Properties

Label 6.9.b.a
Level $6$
Weight $9$
Character orbit 6.b
Analytic conductor $2.444$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6,9,Mod(5,6)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6.5");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 6.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: \(2.44427166037\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 4\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{2} + (9 \beta - 63) q^{3} - 128 q^{4} + 102 \beta q^{5} + ( - 126 \beta - 576) q^{6} + 2786 q^{7} - 256 \beta q^{8} + ( - 1134 \beta + 1377) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{2} + (9 \beta - 63) q^{3} - 128 q^{4} + 102 \beta q^{5} + ( - 126 \beta - 576) q^{6} + 2786 q^{7} - 256 \beta q^{8} + ( - 1134 \beta + 1377) q^{9} - 6528 q^{10} + 3966 \beta q^{11} + ( - 1152 \beta + 8064) q^{12} - 13150 q^{13} + 5572 \beta q^{14} + ( - 6426 \beta - 29376) q^{15} + 16384 q^{16} - 11736 \beta q^{17} + (2754 \beta + 72576) q^{18} + 144002 q^{19} - 13056 \beta q^{20} + (25074 \beta - 175518) q^{21} - 253824 q^{22} + 8724 \beta q^{23} + (16128 \beta + 73728) q^{24} + 57697 q^{25} - 26300 \beta q^{26} + (83835 \beta + 239841) q^{27} - 356608 q^{28} - 110910 \beta q^{29} + ( - 58752 \beta + 411264) q^{30} + 728738 q^{31} + 32768 \beta q^{32} + ( - 249858 \beta - 1142208) q^{33} + 751104 q^{34} + 284172 \beta q^{35} + (145152 \beta - 176256) q^{36} - 1964446 q^{37} + 288004 \beta q^{38} + ( - 118350 \beta + 828450) q^{39} + 835584 q^{40} + 174324 \beta q^{41} + ( - 351036 \beta - 1604736) q^{42} - 78142 q^{43} - 507648 \beta q^{44} + (140454 \beta + 3701376) q^{45} - 558336 q^{46} - 622200 \beta q^{47} + (147456 \beta - 1032192) q^{48} + 1996995 q^{49} + 115394 \beta q^{50} + (739368 \beta + 3379968) q^{51} + 1683200 q^{52} + 92286 \beta q^{53} + (479682 \beta - 5365440) q^{54} - 12945024 q^{55} - 713216 \beta q^{56} + (1296018 \beta - 9072126) q^{57} + 7098240 q^{58} - 884634 \beta q^{59} + (822528 \beta + 3760128) q^{60} + 17578274 q^{61} + 1457476 \beta q^{62} + ( - 3159324 \beta + 3836322) q^{63} - 2097152 q^{64} - 1341300 \beta q^{65} + ( - 2284416 \beta + 15990912) q^{66} - 17136766 q^{67} + 1502208 \beta q^{68} + ( - 549612 \beta - 2512512) q^{69} - 18187008 q^{70} + 4576860 \beta q^{71} + ( - 352512 \beta - 9289728) q^{72} + 28139330 q^{73} - 3928892 \beta q^{74} + (519273 \beta - 3634911) q^{75} - 18432256 q^{76} + 11049276 \beta q^{77} + (1656900 \beta + 7574400) q^{78} + 9182498 q^{79} + 1671168 \beta q^{80} + ( - 3123036 \beta - 39254463) q^{81} - 11156736 q^{82} - 15398742 \beta q^{83} + ( - 3209472 \beta + 22466304) q^{84} + 38306304 q^{85} - 156284 \beta q^{86} + (6987330 \beta + 31942080) q^{87} + 32489472 q^{88} - 14363604 \beta q^{89} + (7402752 \beta - 8989056) q^{90} - 36635900 q^{91} - 1116672 \beta q^{92} + (6558642 \beta - 45910494) q^{93} + 39820800 q^{94} + 14688204 \beta q^{95} + ( - 2064384 \beta - 9437184) q^{96} - 128722558 q^{97} + 3993990 \beta q^{98} + (5461182 \beta + 143918208) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 126 q^{3} - 256 q^{4} - 1152 q^{6} + 5572 q^{7} + 2754 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 126 q^{3} - 256 q^{4} - 1152 q^{6} + 5572 q^{7} + 2754 q^{9} - 13056 q^{10} + 16128 q^{12} - 26300 q^{13} - 58752 q^{15} + 32768 q^{16} + 145152 q^{18} + 288004 q^{19} - 351036 q^{21} - 507648 q^{22} + 147456 q^{24} + 115394 q^{25} + 479682 q^{27} - 713216 q^{28} + 822528 q^{30} + 1457476 q^{31} - 2284416 q^{33} + 1502208 q^{34} - 352512 q^{36} - 3928892 q^{37} + 1656900 q^{39} + 1671168 q^{40} - 3209472 q^{42} - 156284 q^{43} + 7402752 q^{45} - 1116672 q^{46} - 2064384 q^{48} + 3993990 q^{49} + 6759936 q^{51} + 3366400 q^{52} - 10730880 q^{54} - 25890048 q^{55} - 18144252 q^{57} + 14196480 q^{58} + 7520256 q^{60} + 35156548 q^{61} + 7672644 q^{63} - 4194304 q^{64} + 31981824 q^{66} - 34273532 q^{67} - 5025024 q^{69} - 36374016 q^{70} - 18579456 q^{72} + 56278660 q^{73} - 7269822 q^{75} - 36864512 q^{76} + 15148800 q^{78} + 18364996 q^{79} - 78508926 q^{81} - 22313472 q^{82} + 44932608 q^{84} + 76612608 q^{85} + 63884160 q^{87} + 64978944 q^{88} - 17978112 q^{90} - 73271800 q^{91} - 91820988 q^{93} + 79641600 q^{94} - 18874368 q^{96} - 257445116 q^{97} + 287836416 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
1.41421i
1.41421i
11.3137i −63.0000 50.9117i −128.000 576.999i −576.000 + 712.764i 2786.00 1448.15i 1377.00 + 6414.87i −6528.00
5.2 11.3137i −63.0000 + 50.9117i −128.000 576.999i −576.000 712.764i 2786.00 1448.15i 1377.00 6414.87i −6528.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.9.b.a 2
3.b odd 2 1 inner 6.9.b.a 2
4.b odd 2 1 48.9.e.d 2
5.b even 2 1 150.9.d.a 2
5.c odd 4 2 150.9.b.a 4
8.b even 2 1 192.9.e.h 2
8.d odd 2 1 192.9.e.c 2
9.c even 3 2 162.9.d.a 4
9.d odd 6 2 162.9.d.a 4
12.b even 2 1 48.9.e.d 2
15.d odd 2 1 150.9.d.a 2
15.e even 4 2 150.9.b.a 4
24.f even 2 1 192.9.e.c 2
24.h odd 2 1 192.9.e.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.9.b.a 2 1.a even 1 1 trivial
6.9.b.a 2 3.b odd 2 1 inner
48.9.e.d 2 4.b odd 2 1
48.9.e.d 2 12.b even 2 1
150.9.b.a 4 5.c odd 4 2
150.9.b.a 4 15.e even 4 2
150.9.d.a 2 5.b even 2 1
150.9.d.a 2 15.d odd 2 1
162.9.d.a 4 9.c even 3 2
162.9.d.a 4 9.d odd 6 2
192.9.e.c 2 8.d odd 2 1
192.9.e.c 2 24.f even 2 1
192.9.e.h 2 8.b even 2 1
192.9.e.h 2 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(6, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 128 \) Copy content Toggle raw display
$3$ \( T^{2} + 126T + 6561 \) Copy content Toggle raw display
$5$ \( T^{2} + 332928 \) Copy content Toggle raw display
$7$ \( (T - 2786)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 503332992 \) Copy content Toggle raw display
$13$ \( (T + 13150)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4407478272 \) Copy content Toggle raw display
$19$ \( (T - 144002)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2435461632 \) Copy content Toggle raw display
$29$ \( T^{2} + 393632899200 \) Copy content Toggle raw display
$31$ \( (T - 728738)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1964446)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 972443423232 \) Copy content Toggle raw display
$43$ \( (T + 78142)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12388250880000 \) Copy content Toggle raw display
$53$ \( T^{2} + 272534585472 \) Copy content Toggle raw display
$59$ \( T^{2} + 25042474046592 \) Copy content Toggle raw display
$61$ \( (T - 17578274)^{2} \) Copy content Toggle raw display
$67$ \( (T + 17136766)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 670324718707200 \) Copy content Toggle raw display
$73$ \( (T - 28139330)^{2} \) Copy content Toggle raw display
$79$ \( (T - 9182498)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 75\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{2} + 66\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( (T + 128722558)^{2} \) Copy content Toggle raw display
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