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Properties

Label 169.4.c.c
Level $169$
Weight $4$
Character orbit 169.c
Analytic conductor $9.971$
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] = [169,4,Mod(22,169)]
 
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([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.22");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 3) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + 9 q^{5} - 3 \zeta_{6} q^{6} + 15 \zeta_{6} q^{7} + 21 q^{8} + 26 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \zeta_{6} + 3) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + 9 q^{5} - 3 \zeta_{6} q^{6} + 15 \zeta_{6} q^{7} + 21 q^{8} + 26 \zeta_{6} q^{9} + ( - 27 \zeta_{6} + 27) q^{10} + (48 \zeta_{6} - 48) q^{11} - q^{12} + 45 q^{14} + ( - 9 \zeta_{6} + 9) q^{15} + ( - 71 \zeta_{6} + 71) q^{16} - 45 \zeta_{6} q^{17} + 78 q^{18} + 6 \zeta_{6} q^{19} - 9 \zeta_{6} q^{20} + 15 q^{21} + 144 \zeta_{6} q^{22} + ( - 162 \zeta_{6} + 162) q^{23} + ( - 21 \zeta_{6} + 21) q^{24} - 44 q^{25} + 53 q^{27} + ( - 15 \zeta_{6} + 15) q^{28} + ( - 144 \zeta_{6} + 144) q^{29} - 27 \zeta_{6} q^{30} - 264 q^{31} - 45 \zeta_{6} q^{32} + 48 \zeta_{6} q^{33} - 135 q^{34} + 135 \zeta_{6} q^{35} + ( - 26 \zeta_{6} + 26) q^{36} + ( - 303 \zeta_{6} + 303) q^{37} + 18 q^{38} + 189 q^{40} + (192 \zeta_{6} - 192) q^{41} + ( - 45 \zeta_{6} + 45) q^{42} - 97 \zeta_{6} q^{43} + 48 q^{44} + 234 \zeta_{6} q^{45} - 486 \zeta_{6} q^{46} - 111 q^{47} - 71 \zeta_{6} q^{48} + ( - 118 \zeta_{6} + 118) q^{49} + (132 \zeta_{6} - 132) q^{50} - 45 q^{51} - 414 q^{53} + ( - 159 \zeta_{6} + 159) q^{54} + (432 \zeta_{6} - 432) q^{55} + 315 \zeta_{6} q^{56} + 6 q^{57} - 432 \zeta_{6} q^{58} + 522 \zeta_{6} q^{59} - 9 q^{60} - 376 \zeta_{6} q^{61} + (792 \zeta_{6} - 792) q^{62} + (390 \zeta_{6} - 390) q^{63} + 433 q^{64} + 144 q^{66} + (36 \zeta_{6} - 36) q^{67} + (45 \zeta_{6} - 45) q^{68} - 162 \zeta_{6} q^{69} + 405 q^{70} + 357 \zeta_{6} q^{71} + 546 \zeta_{6} q^{72} + 1098 q^{73} - 909 \zeta_{6} q^{74} + (44 \zeta_{6} - 44) q^{75} + ( - 6 \zeta_{6} + 6) q^{76} - 720 q^{77} - 830 q^{79} + ( - 639 \zeta_{6} + 639) q^{80} + (649 \zeta_{6} - 649) q^{81} + 576 \zeta_{6} q^{82} + 438 q^{83} - 15 \zeta_{6} q^{84} - 405 \zeta_{6} q^{85} - 291 q^{86} - 144 \zeta_{6} q^{87} + (1008 \zeta_{6} - 1008) q^{88} + (438 \zeta_{6} - 438) q^{89} + 702 q^{90} - 162 q^{92} + (264 \zeta_{6} - 264) q^{93} + (333 \zeta_{6} - 333) q^{94} + 54 \zeta_{6} q^{95} - 45 q^{96} - 852 \zeta_{6} q^{97} - 354 \zeta_{6} q^{98} - 1248 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + q^{3} - q^{4} + 18 q^{5} - 3 q^{6} + 15 q^{7} + 42 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + q^{3} - q^{4} + 18 q^{5} - 3 q^{6} + 15 q^{7} + 42 q^{8} + 26 q^{9} + 27 q^{10} - 48 q^{11} - 2 q^{12} + 90 q^{14} + 9 q^{15} + 71 q^{16} - 45 q^{17} + 156 q^{18} + 6 q^{19} - 9 q^{20} + 30 q^{21} + 144 q^{22} + 162 q^{23} + 21 q^{24} - 88 q^{25} + 106 q^{27} + 15 q^{28} + 144 q^{29} - 27 q^{30} - 528 q^{31} - 45 q^{32} + 48 q^{33} - 270 q^{34} + 135 q^{35} + 26 q^{36} + 303 q^{37} + 36 q^{38} + 378 q^{40} - 192 q^{41} + 45 q^{42} - 97 q^{43} + 96 q^{44} + 234 q^{45} - 486 q^{46} - 222 q^{47} - 71 q^{48} + 118 q^{49} - 132 q^{50} - 90 q^{51} - 828 q^{53} + 159 q^{54} - 432 q^{55} + 315 q^{56} + 12 q^{57} - 432 q^{58} + 522 q^{59} - 18 q^{60} - 376 q^{61} - 792 q^{62} - 390 q^{63} + 866 q^{64} + 288 q^{66} - 36 q^{67} - 45 q^{68} - 162 q^{69} + 810 q^{70} + 357 q^{71} + 546 q^{72} + 2196 q^{73} - 909 q^{74} - 44 q^{75} + 6 q^{76} - 1440 q^{77} - 1660 q^{79} + 639 q^{80} - 649 q^{81} + 576 q^{82} + 876 q^{83} - 15 q^{84} - 405 q^{85} - 582 q^{86} - 144 q^{87} - 1008 q^{88} - 438 q^{89} + 1404 q^{90} - 324 q^{92} - 264 q^{93} - 333 q^{94} + 54 q^{95} - 90 q^{96} - 852 q^{97} - 354 q^{98} - 2496 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.500000 + 0.866025i
0.500000 − 0.866025i
1.50000 − 2.59808i 0.500000 − 0.866025i −0.500000 − 0.866025i 9.00000 −1.50000 − 2.59808i 7.50000 + 12.9904i 21.0000 13.0000 + 22.5167i 13.5000 − 23.3827i
146.1 1.50000 + 2.59808i 0.500000 + 0.866025i −0.500000 + 0.866025i 9.00000 −1.50000 + 2.59808i 7.50000 − 12.9904i 21.0000 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.c even 3 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( (T - 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$11$ \( T^{2} + 48T + 2304 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 45T + 2025 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 162T + 26244 \) Copy content Toggle raw display
$29$ \( T^{2} - 144T + 20736 \) Copy content Toggle raw display
$31$ \( (T + 264)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 303T + 91809 \) Copy content Toggle raw display
$41$ \( T^{2} + 192T + 36864 \) Copy content Toggle raw display
$43$ \( T^{2} + 97T + 9409 \) Copy content Toggle raw display
$47$ \( (T + 111)^{2} \) Copy content Toggle raw display
$53$ \( (T + 414)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 522T + 272484 \) Copy content Toggle raw display
$61$ \( T^{2} + 376T + 141376 \) Copy content Toggle raw display
$67$ \( T^{2} + 36T + 1296 \) Copy content Toggle raw display
$71$ \( T^{2} - 357T + 127449 \) Copy content Toggle raw display
$73$ \( (T - 1098)^{2} \) Copy content Toggle raw display
$79$ \( (T + 830)^{2} \) Copy content Toggle raw display
$83$ \( (T - 438)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 438T + 191844 \) Copy content Toggle raw display
$97$ \( T^{2} + 852T + 725904 \) Copy content Toggle raw display
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