Properties

Label 169.4.c.e
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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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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

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.

\(f(q)\) \(=\) \( q + ( - 5 \zeta_{6} + 5) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} - 17 \zeta_{6} q^{4} - 7 q^{5} - 35 \zeta_{6} q^{6} + 13 \zeta_{6} q^{7} - 45 q^{8} - 22 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \zeta_{6} + 5) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} - 17 \zeta_{6} q^{4} - 7 q^{5} - 35 \zeta_{6} q^{6} + 13 \zeta_{6} q^{7} - 45 q^{8} - 22 \zeta_{6} q^{9} + (35 \zeta_{6} - 35) q^{10} + ( - 26 \zeta_{6} + 26) q^{11} - 119 q^{12} + 65 q^{14} + (49 \zeta_{6} - 49) q^{15} + (89 \zeta_{6} - 89) q^{16} - 77 \zeta_{6} q^{17} - 110 q^{18} + 126 \zeta_{6} q^{19} + 119 \zeta_{6} q^{20} + 91 q^{21} - 130 \zeta_{6} q^{22} + ( - 96 \zeta_{6} + 96) q^{23} + (315 \zeta_{6} - 315) q^{24} - 76 q^{25} + 35 q^{27} + ( - 221 \zeta_{6} + 221) q^{28} + ( - 82 \zeta_{6} + 82) q^{29} + 245 \zeta_{6} q^{30} + 196 q^{31} + 85 \zeta_{6} q^{32} - 182 \zeta_{6} q^{33} - 385 q^{34} - 91 \zeta_{6} q^{35} + (374 \zeta_{6} - 374) q^{36} + ( - 131 \zeta_{6} + 131) q^{37} + 630 q^{38} + 315 q^{40} + (336 \zeta_{6} - 336) q^{41} + ( - 455 \zeta_{6} + 455) q^{42} + 201 \zeta_{6} q^{43} - 442 q^{44} + 154 \zeta_{6} q^{45} - 480 \zeta_{6} q^{46} - 105 q^{47} + 623 \zeta_{6} q^{48} + ( - 174 \zeta_{6} + 174) q^{49} + (380 \zeta_{6} - 380) q^{50} - 539 q^{51} - 432 q^{53} + ( - 175 \zeta_{6} + 175) q^{54} + (182 \zeta_{6} - 182) q^{55} - 585 \zeta_{6} q^{56} + 882 q^{57} - 410 \zeta_{6} q^{58} + 294 \zeta_{6} q^{59} + 833 q^{60} + 56 \zeta_{6} q^{61} + ( - 980 \zeta_{6} + 980) q^{62} + ( - 286 \zeta_{6} + 286) q^{63} - 287 q^{64} - 910 q^{66} + (478 \zeta_{6} - 478) q^{67} + (1309 \zeta_{6} - 1309) q^{68} - 672 \zeta_{6} q^{69} - 455 q^{70} - 9 \zeta_{6} q^{71} + 990 \zeta_{6} q^{72} + 98 q^{73} - 655 \zeta_{6} q^{74} + (532 \zeta_{6} - 532) q^{75} + ( - 2142 \zeta_{6} + 2142) q^{76} + 338 q^{77} + 1304 q^{79} + ( - 623 \zeta_{6} + 623) q^{80} + ( - 839 \zeta_{6} + 839) q^{81} + 1680 \zeta_{6} q^{82} - 308 q^{83} - 1547 \zeta_{6} q^{84} + 539 \zeta_{6} q^{85} + 1005 q^{86} - 574 \zeta_{6} q^{87} + (1170 \zeta_{6} - 1170) q^{88} + ( - 1190 \zeta_{6} + 1190) q^{89} + 770 q^{90} - 1632 q^{92} + ( - 1372 \zeta_{6} + 1372) q^{93} + (525 \zeta_{6} - 525) q^{94} - 882 \zeta_{6} q^{95} + 595 q^{96} - 70 \zeta_{6} q^{97} - 870 \zeta_{6} q^{98} - 572 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} + 7 q^{3} - 17 q^{4} - 14 q^{5} - 35 q^{6} + 13 q^{7} - 90 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{2} + 7 q^{3} - 17 q^{4} - 14 q^{5} - 35 q^{6} + 13 q^{7} - 90 q^{8} - 22 q^{9} - 35 q^{10} + 26 q^{11} - 238 q^{12} + 130 q^{14} - 49 q^{15} - 89 q^{16} - 77 q^{17} - 220 q^{18} + 126 q^{19} + 119 q^{20} + 182 q^{21} - 130 q^{22} + 96 q^{23} - 315 q^{24} - 152 q^{25} + 70 q^{27} + 221 q^{28} + 82 q^{29} + 245 q^{30} + 392 q^{31} + 85 q^{32} - 182 q^{33} - 770 q^{34} - 91 q^{35} - 374 q^{36} + 131 q^{37} + 1260 q^{38} + 630 q^{40} - 336 q^{41} + 455 q^{42} + 201 q^{43} - 884 q^{44} + 154 q^{45} - 480 q^{46} - 210 q^{47} + 623 q^{48} + 174 q^{49} - 380 q^{50} - 1078 q^{51} - 864 q^{53} + 175 q^{54} - 182 q^{55} - 585 q^{56} + 1764 q^{57} - 410 q^{58} + 294 q^{59} + 1666 q^{60} + 56 q^{61} + 980 q^{62} + 286 q^{63} - 574 q^{64} - 1820 q^{66} - 478 q^{67} - 1309 q^{68} - 672 q^{69} - 910 q^{70} - 9 q^{71} + 990 q^{72} + 196 q^{73} - 655 q^{74} - 532 q^{75} + 2142 q^{76} + 676 q^{77} + 2608 q^{79} + 623 q^{80} + 839 q^{81} + 1680 q^{82} - 616 q^{83} - 1547 q^{84} + 539 q^{85} + 2010 q^{86} - 574 q^{87} - 1170 q^{88} + 1190 q^{89} + 1540 q^{90} - 3264 q^{92} + 1372 q^{93} - 525 q^{94} - 882 q^{95} + 1190 q^{96} - 70 q^{97} - 870 q^{98} - 1144 q^{99}+O(q^{100}) \) 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.

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
2.50000 4.33013i 3.50000 6.06218i −8.50000 14.7224i −7.00000 −17.5000 30.3109i 6.50000 + 11.2583i −45.0000 −11.0000 19.0526i −17.5000 + 30.3109i
146.1 2.50000 + 4.33013i 3.50000 + 6.06218i −8.50000 + 14.7224i −7.00000 −17.5000 + 30.3109i 6.50000 11.2583i −45.0000 −11.0000 + 19.0526i −17.5000 30.3109i
\(n\): 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.e 2
13.b even 2 1 169.4.c.a 2
13.c even 3 1 13.4.a.a 1
13.c even 3 1 inner 169.4.c.e 2
13.d odd 4 2 169.4.e.e 4
13.e even 6 1 169.4.a.e 1
13.e even 6 1 169.4.c.a 2
13.f odd 12 2 169.4.b.a 2
13.f odd 12 2 169.4.e.e 4
39.h odd 6 1 1521.4.a.a 1
39.i odd 6 1 117.4.a.b 1
52.j odd 6 1 208.4.a.g 1
65.n even 6 1 325.4.a.d 1
65.q odd 12 2 325.4.b.b 2
91.n odd 6 1 637.4.a.a 1
104.n odd 6 1 832.4.a.a 1
104.r even 6 1 832.4.a.r 1
143.k odd 6 1 1573.4.a.a 1
156.p even 6 1 1872.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 13.c even 3 1
117.4.a.b 1 39.i odd 6 1
169.4.a.e 1 13.e even 6 1
169.4.b.a 2 13.f odd 12 2
169.4.c.a 2 13.b even 2 1
169.4.c.a 2 13.e even 6 1
169.4.c.e 2 1.a even 1 1 trivial
169.4.c.e 2 13.c even 3 1 inner
169.4.e.e 4 13.d odd 4 2
169.4.e.e 4 13.f odd 12 2
208.4.a.g 1 52.j odd 6 1
325.4.a.d 1 65.n even 6 1
325.4.b.b 2 65.q odd 12 2
637.4.a.a 1 91.n odd 6 1
832.4.a.a 1 104.n odd 6 1
832.4.a.r 1 104.r even 6 1
1521.4.a.a 1 39.h odd 6 1
1573.4.a.a 1 143.k odd 6 1
1872.4.a.k 1 156.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5T_{2} + 25 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$3$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$5$ \( (T + 7)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$11$ \( T^{2} - 26T + 676 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 77T + 5929 \) Copy content Toggle raw display
$19$ \( T^{2} - 126T + 15876 \) Copy content Toggle raw display
$23$ \( T^{2} - 96T + 9216 \) Copy content Toggle raw display
$29$ \( T^{2} - 82T + 6724 \) Copy content Toggle raw display
$31$ \( (T - 196)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 131T + 17161 \) Copy content Toggle raw display
$41$ \( T^{2} + 336T + 112896 \) Copy content Toggle raw display
$43$ \( T^{2} - 201T + 40401 \) Copy content Toggle raw display
$47$ \( (T + 105)^{2} \) Copy content Toggle raw display
$53$ \( (T + 432)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 294T + 86436 \) Copy content Toggle raw display
$61$ \( T^{2} - 56T + 3136 \) Copy content Toggle raw display
$67$ \( T^{2} + 478T + 228484 \) Copy content Toggle raw display
$71$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$73$ \( (T - 98)^{2} \) Copy content Toggle raw display
$79$ \( (T - 1304)^{2} \) Copy content Toggle raw display
$83$ \( (T + 308)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1190 T + 1416100 \) Copy content Toggle raw display
$97$ \( T^{2} + 70T + 4900 \) Copy content Toggle raw display
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