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Properties

Label 136.4.k.b
Level $136$
Weight $4$
Character orbit 136.k
Analytic conductor $8.024$
Analytic rank $0$
Dimension $14$
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] = [136,4,Mod(81,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.k (of order \(4\), degree \(2\), 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: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 119x^{12} + 5319x^{10} + 112122x^{8} + 1120191x^{6} + 4382607x^{4} + 1699337x^{2} + 2704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - \beta_{7} q^{5} + ( - \beta_{8} - \beta_{5} + 1) q^{7} + ( - \beta_{11} + 6 \beta_{5} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - \beta_{7} q^{5} + ( - \beta_{8} - \beta_{5} + 1) q^{7} + ( - \beta_{11} + 6 \beta_{5} + \cdots + \beta_{2}) q^{9}+ \cdots + (12 \beta_{13} - 19 \beta_{11} + \cdots + 211) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 6 q^{3} + 6 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 6 q^{3} + 6 q^{5} + 10 q^{7} - 66 q^{11} - 124 q^{13} + 130 q^{17} - 148 q^{21} - 162 q^{23} - 204 q^{27} + 158 q^{29} - 350 q^{31} + 116 q^{33} + 236 q^{35} + 582 q^{37} - 320 q^{39} + 878 q^{41} - 26 q^{45} - 448 q^{47} - 590 q^{51} + 1460 q^{55} + 292 q^{57} - 1130 q^{61} + 114 q^{63} + 20 q^{65} - 64 q^{67} + 2076 q^{69} - 3274 q^{71} - 1426 q^{73} + 946 q^{75} + 1718 q^{79} + 166 q^{81} - 2018 q^{85} + 476 q^{89} + 2128 q^{91} - 2444 q^{95} + 1926 q^{97} + 3870 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 119x^{12} + 5319x^{10} + 112122x^{8} + 1120191x^{6} + 4382607x^{4} + 1699337x^{2} + 2704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16553 \nu^{12} - 2958786 \nu^{10} - 553258413 \nu^{8} - 35042935761 \nu^{6} + \cdots - 1462819625392 ) / 53247141288 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9031 \nu^{12} + 574494 \nu^{10} - 2543421 \nu^{8} - 708095937 \nu^{6} - 12509813490 \nu^{4} + \cdots - 1244553232 ) / 29043895248 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9031 \nu^{12} + 574494 \nu^{10} - 2543421 \nu^{8} - 708095937 \nu^{6} - 12509813490 \nu^{4} + \cdots - 1244553232 ) / 29043895248 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9031 \nu^{12} - 574494 \nu^{10} + 2543421 \nu^{8} + 708095937 \nu^{6} + 12509813490 \nu^{4} + \cdots + 480468824824 ) / 14521947624 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5983429 \nu^{13} - 712145454 \nu^{11} - 31833327273 \nu^{9} - 670840961865 \nu^{7} + \cdots - 9487413190220 \nu ) / 377570638224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16718259 \nu^{13} - 19735339 \nu^{12} - 1997695269 \nu^{11} - 2196992499 \nu^{10} + \cdots - 2125260748364 ) / 1038319255116 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16718259 \nu^{13} + 19735339 \nu^{12} - 1997695269 \nu^{11} + 2196992499 \nu^{10} + \cdots + 2125260748364 ) / 1038319255116 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 394782247 \nu^{13} + 30952792 \nu^{12} + 46939392582 \nu^{11} + 2586850656 \nu^{10} + \cdots + 21859444421360 ) / 4153277020464 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 394782247 \nu^{13} - 30952792 \nu^{12} + 46939392582 \nu^{11} - 2586850656 \nu^{10} + \cdots - 21859444421360 ) / 4153277020464 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 778104491 \nu^{13} - 93193189950 \nu^{11} - 4199517040215 \nu^{9} + \cdots - 14\!\cdots\!88 \nu ) / 2076638510232 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 197687963 \nu^{13} - 23515736826 \nu^{11} - 1050433671063 \nu^{9} + \cdots - 313807418169676 \nu ) / 377570638224 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 143212757 \nu^{13} - 11120005 \nu^{12} - 17048541404 \nu^{11} - 1318184036 \nu^{10} + \cdots - 10332452117624 ) / 230737612248 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 143212757 \nu^{13} - 11120005 \nu^{12} + 17048541404 \nu^{11} - 1318184036 \nu^{10} + \cdots - 10332452117624 ) / 230737612248 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} + \beta_{2} - 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{13} + \beta_{12} - 4 \beta_{11} - 4 \beta_{9} - 4 \beta_{8} - 5 \beta_{7} + \cdots + 60 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 7 \beta_{13} - 7 \beta_{12} - 16 \beta_{9} + 16 \beta_{8} - 23 \beta_{7} + 23 \beta_{6} - 75 \beta_{4} + \cdots + 1851 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 59 \beta_{13} - 59 \beta_{12} + 267 \beta_{11} + 9 \beta_{10} + 212 \beta_{9} + 212 \beta_{8} + \cdots - 2104 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 400 \beta_{13} + 400 \beta_{12} + 1000 \beta_{9} - 1000 \beta_{8} + 1328 \beta_{7} - 1328 \beta_{6} + \cdots - 61061 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1389 \beta_{13} + 1389 \beta_{12} - 6950 \beta_{11} - 336 \beta_{10} - 4776 \beta_{9} + \cdots + 40585 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 9472 \beta_{13} - 9472 \beta_{12} - 25150 \beta_{9} + 25150 \beta_{8} - 31688 \beta_{7} + \cdots + 1124836 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 125251 \beta_{13} - 125251 \beta_{12} + 665760 \beta_{11} + 37224 \beta_{10} + 415996 \beta_{9} + \cdots - 3320388 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 855515 \beta_{13} + 855515 \beta_{12} + 2370092 \beta_{9} - 2370092 \beta_{8} + 2880691 \beta_{7} + \cdots - 89151457 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5594021 \beta_{13} + 5594021 \beta_{12} - 30731683 \beta_{11} - 1859493 \beta_{10} + \cdots + 140628400 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 38091144 \beta_{13} - 38091144 \beta_{12} - 108692148 \beta_{9} + 108692148 \beta_{8} + \cdots + 3699241435 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 124623174 \beta_{13} - 124623174 \beta_{12} + 696412656 \beta_{11} + 44274816 \beta_{10} + \cdots - 3038627173 \beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
5.11455i
4.87012i
0.657265i
− 0.0399724i
− 3.13314i
− 3.80701i
− 6.66182i
− 5.11455i
− 4.87012i
− 0.657265i
0.0399724i
3.13314i
3.80701i
6.66182i
0 −5.11455 − 5.11455i 0 7.20340 + 7.20340i 0 19.2581 − 19.2581i 0 25.3172i 0
81.2 0 −4.87012 − 4.87012i 0 −8.46115 − 8.46115i 0 −8.46698 + 8.46698i 0 20.4362i 0
81.3 0 −0.657265 − 0.657265i 0 13.8302 + 13.8302i 0 −14.0096 + 14.0096i 0 − 26.1360i 0
81.4 0 0.0399724 + 0.0399724i 0 −1.97903 − 1.97903i 0 −4.30175 + 4.30175i 0 − 26.9968i 0
81.5 0 3.13314 + 3.13314i 0 −13.6954 − 13.6954i 0 −0.366316 + 0.366316i 0 − 7.36691i 0
81.6 0 3.80701 + 3.80701i 0 2.78003 + 2.78003i 0 25.7370 − 25.7370i 0 1.98658i 0
81.7 0 6.66182 + 6.66182i 0 3.32200 + 3.32200i 0 −12.8505 + 12.8505i 0 61.7597i 0
89.1 0 −5.11455 + 5.11455i 0 7.20340 − 7.20340i 0 19.2581 + 19.2581i 0 − 25.3172i 0
89.2 0 −4.87012 + 4.87012i 0 −8.46115 + 8.46115i 0 −8.46698 − 8.46698i 0 − 20.4362i 0
89.3 0 −0.657265 + 0.657265i 0 13.8302 − 13.8302i 0 −14.0096 − 14.0096i 0 26.1360i 0
89.4 0 0.0399724 − 0.0399724i 0 −1.97903 + 1.97903i 0 −4.30175 − 4.30175i 0 26.9968i 0
89.5 0 3.13314 − 3.13314i 0 −13.6954 + 13.6954i 0 −0.366316 − 0.366316i 0 7.36691i 0
89.6 0 3.80701 − 3.80701i 0 2.78003 − 2.78003i 0 25.7370 + 25.7370i 0 − 1.98658i 0
89.7 0 6.66182 − 6.66182i 0 3.32200 − 3.32200i 0 −12.8505 − 12.8505i 0 − 61.7597i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.k.b ✓ 14
4.b odd 2 1 272.4.o.f 14
17.c even 4 1 inner 136.4.k.b ✓ 14
17.d even 8 2 2312.4.a.l 14
68.f odd 4 1 272.4.o.f 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.k.b ✓ 14 1.a even 1 1 trivial
136.4.k.b ✓ 14 17.c even 4 1 inner
272.4.o.f 14 4.b odd 2 1
272.4.o.f 14 68.f odd 4 1
2312.4.a.l 14 17.d even 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 6 T_{3}^{13} + 18 T_{3}^{12} + 140 T_{3}^{11} + 6044 T_{3}^{10} - 26400 T_{3}^{9} + \cdots + 346112 \) acting on \(S_{4}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 6 T^{13} + \cdots + 346112 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 5698471938048 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 181426562039808 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 27\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( (T^{7} + 62 T^{6} + \cdots - 237436285696)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 69\!\cdots\!17 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 88\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 58\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 20\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 20\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots + 16\!\cdots\!60)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 84\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 77\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots - 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 12\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 80\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 72\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots - 34\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 53\!\cdots\!08 \) Copy content Toggle raw display
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