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Properties

Label 108.4.b.b
Level $108$
Weight $4$
Character orbit 108.b
Analytic conductor $6.372$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [108,4,Mod(107,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.107");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 108.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(6.37220628062\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + \cdots + 9496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{6} q^{4} + \beta_{5} q^{5} - \beta_{4} q^{7} + (\beta_{9} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{6} q^{4} + \beta_{5} q^{5} - \beta_{4} q^{7} + (\beta_{9} - \beta_1) q^{8} + (\beta_{6} - \beta_{4} + \beta_{3} + 4) q^{10} + (\beta_{9} + \beta_{2} - \beta_1) q^{11} + (\beta_{10} - \beta_{6} - 6) q^{13} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots - \beta_1) q^{14}+ \cdots + (22 \beta_{9} - 38 \beta_{8} + \cdots - 24 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} + 42 q^{10} - 72 q^{13} + 114 q^{16} + 66 q^{22} - 384 q^{25} - 282 q^{28} - 324 q^{34} - 240 q^{37} + 774 q^{40} + 1752 q^{46} + 288 q^{49} + 924 q^{52} - 948 q^{58} + 144 q^{61} - 3066 q^{64} - 3558 q^{70} + 156 q^{73} + 576 q^{76} + 5832 q^{82} - 168 q^{85} + 5022 q^{88} - 3444 q^{94} + 516 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + \cdots + 9496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 33253968445 \nu^{11} - 2668198080286 \nu^{10} + 792575592373 \nu^{9} + \cdots - 56\!\cdots\!84 ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16345777051 \nu^{11} - 1455335526278 \nu^{10} - 1946811804847 \nu^{9} + \cdots - 28\!\cdots\!40 ) / 112314071783672 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2029314856 \nu^{11} - 20657394999 \nu^{10} - 72348050754 \nu^{9} - 44845772052 \nu^{8} + \cdots - 67923831391169 ) / 6381481351345 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2849350 \nu^{11} + 6585588 \nu^{10} - 7900308 \nu^{9} + 8937888 \nu^{8} - 274261062 \nu^{7} + \cdots - 3774567143 ) / 3046053151 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 454920367261 \nu^{11} - 699408466494 \nu^{10} - 909094946747 \nu^{9} + \cdots + 873663409442004 ) / 449256287134688 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23425387202 \nu^{11} - 19598582637 \nu^{10} - 48064283082 \nu^{9} + \cdots - 38283473678982 ) / 12762962702690 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1144977542195 \nu^{11} + 4352732408014 \nu^{10} - 29629791457 \nu^{9} + \cdots + 19\!\cdots\!76 ) / 561570358918360 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 159260070855 \nu^{11} + 117861110662 \nu^{10} + 438728620059 \nu^{9} + \cdots - 385589858796132 ) / 70196294864795 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 716000535355 \nu^{11} - 22806431518 \nu^{10} + 1186916251229 \nu^{9} + \cdots - 24\!\cdots\!72 ) / 102103701621520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 114943514282 \nu^{11} + 101958949683 \nu^{10} + 310110651558 \nu^{9} + \cdots + 11\!\cdots\!98 ) / 12762962702690 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29835469322 \nu^{11} + 31182551757 \nu^{10} + 48231870066 \nu^{9} - 289609225533 \nu^{8} + \cdots + 58683428040738 ) / 2552592540538 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( - 2 \beta_{11} - \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} - 9 \beta_{6} + 8 \beta_{5} + \cdots - 4 ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{10} - 6 \beta_{9} + 6 \beta_{8} - 2 \beta_{7} - 11 \beta_{6} + 6 \beta_{5} - 2 \beta_{4} + \cdots - 24 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13 \beta_{11} - \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 13 \beta_{7} + 30 \beta_{6} + 46 \beta_{5} + \cdots + 224 ) / 72 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{11} + 3 \beta_{10} + 18 \beta_{9} - 32 \beta_{8} + 26 \beta_{7} - 61 \beta_{6} + 44 \beta_{5} + \cdots - 850 ) / 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 119 \beta_{11} + 23 \beta_{10} - 129 \beta_{9} - 149 \beta_{8} - 387 \beta_{7} + 392 \beta_{6} + \cdots - 572 ) / 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 167 \beta_{11} + 172 \beta_{10} + 297 \beta_{9} - 633 \beta_{8} - 183 \beta_{7} + 627 \beta_{6} + \cdots - 5210 ) / 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 75 \beta_{11} + 683 \beta_{10} + 525 \beta_{9} + 543 \beta_{8} - 249 \beta_{7} + 2386 \beta_{6} + \cdots - 35724 ) / 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1272 \beta_{11} + 529 \beta_{10} + 72 \beta_{9} - 1448 \beta_{8} - 3032 \beta_{7} + 11519 \beta_{6} + \cdots + 30702 ) / 36 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 7085 \beta_{11} + 5021 \beta_{10} + 15741 \beta_{9} + 1455 \beta_{8} + 20007 \beta_{7} + \cdots - 156028 ) / 72 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 12715 \beta_{11} - 7950 \beta_{10} - 17487 \beta_{9} + 56011 \beta_{8} + 8305 \beta_{7} + \cdots + 415114 ) / 36 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 22857 \beta_{11} - 49075 \beta_{10} + 2019 \beta_{9} + 26941 \beta_{8} + 102273 \beta_{7} + \cdots + 3750276 ) / 72 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
2.61836 + 1.60260i
2.61836 − 1.60260i
−1.29835 − 1.36719i
−1.29835 + 1.36719i
−0.453986 + 2.07664i
−0.453986 − 2.07664i
−2.18604 + 2.07664i
−2.18604 − 2.07664i
0.433705 − 1.36719i
0.433705 + 1.36719i
0.886307 + 1.60260i
0.886307 − 1.60260i
−2.72048 − 0.773954i 0 6.80199 + 4.21105i 20.8488i 0 − 13.9048i −15.2455 − 16.7205i 0 16.1360 − 56.7186i
107.2 −2.72048 + 0.773954i 0 6.80199 − 4.21105i − 20.8488i 0 13.9048i −15.2455 + 16.7205i 0 16.1360 + 56.7186i
107.3 −2.27435 − 1.68146i 0 2.34537 + 7.64848i − 5.83890i 0 8.83113i 7.52641 − 21.3390i 0 −9.81789 + 13.2797i
107.4 −2.27435 + 1.68146i 0 2.34537 − 7.64848i 5.83890i 0 − 8.83113i 7.52641 + 21.3390i 0 −9.81789 − 13.2797i
107.5 −0.419903 − 2.79708i 0 −7.64736 + 2.34901i 1.49508i 0 26.1852i 9.78152 + 20.4040i 0 4.18188 − 0.627790i
107.6 −0.419903 + 2.79708i 0 −7.64736 − 2.34901i − 1.49508i 0 − 26.1852i 9.78152 − 20.4040i 0 4.18188 + 0.627790i
107.7 0.419903 − 2.79708i 0 −7.64736 − 2.34901i 1.49508i 0 − 26.1852i −9.78152 + 20.4040i 0 4.18188 + 0.627790i
107.8 0.419903 + 2.79708i 0 −7.64736 + 2.34901i − 1.49508i 0 26.1852i −9.78152 − 20.4040i 0 4.18188 − 0.627790i
107.9 2.27435 − 1.68146i 0 2.34537 − 7.64848i − 5.83890i 0 − 8.83113i −7.52641 − 21.3390i 0 −9.81789 − 13.2797i
107.10 2.27435 + 1.68146i 0 2.34537 + 7.64848i 5.83890i 0 8.83113i −7.52641 + 21.3390i 0 −9.81789 + 13.2797i
107.11 2.72048 − 0.773954i 0 6.80199 − 4.21105i 20.8488i 0 13.9048i 15.2455 − 16.7205i 0 16.1360 + 56.7186i
107.12 2.72048 + 0.773954i 0 6.80199 + 4.21105i − 20.8488i 0 − 13.9048i 15.2455 + 16.7205i 0 16.1360 − 56.7186i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.b.b ✓ 12
3.b odd 2 1 inner 108.4.b.b ✓ 12
4.b odd 2 1 inner 108.4.b.b ✓ 12
8.b even 2 1 1728.4.c.j 12
8.d odd 2 1 1728.4.c.j 12
12.b even 2 1 inner 108.4.b.b ✓ 12
24.f even 2 1 1728.4.c.j 12
24.h odd 2 1 1728.4.c.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.b.b ✓ 12 1.a even 1 1 trivial
108.4.b.b ✓ 12 3.b odd 2 1 inner
108.4.b.b ✓ 12 4.b odd 2 1 inner
108.4.b.b ✓ 12 12.b even 2 1 inner
1728.4.c.j 12 8.b even 2 1
1728.4.c.j 12 8.d odd 2 1
1728.4.c.j 12 24.f even 2 1
1728.4.c.j 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 471T_{5}^{4} + 15867T_{5}^{2} + 33125 \) acting on \(S_{4}^{\mathrm{new}}(108, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 471 T^{4} + \cdots + 33125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 957 T^{4} + \cdots + 10338975)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 4929 T^{4} + \cdots - 2112318675)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 18 T^{2} + \cdots - 70760)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 17532 T^{4} + \cdots + 42531205952)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 1093873434624)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 20688 T^{4} + \cdots - 15885545472)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 28380 T^{4} + \cdots + 251748219200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 37183780589679)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 60 T^{2} + \cdots - 8168000)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 223920 T^{4} + \cdots + 855375564800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 62306621744832)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 820610118691973)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 36 T^{2} + \cdots - 68226752)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 467529795115200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 39 T^{2} + \cdots + 14711755)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 49\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 39\!\cdots\!43)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 129 T^{2} + \cdots - 298155155)^{4} \) Copy content Toggle raw display
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