↑

Properties

Label 169.4.e.d
Level $169$
Weight $4$
Character orbit 169.e
Analytic conductor $9.971$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,4,Mod(23,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.e (of order \(6\), degree \(2\), not 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: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + 1) q^{3} + \beta_{2} q^{4} + 3 \beta_{3} q^{5} + ( - \beta_{3} + \beta_1) q^{6} + ( - 5 \beta_{3} + 5 \beta_1) q^{7} - 7 \beta_{3} q^{8} + 26 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + 1) q^{3} + \beta_{2} q^{4} + 3 \beta_{3} q^{5} + ( - \beta_{3} + \beta_1) q^{6} + ( - 5 \beta_{3} + 5 \beta_1) q^{7} - 7 \beta_{3} q^{8} + 26 \beta_{2} q^{9} + (27 \beta_{2} - 27) q^{10} + 16 \beta_1 q^{11} + q^{12} + 45 q^{14} + 3 \beta_1 q^{15} + ( - 71 \beta_{2} + 71) q^{16} + 45 \beta_{2} q^{17} + 26 \beta_{3} q^{18} + (2 \beta_{3} - 2 \beta_1) q^{19} + (3 \beta_{3} - 3 \beta_1) q^{20} - 5 \beta_{3} q^{21} + 144 \beta_{2} q^{22} + (162 \beta_{2} - 162) q^{23} - 7 \beta_1 q^{24} + 44 q^{25} + 53 q^{27} + 5 \beta_1 q^{28} + ( - 144 \beta_{2} + 144) q^{29} + 27 \beta_{2} q^{30} - 88 \beta_{3} q^{31} + ( - 15 \beta_{3} + 15 \beta_1) q^{32} + ( - 16 \beta_{3} + 16 \beta_1) q^{33} + 45 \beta_{3} q^{34} + 135 \beta_{2} q^{35} + (26 \beta_{2} - 26) q^{36} - 101 \beta_1 q^{37} - 18 q^{38} + 189 q^{40} - 64 \beta_1 q^{41} + ( - 45 \beta_{2} + 45) q^{42} + 97 \beta_{2} q^{43} + 16 \beta_{3} q^{44} + (78 \beta_{3} - 78 \beta_1) q^{45} + (162 \beta_{3} - 162 \beta_1) q^{46} + 37 \beta_{3} q^{47} - 71 \beta_{2} q^{48} + (118 \beta_{2} - 118) q^{49} + 44 \beta_1 q^{50} + 45 q^{51} - 414 q^{53} + 53 \beta_1 q^{54} + (432 \beta_{2} - 432) q^{55} - 315 \beta_{2} q^{56} + 2 \beta_{3} q^{57} + ( - 144 \beta_{3} + 144 \beta_1) q^{58} + ( - 174 \beta_{3} + 174 \beta_1) q^{59} + 3 \beta_{3} q^{60} - 376 \beta_{2} q^{61} + ( - 792 \beta_{2} + 792) q^{62} + 130 \beta_1 q^{63} - 433 q^{64} + 144 q^{66} - 12 \beta_1 q^{67} + (45 \beta_{2} - 45) q^{68} + 162 \beta_{2} q^{69} + 135 \beta_{3} q^{70} + (119 \beta_{3} - 119 \beta_1) q^{71} + ( - 182 \beta_{3} + 182 \beta_1) q^{72} - 366 \beta_{3} q^{73} - 909 \beta_{2} q^{74} + ( - 44 \beta_{2} + 44) q^{75} - 2 \beta_1 q^{76} + 720 q^{77} - 830 q^{79} + 213 \beta_1 q^{80} + (649 \beta_{2} - 649) q^{81} - 576 \beta_{2} q^{82} + 146 \beta_{3} q^{83} + ( - 5 \beta_{3} + 5 \beta_1) q^{84} + (135 \beta_{3} - 135 \beta_1) q^{85} + 97 \beta_{3} q^{86} - 144 \beta_{2} q^{87} + ( - 1008 \beta_{2} + 1008) q^{88} + 146 \beta_1 q^{89} - 702 q^{90} - 162 q^{92} - 88 \beta_1 q^{93} + (333 \beta_{2} - 333) q^{94} - 54 \beta_{2} q^{95} - 15 \beta_{3} q^{96} + ( - 284 \beta_{3} + 284 \beta_1) q^{97} + (118 \beta_{3} - 118 \beta_1) q^{98} + 416 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{4} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 2 q^{4} + 52 q^{9} - 54 q^{10} + 4 q^{12} + 180 q^{14} + 142 q^{16} + 90 q^{17} + 288 q^{22} - 324 q^{23} + 176 q^{25} + 212 q^{27} + 288 q^{29} + 54 q^{30} + 270 q^{35} - 52 q^{36} - 72 q^{38} + 756 q^{40} + 90 q^{42} + 194 q^{43} - 142 q^{48} - 236 q^{49} + 180 q^{51} - 1656 q^{53} - 864 q^{55} - 630 q^{56} - 752 q^{61} + 1584 q^{62} - 1732 q^{64} + 576 q^{66} - 90 q^{68} + 324 q^{69} - 1818 q^{74} + 88 q^{75} + 2880 q^{77} - 3320 q^{79} - 1298 q^{81} - 1152 q^{82} - 288 q^{87} + 2016 q^{88} - 2808 q^{90} - 648 q^{92} - 666 q^{94} - 108 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 3\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\zeta_{12}^{3} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−2.59808 + 1.50000i 0.500000 + 0.866025i 0.500000 − 0.866025i 9.00000i −2.59808 − 1.50000i −12.9904 − 7.50000i − 21.0000i 13.0000 − 22.5167i −13.5000 − 23.3827i
23.2 2.59808 − 1.50000i 0.500000 + 0.866025i 0.500000 − 0.866025i − 9.00000i 2.59808 + 1.50000i 12.9904 + 7.50000i 21.0000i 13.0000 − 22.5167i −13.5000 − 23.3827i
147.1 −2.59808 − 1.50000i 0.500000 − 0.866025i 0.500000 + 0.866025i − 9.00000i −2.59808 + 1.50000i −12.9904 + 7.50000i 21.0000i 13.0000 + 22.5167i −13.5000 + 23.3827i
147.2 2.59808 + 1.50000i 0.500000 − 0.866025i 0.500000 + 0.866025i 9.00000i 2.59808 − 1.50000i 12.9904 − 7.50000i − 21.0000i 13.0000 + 22.5167i −13.5000 + 23.3827i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.e.d 4
13.b even 2 1 inner 169.4.e.d 4
13.c even 3 1 13.4.b.a ✓ 2
13.c even 3 1 inner 169.4.e.d 4
13.d odd 4 1 169.4.c.b 2
13.d odd 4 1 169.4.c.c 2
13.e even 6 1 13.4.b.a ✓ 2
13.e even 6 1 inner 169.4.e.d 4
13.f odd 12 1 169.4.a.b 1
13.f odd 12 1 169.4.a.c 1
13.f odd 12 1 169.4.c.b 2
13.f odd 12 1 169.4.c.c 2
39.h odd 6 1 117.4.b.a 2
39.i odd 6 1 117.4.b.a 2
39.k even 12 1 1521.4.a.d 1
39.k even 12 1 1521.4.a.i 1
52.i odd 6 1 208.4.f.b 2
52.j odd 6 1 208.4.f.b 2
65.l even 6 1 325.4.c.b 2
65.n even 6 1 325.4.c.b 2
65.q odd 12 1 325.4.d.a 2
65.q odd 12 1 325.4.d.b 2
65.r odd 12 1 325.4.d.a 2
65.r odd 12 1 325.4.d.b 2
104.n odd 6 1 832.4.f.c 2
104.p odd 6 1 832.4.f.c 2
104.r even 6 1 832.4.f.e 2
104.s even 6 1 832.4.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a ✓ 2 13.c even 3 1
13.4.b.a ✓ 2 13.e even 6 1
117.4.b.a 2 39.h odd 6 1
117.4.b.a 2 39.i odd 6 1
169.4.a.b 1 13.f odd 12 1
169.4.a.c 1 13.f odd 12 1
169.4.c.b 2 13.d odd 4 1
169.4.c.b 2 13.f odd 12 1
169.4.c.c 2 13.d odd 4 1
169.4.c.c 2 13.f odd 12 1
169.4.e.d 4 1.a even 1 1 trivial
169.4.e.d 4 13.b even 2 1 inner
169.4.e.d 4 13.c even 3 1 inner
169.4.e.d 4 13.e even 6 1 inner
208.4.f.b 2 52.i odd 6 1
208.4.f.b 2 52.j odd 6 1
325.4.c.b 2 65.l even 6 1
325.4.c.b 2 65.n even 6 1
325.4.d.a 2 65.q odd 12 1
325.4.d.a 2 65.r odd 12 1
325.4.d.b 2 65.q odd 12 1
325.4.d.b 2 65.r odd 12 1
832.4.f.c 2 104.n odd 6 1
832.4.f.c 2 104.p odd 6 1
832.4.f.e 2 104.r even 6 1
832.4.f.e 2 104.s even 6 1
1521.4.a.d 1 39.k even 12 1
1521.4.a.i 1 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 9T_{2}^{2} + 81 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 225 T^{2} + 50625 \) Copy content Toggle raw display
$11$ \( T^{4} - 2304 T^{2} + \cdots + 5308416 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 45 T + 2025)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$23$ \( (T^{2} + 162 T + 26244)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 144 T + 20736)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 69696)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 91809 T^{2} + \cdots + 8428892481 \) Copy content Toggle raw display
$41$ \( T^{4} - 36864 T^{2} + \cdots + 1358954496 \) Copy content Toggle raw display
$43$ \( (T^{2} - 97 T + 9409)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 12321)^{2} \) Copy content Toggle raw display
$53$ \( (T + 414)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 272484 T^{2} + \cdots + 74247530256 \) Copy content Toggle raw display
$61$ \( (T^{2} + 376 T + 141376)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 1296 T^{2} + \cdots + 1679616 \) Copy content Toggle raw display
$71$ \( T^{4} - 127449 T^{2} + \cdots + 16243247601 \) Copy content Toggle raw display
$73$ \( (T^{2} + 1205604)^{2} \) Copy content Toggle raw display
$79$ \( (T + 830)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 191844)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 191844 T^{2} + \cdots + 36804120336 \) Copy content Toggle raw display
$97$ \( T^{4} - 725904 T^{2} + \cdots + 526936617216 \) Copy content Toggle raw display
show more
show less