Properties

Label 169.4.e.b
Level $169$
Weight $4$
Character orbit 169.e
Analytic conductor $9.971$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} + 2) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + 4 \zeta_{6} q^{4} + ( - 16 \zeta_{6} + 8) q^{5} + ( - 14 \zeta_{6} + 28) q^{6} + (13 \zeta_{6} - 26) q^{7} + ( - 16 \zeta_{6} + 8) q^{8} - 22 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} + 2) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + 4 \zeta_{6} q^{4} + ( - 16 \zeta_{6} + 8) q^{5} + ( - 14 \zeta_{6} + 28) q^{6} + (13 \zeta_{6} - 26) q^{7} + ( - 16 \zeta_{6} + 8) q^{8} - 22 \zeta_{6} q^{9} + ( - 48 \zeta_{6} + 48) q^{10} + (13 \zeta_{6} + 13) q^{11} + 28 q^{12} - 78 q^{14} + ( - 56 \zeta_{6} - 56) q^{15} + ( - 80 \zeta_{6} + 80) q^{16} - 27 \zeta_{6} q^{17} + ( - 88 \zeta_{6} + 44) q^{18} + ( - 51 \zeta_{6} + 102) q^{19} + ( - 32 \zeta_{6} + 64) q^{20} + (182 \zeta_{6} - 91) q^{21} + 78 \zeta_{6} q^{22} + (57 \zeta_{6} - 57) q^{23} + ( - 56 \zeta_{6} - 56) q^{24} - 67 q^{25} + 35 q^{27} + ( - 52 \zeta_{6} - 52) q^{28} + ( - 69 \zeta_{6} + 69) q^{29} - 336 \zeta_{6} q^{30} + (84 \zeta_{6} - 42) q^{31} + ( - 96 \zeta_{6} + 192) q^{32} + ( - 91 \zeta_{6} + 182) q^{33} + ( - 108 \zeta_{6} + 54) q^{34} + 312 \zeta_{6} q^{35} + ( - 88 \zeta_{6} + 88) q^{36} + (23 \zeta_{6} + 23) q^{37} + 306 q^{38} - 192 q^{40} + (227 \zeta_{6} + 227) q^{41} + (546 \zeta_{6} - 546) q^{42} + 85 \zeta_{6} q^{43} + (104 \zeta_{6} - 52) q^{44} + (176 \zeta_{6} - 352) q^{45} + (114 \zeta_{6} - 228) q^{46} + (396 \zeta_{6} - 198) q^{47} - 560 \zeta_{6} q^{48} + ( - 164 \zeta_{6} + 164) q^{49} + ( - 134 \zeta_{6} - 134) q^{50} - 189 q^{51} + 426 q^{53} + (70 \zeta_{6} + 70) q^{54} + ( - 312 \zeta_{6} + 312) q^{55} + 312 \zeta_{6} q^{56} + ( - 714 \zeta_{6} + 357) q^{57} + ( - 138 \zeta_{6} + 276) q^{58} + ( - 11 \zeta_{6} + 22) q^{59} + ( - 448 \zeta_{6} + 224) q^{60} + 17 \zeta_{6} q^{61} + (252 \zeta_{6} - 252) q^{62} + (286 \zeta_{6} + 286) q^{63} - 64 q^{64} + 546 q^{66} + ( - 95 \zeta_{6} - 95) q^{67} + ( - 108 \zeta_{6} + 108) q^{68} + 399 \zeta_{6} q^{69} + (1248 \zeta_{6} - 624) q^{70} + (337 \zeta_{6} - 674) q^{71} + (176 \zeta_{6} - 352) q^{72} + ( - 1160 \zeta_{6} + 580) q^{73} + 138 \zeta_{6} q^{74} + (469 \zeta_{6} - 469) q^{75} + (204 \zeta_{6} + 204) q^{76} - 507 q^{77} - 1244 q^{79} + ( - 640 \zeta_{6} - 640) q^{80} + ( - 839 \zeta_{6} + 839) q^{81} + 1362 \zeta_{6} q^{82} + (492 \zeta_{6} - 246) q^{83} + (364 \zeta_{6} - 728) q^{84} + (216 \zeta_{6} - 432) q^{85} + (340 \zeta_{6} - 170) q^{86} - 483 \zeta_{6} q^{87} + ( - 312 \zeta_{6} + 312) q^{88} + ( - 177 \zeta_{6} - 177) q^{89} - 1056 q^{90} - 228 q^{92} + (294 \zeta_{6} + 294) q^{93} + (1188 \zeta_{6} - 1188) q^{94} - 1224 \zeta_{6} q^{95} + ( - 1344 \zeta_{6} + 672) q^{96} + (713 \zeta_{6} - 1426) q^{97} + ( - 328 \zeta_{6} + 656) q^{98} + ( - 572 \zeta_{6} + 286) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 7 q^{3} + 4 q^{4} + 42 q^{6} - 39 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{2} + 7 q^{3} + 4 q^{4} + 42 q^{6} - 39 q^{7} - 22 q^{9} + 48 q^{10} + 39 q^{11} + 56 q^{12} - 156 q^{14} - 168 q^{15} + 80 q^{16} - 27 q^{17} + 153 q^{19} + 96 q^{20} + 78 q^{22} - 57 q^{23} - 168 q^{24} - 134 q^{25} + 70 q^{27} - 156 q^{28} + 69 q^{29} - 336 q^{30} + 288 q^{32} + 273 q^{33} + 312 q^{35} + 88 q^{36} + 69 q^{37} + 612 q^{38} - 384 q^{40} + 681 q^{41} - 546 q^{42} + 85 q^{43} - 528 q^{45} - 342 q^{46} - 560 q^{48} + 164 q^{49} - 402 q^{50} - 378 q^{51} + 852 q^{53} + 210 q^{54} + 312 q^{55} + 312 q^{56} + 414 q^{58} + 33 q^{59} + 17 q^{61} - 252 q^{62} + 858 q^{63} - 128 q^{64} + 1092 q^{66} - 285 q^{67} + 108 q^{68} + 399 q^{69} - 1011 q^{71} - 528 q^{72} + 138 q^{74} - 469 q^{75} + 612 q^{76} - 1014 q^{77} - 2488 q^{79} - 1920 q^{80} + 839 q^{81} + 1362 q^{82} - 1092 q^{84} - 648 q^{85} - 483 q^{87} + 312 q^{88} - 531 q^{89} - 2112 q^{90} - 456 q^{92} + 882 q^{93} - 1188 q^{94} - 1224 q^{95} - 2139 q^{97} + 984 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
3.00000 1.73205i 3.50000 + 6.06218i 2.00000 3.46410i 13.8564i 21.0000 + 12.1244i −19.5000 11.2583i 13.8564i −11.0000 + 19.0526i 24.0000 + 41.5692i
147.1 3.00000 + 1.73205i 3.50000 6.06218i 2.00000 + 3.46410i 13.8564i 21.0000 12.1244i −19.5000 + 11.2583i 13.8564i −11.0000 19.0526i 24.0000 41.5692i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.b 2
13.b even 2 1 13.4.e.a 2
13.c even 3 1 13.4.e.a 2
13.c even 3 1 169.4.b.b 2
13.d odd 4 2 169.4.c.i 4
13.e even 6 1 169.4.b.b 2
13.e even 6 1 inner 169.4.e.b 2
13.f odd 12 2 169.4.a.h 2
13.f odd 12 2 169.4.c.i 4
39.d odd 2 1 117.4.q.c 2
39.i odd 6 1 117.4.q.c 2
39.k even 12 2 1521.4.a.q 2
52.b odd 2 1 208.4.w.a 2
52.j odd 6 1 208.4.w.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.a 2 13.b even 2 1
13.4.e.a 2 13.c even 3 1
117.4.q.c 2 39.d odd 2 1
117.4.q.c 2 39.i odd 6 1
169.4.a.h 2 13.f odd 12 2
169.4.b.b 2 13.c even 3 1
169.4.b.b 2 13.e even 6 1
169.4.c.i 4 13.d odd 4 2
169.4.c.i 4 13.f odd 12 2
169.4.e.b 2 1.a even 1 1 trivial
169.4.e.b 2 13.e even 6 1 inner
208.4.w.a 2 52.b odd 2 1
208.4.w.a 2 52.j odd 6 1
1521.4.a.q 2 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 6T_{2} + 12 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$3$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$5$ \( T^{2} + 192 \) Copy content Toggle raw display
$7$ \( T^{2} + 39T + 507 \) Copy content Toggle raw display
$11$ \( T^{2} - 39T + 507 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 27T + 729 \) Copy content Toggle raw display
$19$ \( T^{2} - 153T + 7803 \) Copy content Toggle raw display
$23$ \( T^{2} + 57T + 3249 \) Copy content Toggle raw display
$29$ \( T^{2} - 69T + 4761 \) Copy content Toggle raw display
$31$ \( T^{2} + 5292 \) Copy content Toggle raw display
$37$ \( T^{2} - 69T + 1587 \) Copy content Toggle raw display
$41$ \( T^{2} - 681T + 154587 \) Copy content Toggle raw display
$43$ \( T^{2} - 85T + 7225 \) Copy content Toggle raw display
$47$ \( T^{2} + 117612 \) Copy content Toggle raw display
$53$ \( (T - 426)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 33T + 363 \) Copy content Toggle raw display
$61$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$67$ \( T^{2} + 285T + 27075 \) Copy content Toggle raw display
$71$ \( T^{2} + 1011 T + 340707 \) Copy content Toggle raw display
$73$ \( T^{2} + 1009200 \) Copy content Toggle raw display
$79$ \( (T + 1244)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 181548 \) Copy content Toggle raw display
$89$ \( T^{2} + 531T + 93987 \) Copy content Toggle raw display
$97$ \( T^{2} + 2139 T + 1525107 \) Copy content Toggle raw display
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