↑

Properties

Label 136.2.n.c
Level $136$
Weight $2$
Character orbit 136.n
Analytic conductor $1.086$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [136,2,Mod(9,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 136.n (of order \(8\), degree \(4\), 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: \(1.08596546749\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 28x^{10} + 258x^{8} + 880x^{6} + 1033x^{4} + 132x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{8} q^{5} + (\beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{3} + 1) q^{7} + ( - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{8} q^{5} + (\beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{3} + 1) q^{7} + ( - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4}) q^{9} + (\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_1 - 1) q^{11} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{11} + \beta_{9} - \beta_{5} - \beta_{4} + \beta_1 + 1) q^{15} + (\beta_{9} + \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{3}) q^{17} + ( - \beta_{11} + \beta_{9} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{19} + (\beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2} + 2 \beta_1) q^{21} + ( - \beta_{10} - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 - 1) q^{23} + ( - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{25} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_{2} + 2) q^{27} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{29} + ( - \beta_{11} - 3 \beta_{7} - \beta_{4} - \beta_{3} - \beta_{2} - 3) q^{31} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{33} + (\beta_{11} + \beta_{10} + 3 \beta_{7} + 3 \beta_{6} + \beta_{5} + \beta_{3} - 2) q^{35} + ( - \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{6} - 2 \beta_1) q^{37} + ( - 2 \beta_{8} + 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{39} + ( - \beta_{11} + 2 \beta_{9} + \beta_{8} + 3 \beta_{7} + 2 \beta_{5} - 2 \beta_{2} - 3 \beta_1) q^{41} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{3} - \beta_{2}) q^{43} + ( - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - 5 \beta_{6} + \beta_{5} + \beta_{2} + \beta_1 - 5) q^{45} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + \cdots - 2 \beta_1) q^{47}+ \cdots + (2 \beta_{11} - \beta_{10} - \beta_{9} - 5 \beta_{7} - 7 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{5} - 8 q^{9} - 12 q^{11} + 20 q^{15} + 4 q^{17} - 4 q^{19} - 8 q^{23} - 16 q^{25} + 24 q^{27} + 8 q^{29} - 32 q^{31} - 24 q^{33} - 32 q^{35} + 4 q^{37} - 8 q^{39} + 16 q^{41} + 8 q^{43} - 64 q^{45} + 44 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} + 16 q^{59} + 44 q^{61} + 100 q^{63} - 20 q^{65} - 40 q^{67} + 56 q^{69} + 32 q^{71} + 8 q^{73} + 92 q^{75} - 12 q^{77} - 8 q^{79} + 40 q^{83} + 40 q^{85} - 84 q^{87} - 40 q^{91} - 76 q^{93} + 28 q^{95} - 16 q^{97} - 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 28x^{10} + 258x^{8} + 880x^{6} + 1033x^{4} + 132x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -9\nu^{11} - 250\nu^{9} - 2274\nu^{7} - 7596\nu^{5} - 8901\nu^{3} - 1658\nu ) / 272 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 39 \nu^{11} + 17 \nu^{10} - 1089 \nu^{9} + 459 \nu^{8} - 9973 \nu^{7} + 3927 \nu^{6} - 33443 \nu^{5} + 11305 \nu^{4} - 37228 \nu^{3} + 10064 \nu^{2} - 3524 \nu + 612 ) / 1088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39 \nu^{11} - 25 \nu^{10} + 1089 \nu^{9} - 651 \nu^{8} + 9973 \nu^{7} - 5223 \nu^{6} + 33443 \nu^{5} - 12889 \nu^{4} + 37228 \nu^{3} - 8184 \nu^{2} + 2436 \nu - 756 ) / 1088 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 39 \nu^{11} + 17 \nu^{10} + 1089 \nu^{9} + 459 \nu^{8} + 9973 \nu^{7} + 3927 \nu^{6} + 33443 \nu^{5} + 11305 \nu^{4} + 37228 \nu^{3} + 10064 \nu^{2} + 3524 \nu + 612 ) / 1088 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39 \nu^{11} - 25 \nu^{10} - 1089 \nu^{9} - 651 \nu^{8} - 9973 \nu^{7} - 5223 \nu^{6} - 33443 \nu^{5} - 12889 \nu^{4} - 37228 \nu^{3} - 8184 \nu^{2} - 2436 \nu - 756 ) / 1088 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 153 \nu^{11} - 39 \nu^{10} + 4267 \nu^{9} - 1089 \nu^{8} + 39015 \nu^{7} - 9973 \nu^{6} + 130713 \nu^{5} - 33443 \nu^{4} + 146744 \nu^{3} - 37228 \nu^{2} + 10132 \nu - 2436 ) / 1088 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 153 \nu^{11} - 39 \nu^{10} - 4267 \nu^{9} - 1089 \nu^{8} - 39015 \nu^{7} - 9973 \nu^{6} - 130713 \nu^{5} - 33443 \nu^{4} - 146744 \nu^{3} - 37228 \nu^{2} - 10132 \nu - 2436 ) / 1088 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 119 \nu^{11} - 27 \nu^{10} + 3315 \nu^{9} - 767 \nu^{8} + 30243 \nu^{7} - 7247 \nu^{6} + 100725 \nu^{5} - 25865 \nu^{4} + 110670 \nu^{3} - 31582 \nu^{2} + 3808 \nu - 2016 ) / 544 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 119 \nu^{11} - 27 \nu^{10} - 3315 \nu^{9} - 767 \nu^{8} - 30243 \nu^{7} - 7247 \nu^{6} - 100725 \nu^{5} - 25865 \nu^{4} - 110670 \nu^{3} - 31582 \nu^{2} - 3808 \nu - 2016 ) / 544 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 123 \nu^{11} - 41 \nu^{10} + 3445 \nu^{9} - 1137 \nu^{8} + 31741 \nu^{7} - 10297 \nu^{6} + 107943 \nu^{5} - 33839 \nu^{4} + 123568 \nu^{3} - 36486 \nu^{2} + 8300 \nu - 1384 ) / 544 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 123 \nu^{11} - 41 \nu^{10} - 3445 \nu^{9} - 1137 \nu^{8} - 31741 \nu^{7} - 10297 \nu^{6} - 107943 \nu^{5} - 33839 \nu^{4} - 123568 \nu^{3} - 36486 \nu^{2} - 8300 \nu - 1384 ) / 544 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} + 2\beta_{10} - 4\beta_{7} - 4\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} - 13\beta_{5} - 7\beta_{4} + 13\beta_{3} + 7\beta_{2} - 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 20 \beta_{11} - 20 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} + 52 \beta_{7} + 52 \beta_{6} + 9 \beta_{5} + 17 \beta_{4} + 9 \beta_{3} + 17 \beta_{2} + 80 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} - 6 \beta_{8} - 42 \beta_{7} + 42 \beta_{6} + 155 \beta_{5} + 69 \beta_{4} - 155 \beta_{3} - 69 \beta_{2} + 64 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 218 \beta_{11} + 218 \beta_{10} + 84 \beta_{9} + 84 \beta_{8} - 612 \beta_{7} - 612 \beta_{6} - 107 \beta_{5} - 243 \beta_{4} - 107 \beta_{3} - 243 \beta_{2} - 884 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 52 \beta_{11} + 52 \beta_{10} - 132 \beta_{9} + 132 \beta_{8} + 670 \beta_{7} - 670 \beta_{6} - 1817 \beta_{5} - 767 \beta_{4} + 1817 \beta_{3} + 767 \beta_{2} - 964 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2452 \beta_{11} - 2452 \beta_{10} - 998 \beta_{9} - 998 \beta_{8} + 7084 \beta_{7} + 7084 \beta_{6} + 1381 \beta_{5} + 3285 \beta_{4} + 1381 \beta_{3} + 3285 \beta_{2} + 10072 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 906 \beta_{11} - 906 \beta_{10} + 2214 \beta_{9} - 2214 \beta_{8} - 9690 \beta_{7} + 9690 \beta_{6} + 21307 \beta_{5} + 8869 \beta_{4} - 21307 \beta_{3} - 8869 \beta_{2} + 13760 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 27962 \beta_{11} + 27962 \beta_{10} + 11532 \beta_{9} + 11532 \beta_{8} - 82108 \beta_{7} - 82108 \beta_{6} - 17963 \beta_{5} - 43211 \beta_{4} - 17963 \beta_{3} - 43211 \beta_{2} - 116276 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 13716 \beta_{11} + 13716 \beta_{10} - 33212 \beta_{9} + 33212 \beta_{8} + 133350 \beta_{7} - 133350 \beta_{6} - 250913 \beta_{5} - 104063 \beta_{4} + 250913 \beta_{3} + 104063 \beta_{2} - 188772 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
3.18938i
0.306239i
− 3.49562i
− 1.75800i
0.216105i
1.54190i
1.75800i
− 0.216105i
− 1.54190i
− 3.18938i
− 0.306239i
3.49562i
0 −2.25523 − 0.934148i 0 −1.35305 + 3.26655i 0 0.666590 + 1.60929i 0 2.09212 + 2.09212i 0
9.2 0 −0.216544 − 0.0896953i 0 1.33137 − 3.21420i 0 0.934059 + 2.25502i 0 −2.08247 − 2.08247i 0
9.3 0 2.47178 + 1.02384i 0 −0.392531 + 0.947653i 0 −0.893542 − 2.15720i 0 2.94010 + 2.94010i 0
25.1 0 −1.24310 + 3.00110i 0 0.194339 + 0.0804980i 0 −1.76317 + 0.730328i 0 −5.33998 − 5.33998i 0
25.2 0 0.152809 − 0.368914i 0 −1.09698 − 0.454383i 0 4.72436 − 1.95689i 0 2.00857 + 2.00857i 0
25.3 0 1.09029 − 2.63218i 0 3.31685 + 1.37389i 0 −3.66830 + 1.51946i 0 −3.61834 − 3.61834i 0
49.1 0 −1.24310 − 3.00110i 0 0.194339 − 0.0804980i 0 −1.76317 − 0.730328i 0 −5.33998 + 5.33998i 0
49.2 0 0.152809 + 0.368914i 0 −1.09698 + 0.454383i 0 4.72436 + 1.95689i 0 2.00857 − 2.00857i 0
49.3 0 1.09029 + 2.63218i 0 3.31685 − 1.37389i 0 −3.66830 − 1.51946i 0 −3.61834 + 3.61834i 0
121.1 0 −2.25523 + 0.934148i 0 −1.35305 − 3.26655i 0 0.666590 − 1.60929i 0 2.09212 − 2.09212i 0
121.2 0 −0.216544 + 0.0896953i 0 1.33137 + 3.21420i 0 0.934059 − 2.25502i 0 −2.08247 + 2.08247i 0
121.3 0 2.47178 − 1.02384i 0 −0.392531 − 0.947653i 0 −0.893542 + 2.15720i 0 2.94010 − 2.94010i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.2.n.c ✓ 12
3.b odd 2 1 1224.2.bq.c 12
4.b odd 2 1 272.2.v.f 12
17.d even 8 1 inner 136.2.n.c ✓ 12
17.e odd 16 2 2312.2.a.w 12
17.e odd 16 2 2312.2.b.n 12
51.g odd 8 1 1224.2.bq.c 12
68.g odd 8 1 272.2.v.f 12
68.i even 16 2 4624.2.a.bt 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.n.c ✓ 12 1.a even 1 1 trivial
136.2.n.c ✓ 12 17.d even 8 1 inner
272.2.v.f 12 4.b odd 2 1
272.2.v.f 12 68.g odd 8 1
1224.2.bq.c 12 3.b odd 2 1
1224.2.bq.c 12 51.g odd 8 1
2312.2.a.w 12 17.e odd 16 2
2312.2.b.n 12 17.e odd 16 2
4624.2.a.bt 12 68.i even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 4 T_{3}^{10} - 8 T_{3}^{9} + 8 T_{3}^{8} + 40 T_{3}^{7} - 224 T_{3}^{6} + 96 T_{3}^{5} + 3652 T_{3}^{4} + 464 T_{3}^{3} + 304 T_{3}^{2} + 192 T_{3} + 32 \) acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{10} - 8 T^{9} + 8 T^{8} + \cdots + 32 \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + 16 T^{10} - 56 T^{9} + \cdots + 128 \) Copy content Toggle raw display
$7$ \( T^{12} - 22 T^{10} - 12 T^{9} + \cdots + 147968 \) Copy content Toggle raw display
$11$ \( T^{12} + 12 T^{11} + 48 T^{10} + \cdots + 16928 \) Copy content Toggle raw display
$13$ \( T^{12} + 92 T^{10} + 3060 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{11} - 40 T^{10} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( T^{12} + 4 T^{11} + 8 T^{10} + 8 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{12} + 8 T^{11} - 6 T^{10} + \cdots + 43655168 \) Copy content Toggle raw display
$29$ \( T^{12} - 8 T^{11} + 20 T^{10} + \cdots + 118210688 \) Copy content Toggle raw display
$31$ \( T^{12} + 32 T^{11} + \cdots + 103219712 \) Copy content Toggle raw display
$37$ \( T^{12} - 4 T^{11} + 56 T^{10} - 328 T^{9} + \cdots + 512 \) Copy content Toggle raw display
$41$ \( T^{12} - 16 T^{11} + \cdots + 591542408 \) Copy content Toggle raw display
$43$ \( T^{12} - 8 T^{11} + 32 T^{10} + \cdots + 148254976 \) Copy content Toggle raw display
$47$ \( T^{12} + 360 T^{10} + \cdots + 440664064 \) Copy content Toggle raw display
$53$ \( T^{12} + 8 T^{11} + 32 T^{10} + \cdots + 25080064 \) Copy content Toggle raw display
$59$ \( T^{12} - 16 T^{11} + 128 T^{10} + \cdots + 9339136 \) Copy content Toggle raw display
$61$ \( T^{12} - 44 T^{11} + \cdots + 118949888 \) Copy content Toggle raw display
$67$ \( (T^{6} + 20 T^{5} + 58 T^{4} - 880 T^{3} + \cdots + 28928)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 32 T^{11} + 578 T^{10} + \cdots + 10913792 \) Copy content Toggle raw display
$73$ \( T^{12} - 8 T^{11} + 158 T^{10} + \cdots + 545424392 \) Copy content Toggle raw display
$79$ \( T^{12} + 8 T^{11} + \cdots + 1130596352 \) Copy content Toggle raw display
$83$ \( T^{12} - 40 T^{11} + 800 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( T^{12} + 552 T^{10} + \cdots + 12558340096 \) Copy content Toggle raw display
$97$ \( T^{12} + 16 T^{11} + 334 T^{10} + \cdots + 30451208 \) Copy content Toggle raw display
show more
show less