Properties

Label 169.4.c.h
Level 169
Weight 4
Character orbit 169.c
Analytic conductor 9.971
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} -2 \zeta_{12}^{2} q^{3} + ( 5 - 5 \zeta_{12}^{2} ) q^{4} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{6} + ( 8 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{7} + ( -26 \zeta_{12} + 13 \zeta_{12}^{3} ) q^{8} + ( 23 - 23 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} -2 \zeta_{12}^{2} q^{3} + ( 5 - 5 \zeta_{12}^{2} ) q^{4} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{6} + ( 8 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{7} + ( -26 \zeta_{12} + 13 \zeta_{12}^{3} ) q^{8} + ( 23 - 23 \zeta_{12}^{2} ) q^{9} -3 \zeta_{12}^{2} q^{10} + ( 8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{11} -10 q^{12} -24 q^{14} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{15} -\zeta_{12}^{2} q^{16} + ( 117 - 117 \zeta_{12}^{2} ) q^{17} + ( -46 \zeta_{12} + 23 \zeta_{12}^{3} ) q^{18} + ( -66 \zeta_{12} + 132 \zeta_{12}^{3} ) q^{19} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{20} + ( -32 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{21} + ( 24 - 24 \zeta_{12}^{2} ) q^{22} -78 \zeta_{12}^{2} q^{23} + ( 26 \zeta_{12} + 26 \zeta_{12}^{3} ) q^{24} -122 q^{25} -100 q^{27} + ( -40 \zeta_{12} - 40 \zeta_{12}^{3} ) q^{28} + 141 \zeta_{12}^{2} q^{29} + ( -6 + 6 \zeta_{12}^{2} ) q^{30} + ( -180 \zeta_{12} + 90 \zeta_{12}^{3} ) q^{31} + ( -105 \zeta_{12} + 210 \zeta_{12}^{3} ) q^{32} + ( 16 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{33} + ( -234 \zeta_{12} + 117 \zeta_{12}^{3} ) q^{34} + ( 24 - 24 \zeta_{12}^{2} ) q^{35} -115 \zeta_{12}^{2} q^{36} + ( -83 \zeta_{12} - 83 \zeta_{12}^{3} ) q^{37} + 198 q^{38} -39 q^{40} + ( -157 \zeta_{12} - 157 \zeta_{12}^{3} ) q^{41} + 48 \zeta_{12}^{2} q^{42} + ( 104 - 104 \zeta_{12}^{2} ) q^{43} + ( 80 \zeta_{12} - 40 \zeta_{12}^{3} ) q^{44} + ( 23 \zeta_{12} - 46 \zeta_{12}^{3} ) q^{45} + ( -78 \zeta_{12} + 156 \zeta_{12}^{3} ) q^{46} + ( 348 \zeta_{12} - 174 \zeta_{12}^{3} ) q^{47} + ( -2 + 2 \zeta_{12}^{2} ) q^{48} + 151 \zeta_{12}^{2} q^{49} + ( 122 \zeta_{12} + 122 \zeta_{12}^{3} ) q^{50} -234 q^{51} + 93 q^{53} + ( 100 \zeta_{12} + 100 \zeta_{12}^{3} ) q^{54} + 24 \zeta_{12}^{2} q^{55} + ( -312 + 312 \zeta_{12}^{2} ) q^{56} + ( 264 \zeta_{12} - 132 \zeta_{12}^{3} ) q^{57} + ( 141 \zeta_{12} - 282 \zeta_{12}^{3} ) q^{58} + ( 164 \zeta_{12} - 328 \zeta_{12}^{3} ) q^{59} + ( -20 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{60} + ( -145 + 145 \zeta_{12}^{2} ) q^{61} + 270 \zeta_{12}^{2} q^{62} + ( -184 \zeta_{12} - 184 \zeta_{12}^{3} ) q^{63} + 307 q^{64} -48 q^{66} + ( 454 \zeta_{12} + 454 \zeta_{12}^{3} ) q^{67} -585 \zeta_{12}^{2} q^{68} + ( -156 + 156 \zeta_{12}^{2} ) q^{69} + ( -48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{70} + ( 610 \zeta_{12} - 1220 \zeta_{12}^{3} ) q^{71} + ( -299 \zeta_{12} + 598 \zeta_{12}^{3} ) q^{72} + ( 530 \zeta_{12} - 265 \zeta_{12}^{3} ) q^{73} + ( -249 + 249 \zeta_{12}^{2} ) q^{74} + 244 \zeta_{12}^{2} q^{75} + ( 330 \zeta_{12} + 330 \zeta_{12}^{3} ) q^{76} + 192 q^{77} + 1276 q^{79} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{80} -421 \zeta_{12}^{2} q^{81} + ( -471 + 471 \zeta_{12}^{2} ) q^{82} + ( 912 \zeta_{12} - 456 \zeta_{12}^{3} ) q^{83} + ( -80 \zeta_{12} + 160 \zeta_{12}^{3} ) q^{84} + ( 117 \zeta_{12} - 234 \zeta_{12}^{3} ) q^{85} + ( -208 \zeta_{12} + 104 \zeta_{12}^{3} ) q^{86} + ( 282 - 282 \zeta_{12}^{2} ) q^{87} -312 \zeta_{12}^{2} q^{88} + ( -564 \zeta_{12} - 564 \zeta_{12}^{3} ) q^{89} -69 q^{90} -390 q^{92} + ( 180 \zeta_{12} + 180 \zeta_{12}^{3} ) q^{93} -522 \zeta_{12}^{2} q^{94} + ( -198 + 198 \zeta_{12}^{2} ) q^{95} + ( 420 \zeta_{12} - 210 \zeta_{12}^{3} ) q^{96} + ( 116 \zeta_{12} - 232 \zeta_{12}^{3} ) q^{97} + ( 151 \zeta_{12} - 302 \zeta_{12}^{3} ) q^{98} + ( 368 \zeta_{12} - 184 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 10q^{4} + 46q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 10q^{4} + 46q^{9} - 6q^{10} - 40q^{12} - 96q^{14} - 2q^{16} + 234q^{17} + 48q^{22} - 156q^{23} - 488q^{25} - 400q^{27} + 282q^{29} - 12q^{30} + 48q^{35} - 230q^{36} + 792q^{38} - 156q^{40} + 96q^{42} + 208q^{43} - 4q^{48} + 302q^{49} - 936q^{51} + 372q^{53} + 48q^{55} - 624q^{56} - 290q^{61} + 540q^{62} + 1228q^{64} - 192q^{66} - 1170q^{68} - 312q^{69} - 498q^{74} + 488q^{75} + 768q^{77} + 5104q^{79} - 842q^{81} - 942q^{82} + 564q^{87} - 624q^{88} - 276q^{90} - 1560q^{92} - 1044q^{94} - 396q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 + \zeta_{12}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i −1.00000 + 1.73205i 2.50000 + 4.33013i 1.73205 −1.73205 3.00000i 6.92820 + 12.0000i −22.5167 11.5000 + 19.9186i −1.50000 + 2.59808i
22.2 0.866025 1.50000i −1.00000 + 1.73205i 2.50000 + 4.33013i −1.73205 1.73205 + 3.00000i −6.92820 12.0000i 22.5167 11.5000 + 19.9186i −1.50000 + 2.59808i
146.1 −0.866025 1.50000i −1.00000 1.73205i 2.50000 4.33013i 1.73205 −1.73205 + 3.00000i 6.92820 12.0000i −22.5167 11.5000 19.9186i −1.50000 2.59808i
146.2 0.866025 + 1.50000i −1.00000 1.73205i 2.50000 4.33013i −1.73205 1.73205 3.00000i −6.92820 + 12.0000i 22.5167 11.5000 19.9186i −1.50000 2.59808i
\(n\): 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.c.h 4
13.b even 2 1 inner 169.4.c.h 4
13.c even 3 1 169.4.a.i 2
13.c even 3 1 inner 169.4.c.h 4
13.d odd 4 1 13.4.e.b 2
13.d odd 4 1 169.4.e.a 2
13.e even 6 1 169.4.a.i 2
13.e even 6 1 inner 169.4.c.h 4
13.f odd 12 1 13.4.e.b 2
13.f odd 12 2 169.4.b.d 2
13.f odd 12 1 169.4.e.a 2
39.f even 4 1 117.4.q.a 2
39.h odd 6 1 1521.4.a.o 2
39.i odd 6 1 1521.4.a.o 2
39.k even 12 1 117.4.q.a 2
52.f even 4 1 208.4.w.b 2
52.l even 12 1 208.4.w.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 13.d odd 4 1
13.4.e.b 2 13.f odd 12 1
117.4.q.a 2 39.f even 4 1
117.4.q.a 2 39.k even 12 1
169.4.a.i 2 13.c even 3 1
169.4.a.i 2 13.e even 6 1
169.4.b.d 2 13.f odd 12 2
169.4.c.h 4 1.a even 1 1 trivial
169.4.c.h 4 13.b even 2 1 inner
169.4.c.h 4 13.c even 3 1 inner
169.4.c.h 4 13.e even 6 1 inner
169.4.e.a 2 13.d odd 4 1
169.4.e.a 2 13.f odd 12 1
208.4.w.b 2 52.f even 4 1
208.4.w.b 2 52.l even 12 1
1521.4.a.o 2 39.h odd 6 1
1521.4.a.o 2 39.i odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 13 T^{2} + 105 T^{4} - 832 T^{6} + 4096 T^{8} \)
$3$ \( ( 1 + 2 T - 23 T^{2} + 54 T^{3} + 729 T^{4} )^{2} \)
$5$ \( ( 1 + 247 T^{2} + 15625 T^{4} )^{2} \)
$7$ \( 1 - 494 T^{2} + 126387 T^{4} - 58118606 T^{6} + 13841287201 T^{8} \)
$11$ \( 1 - 2470 T^{2} + 4329339 T^{4} - 4375755670 T^{6} + 3138428376721 T^{8} \)
$13$ 1
$17$ \( ( 1 - 117 T + 8776 T^{2} - 574821 T^{3} + 24137569 T^{4} )^{2} \)
$19$ \( 1 - 650 T^{2} - 46623381 T^{4} - 30579822650 T^{6} + 2213314919066161 T^{8} \)
$23$ \( ( 1 + 78 T - 6083 T^{2} + 949026 T^{3} + 148035889 T^{4} )^{2} \)
$29$ \( ( 1 - 141 T - 4508 T^{2} - 3438849 T^{3} + 594823321 T^{4} )^{2} \)
$31$ \( ( 1 + 35282 T^{2} + 887503681 T^{4} )^{2} \)
$37$ \( 1 - 80639 T^{2} + 3936921912 T^{4} - 206897611895351 T^{6} + 6582952005840035281 T^{8} \)
$41$ \( 1 - 63895 T^{2} - 667533216 T^{4} - 303507910478695 T^{6} + 22563490300366186081 T^{8} \)
$43$ \( ( 1 - 104 T - 68691 T^{2} - 8268728 T^{3} + 6321363049 T^{4} )^{2} \)
$47$ \( ( 1 + 116818 T^{2} + 10779215329 T^{4} )^{2} \)
$53$ \( ( 1 - 93 T + 148877 T^{2} )^{4} \)
$59$ \( 1 - 330070 T^{2} + 66765671259 T^{4} - 13922528738884870 T^{6} + \)\(17\!\cdots\!81\)\( T^{8} \)
$61$ \( ( 1 + 145 T - 205956 T^{2} + 32912245 T^{3} + 51520374361 T^{4} )^{2} \)
$67$ \( 1 + 16822 T^{2} - 90175402485 T^{4} + 1521690904846918 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \)
$71$ \( 1 + 400478 T^{2} + 32282344563 T^{4} + 51301345504114238 T^{6} + \)\(16\!\cdots\!41\)\( T^{8} \)
$73$ \( ( 1 + 567359 T^{2} + 151334226289 T^{4} )^{2} \)
$79$ \( ( 1 - 1276 T + 493039 T^{2} )^{4} \)
$83$ \( ( 1 + 519766 T^{2} + 326940373369 T^{4} )^{2} \)
$89$ \( 1 - 455650 T^{2} - 289364368461 T^{4} - 226449525226379650 T^{6} + \)\(24\!\cdots\!21\)\( T^{8} \)
$97$ \( 1 - 1784978 T^{2} + 2353174455555 T^{4} - 1486836703414156562 T^{6} + \)\(69\!\cdots\!41\)\( T^{8} \)
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