} + (3824352 \beta + 883034537) q^{49} + (412370 \beta - 4812493960) q^{51} + (3514980 \beta - 1904274962) q^{53} + ( - 3680742 \beta + 4025704984) q^{55} + ( - 11061736 \beta - 2354062768) q^{57} + (4307463 \beta - 3006463292) q^{59} + (4588692 \beta + 4894896454) q^{61} + (13893810 \beta + 13347241656) q^{63} + ( - 2573424 \beta - 3012688172) q^{65} + ( - 21317583 \beta + 7351547612) q^{67} + (17210704 \beta - 10543332128) q^{69} + ( - 20099142 \beta + 2159995544) q^{71} + ( - 3259368 \beta + 5527819738) q^{73} + (28295399 \beta - 41287051708) q^{75} + ( - 28210224 \beta - 9373484064) q^{77} + (41373444 \beta + 25978811632) q^{79} + (20331080 \beta - 4873980151) q^{81} + ( - 13552251 \beta - 54113987956) q^{83} + ( - 51166920 \beta + 60804864860) q^{85} + ( - 112746282 \beta + 23386022184) q^{87} + (90758808 \beta + 35594145930) q^{89} + (65220540 \beta + 41689446288) q^{91} + (31559360 \beta - 11877047680) q^{93} + (113154342 \beta - 18436437784) q^{95} + ( - 64199400 \beta - 849903838) q^{97} + ( - 182725753 \beta + 4157458628) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 56 q^{3} + 7868 q^{5} + 91056 q^{7} + 540202 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 56 q^{3} + 7868 q^{5} + 91056 q^{7} + 540202 q^{9} + 159080 q^{11} + 1050476 q^{13} - 10494832 q^{15} + 1430884 q^{17} - 21866600 q^{19} + 40052544 q^{21} + 35806736 q^{23} + 61878094 q^{25} + 55209392 q^{27} - 228827700 q^{29} + 64722112 q^{31} - 614344864 q^{33} - 91821408 q^{35} + 75558780 q^{37} + 875909072 q^{39} + 1201214196 q^{41} - 45519832 q^{43} + 1525107052 q^{45} - 1229079264 q^{47} + 1766069074 q^{49} - 9624987920 q^{51} - 3808549924 q^{53} + 8051409968 q^{55} - 4708125536 q^{57} - 6012926584 q^{59} + 9789792908 q^{61} + 26694483312 q^{63} - 6025376344 q^{65} + 14703095224 q^{67} - 21086664256 q^{69} + 4319991088 q^{71} + 11055639476 q^{73} - 82574103416 q^{75} - 18746968128 q^{77} + 51957623264 q^{79} - 9747960302 q^{81} - 108227975912 q^{83} + 121609729720 q^{85} + 46772044368 q^{87} + 71188291860 q^{89} + 83378892576 q^{91} - 23754095360 q^{93} - 36872875568 q^{95} - 1699807676 q^{97} + 8314917256 q^{99}+O(q^{100}) \) 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
−4.72015
5.72015
0 −640.180 0 11952.2 0 17464.5 0 232683. 0
1.2 0 696.180 0 −4084.16 0 73591.5 0 307519. 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 8.12.a.b 2
3.b odd 2 1 72.12.a.e 2
4.b odd 2 1 16.12.a.d 2
5.b even 2 1 200.12.a.d 2
5.c odd 4 2 200.12.c.c 4
8.b even 2 1 64.12.a.h 2
8.d odd 2 1 64.12.a.k 2
12.b even 2 1 144.12.a.p 2
16.e even 4 2 256.12.b.h 4
16.f odd 4 2 256.12.b.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.12.a.b 2 1.a even 1 1 trivial
16.12.a.d 2 4.b odd 2 1
64.12.a.h 2 8.b even 2 1
64.12.a.k 2 8.d odd 2 1
72.12.a.e 2 3.b odd 2 1
144.12.a.p 2 12.b even 2 1
200.12.a.d 2 5.b even 2 1
200.12.c.c 4 5.c odd 4 2
256.12.b.h 4 16.e even 4 2
256.12.b.k 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 56T_{3} - 445680 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(8))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 56T - 445680 \) Copy content Toggle raw display
$5$ \( T^{2} - 7868 T - 48814460 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 1285236288 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 208087277936 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 125364026012 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 51795407805500 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 110143192291984 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 47308617068608 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 681230587110400 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 17\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 46\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 80\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 18\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 75\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 14\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 14\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 17\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 25\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 89\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 28\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 24\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 18\!\cdots\!56 \) Copy content Toggle raw display
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