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Properties

Label 169.4.e.e
Level $169$
Weight $4$
Character orbit 169.e
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.e (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + 5 \zeta_{12} q^{2} + ( 7 - 7 \zeta_{12}^{2} ) q^{3} + 17 \zeta_{12}^{2} q^{4} -7 \zeta_{12}^{3} q^{5} + ( 35 \zeta_{12} - 35 \zeta_{12}^{3} ) q^{6} + ( 13 \zeta_{12} - 13 \zeta_{12}^{3} ) q^{7} + 45 \zeta_{12}^{3} q^{8} -22 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + 5 \zeta_{12} q^{2} + ( 7 - 7 \zeta_{12}^{2} ) q^{3} + 17 \zeta_{12}^{2} q^{4} -7 \zeta_{12}^{3} q^{5} + ( 35 \zeta_{12} - 35 \zeta_{12}^{3} ) q^{6} + ( 13 \zeta_{12} - 13 \zeta_{12}^{3} ) q^{7} + 45 \zeta_{12}^{3} q^{8} -22 \zeta_{12}^{2} q^{9} + ( 35 - 35 \zeta_{12}^{2} ) q^{10} -26 \zeta_{12} q^{11} + 119 q^{12} + 65 q^{14} -49 \zeta_{12} q^{15} + ( -89 + 89 \zeta_{12}^{2} ) q^{16} + 77 \zeta_{12}^{2} q^{17} -110 \zeta_{12}^{3} q^{18} + ( -126 \zeta_{12} + 126 \zeta_{12}^{3} ) q^{19} + ( 119 \zeta_{12} - 119 \zeta_{12}^{3} ) q^{20} -91 \zeta_{12}^{3} q^{21} -130 \zeta_{12}^{2} q^{22} + ( -96 + 96 \zeta_{12}^{2} ) q^{23} + 315 \zeta_{12} q^{24} + 76 q^{25} + 35 q^{27} + 221 \zeta_{12} q^{28} + ( 82 - 82 \zeta_{12}^{2} ) q^{29} -245 \zeta_{12}^{2} q^{30} + 196 \zeta_{12}^{3} q^{31} + ( -85 \zeta_{12} + 85 \zeta_{12}^{3} ) q^{32} + ( -182 \zeta_{12} + 182 \zeta_{12}^{3} ) q^{33} + 385 \zeta_{12}^{3} q^{34} -91 \zeta_{12}^{2} q^{35} + ( 374 - 374 \zeta_{12}^{2} ) q^{36} -131 \zeta_{12} q^{37} -630 q^{38} + 315 q^{40} -336 \zeta_{12} q^{41} + ( 455 - 455 \zeta_{12}^{2} ) q^{42} -201 \zeta_{12}^{2} q^{43} -442 \zeta_{12}^{3} q^{44} + ( -154 \zeta_{12} + 154 \zeta_{12}^{3} ) q^{45} + ( -480 \zeta_{12} + 480 \zeta_{12}^{3} ) q^{46} + 105 \zeta_{12}^{3} q^{47} + 623 \zeta_{12}^{2} q^{48} + ( -174 + 174 \zeta_{12}^{2} ) q^{49} + 380 \zeta_{12} q^{50} + 539 q^{51} -432 q^{53} + 175 \zeta_{12} q^{54} + ( -182 + 182 \zeta_{12}^{2} ) q^{55} + 585 \zeta_{12}^{2} q^{56} + 882 \zeta_{12}^{3} q^{57} + ( 410 \zeta_{12} - 410 \zeta_{12}^{3} ) q^{58} + ( 294 \zeta_{12} - 294 \zeta_{12}^{3} ) q^{59} -833 \zeta_{12}^{3} q^{60} + 56 \zeta_{12}^{2} q^{61} + ( -980 + 980 \zeta_{12}^{2} ) q^{62} -286 \zeta_{12} q^{63} + 287 q^{64} -910 q^{66} -478 \zeta_{12} q^{67} + ( -1309 + 1309 \zeta_{12}^{2} ) q^{68} + 672 \zeta_{12}^{2} q^{69} -455 \zeta_{12}^{3} q^{70} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{71} + ( 990 \zeta_{12} - 990 \zeta_{12}^{3} ) q^{72} -98 \zeta_{12}^{3} q^{73} -655 \zeta_{12}^{2} q^{74} + ( 532 - 532 \zeta_{12}^{2} ) q^{75} -2142 \zeta_{12} q^{76} -338 q^{77} + 1304 q^{79} + 623 \zeta_{12} q^{80} + ( 839 - 839 \zeta_{12}^{2} ) q^{81} -1680 \zeta_{12}^{2} q^{82} -308 \zeta_{12}^{3} q^{83} + ( 1547 \zeta_{12} - 1547 \zeta_{12}^{3} ) q^{84} + ( 539 \zeta_{12} - 539 \zeta_{12}^{3} ) q^{85} -1005 \zeta_{12}^{3} q^{86} -574 \zeta_{12}^{2} q^{87} + ( 1170 - 1170 \zeta_{12}^{2} ) q^{88} -1190 \zeta_{12} q^{89} -770 q^{90} -1632 q^{92} + 1372 \zeta_{12} q^{93} + ( -525 + 525 \zeta_{12}^{2} ) q^{94} + 882 \zeta_{12}^{2} q^{95} + 595 \zeta_{12}^{3} q^{96} + ( 70 \zeta_{12} - 70 \zeta_{12}^{3} ) q^{97} + ( -870 \zeta_{12} + 870 \zeta_{12}^{3} ) q^{98} + 572 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 14q^{3} + 34q^{4} - 44q^{9} + O(q^{10}) \) \( 4q + 14q^{3} + 34q^{4} - 44q^{9} + 70q^{10} + 476q^{12} + 260q^{14} - 178q^{16} + 154q^{17} - 260q^{22} - 192q^{23} + 304q^{25} + 140q^{27} + 164q^{29} - 490q^{30} - 182q^{35} + 748q^{36} - 2520q^{38} + 1260q^{40} + 910q^{42} - 402q^{43} + 1246q^{48} - 348q^{49} + 2156q^{51} - 1728q^{53} - 364q^{55} + 1170q^{56} + 112q^{61} - 1960q^{62} + 1148q^{64} - 3640q^{66} - 2618q^{68} + 1344q^{69} - 1310q^{74} + 1064q^{75} - 1352q^{77} + 5216q^{79} + 1678q^{81} - 3360q^{82} - 1148q^{87} + 2340q^{88} - 3080q^{90} - 6528q^{92} - 1050q^{94} + 1764q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{12}^{2}\)

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.

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
−4.33013 + 2.50000i 3.50000 + 6.06218i 8.50000 − 14.7224i − 7.00000i −30.3109 − 17.5000i −11.2583 − 6.50000i 45.0000i −11.0000 + 19.0526i 17.5000 + 30.3109i
23.2 4.33013 − 2.50000i 3.50000 + 6.06218i 8.50000 − 14.7224i 7.00000i 30.3109 + 17.5000i 11.2583 + 6.50000i − 45.0000i −11.0000 + 19.0526i 17.5000 + 30.3109i
147.1 −4.33013 − 2.50000i 3.50000 − 6.06218i 8.50000 + 14.7224i 7.00000i −30.3109 + 17.5000i −11.2583 + 6.50000i − 45.0000i −11.0000 − 19.0526i 17.5000 − 30.3109i
147.2 4.33013 + 2.50000i 3.50000 − 6.06218i 8.50000 + 14.7224i − 7.00000i 30.3109 − 17.5000i 11.2583 − 6.50000i 45.0000i −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.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.e 4
13.b even 2 1 inner 169.4.e.e 4
13.c even 3 1 169.4.b.a 2
13.c even 3 1 inner 169.4.e.e 4
13.d odd 4 1 169.4.c.a 2
13.d odd 4 1 169.4.c.e 2
13.e even 6 1 169.4.b.a 2
13.e even 6 1 inner 169.4.e.e 4
13.f odd 12 1 13.4.a.a ✓ 1
13.f odd 12 1 169.4.a.e 1
13.f odd 12 1 169.4.c.a 2
13.f odd 12 1 169.4.c.e 2
39.k even 12 1 117.4.a.b 1
39.k even 12 1 1521.4.a.a 1
52.l even 12 1 208.4.a.g 1
65.o even 12 1 325.4.b.b 2
65.s odd 12 1 325.4.a.d 1
65.t even 12 1 325.4.b.b 2
91.bc even 12 1 637.4.a.a 1
104.u even 12 1 832.4.a.a 1
104.x odd 12 1 832.4.a.r 1
143.o even 12 1 1573.4.a.a 1
156.v odd 12 1 1872.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a ✓ 1 13.f odd 12 1
117.4.a.b 1 39.k even 12 1
169.4.a.e 1 13.f odd 12 1
169.4.b.a 2 13.c even 3 1
169.4.b.a 2 13.e even 6 1
169.4.c.a 2 13.d odd 4 1
169.4.c.a 2 13.f odd 12 1
169.4.c.e 2 13.d odd 4 1
169.4.c.e 2 13.f odd 12 1
169.4.e.e 4 1.a even 1 1 trivial
169.4.e.e 4 13.b even 2 1 inner
169.4.e.e 4 13.c even 3 1 inner
169.4.e.e 4 13.e even 6 1 inner
208.4.a.g 1 52.l even 12 1
325.4.a.d 1 65.s odd 12 1
325.4.b.b 2 65.o even 12 1
325.4.b.b 2 65.t even 12 1
637.4.a.a 1 91.bc even 12 1
832.4.a.a 1 104.u even 12 1
832.4.a.r 1 104.x odd 12 1
1521.4.a.a 1 39.k even 12 1
1573.4.a.a 1 143.o even 12 1
1872.4.a.k 1 156.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 25 T_{2}^{2} + 625 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 625 - 25 T^{2} + T^{4} \)
$3$ \( ( 49 - 7 T + T^{2} )^{2} \)
$5$ \( ( 49 + T^{2} )^{2} \)
$7$ \( 28561 - 169 T^{2} + T^{4} \)
$11$ \( 456976 - 676 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 5929 - 77 T + T^{2} )^{2} \)
$19$ \( 252047376 - 15876 T^{2} + T^{4} \)
$23$ \( ( 9216 + 96 T + T^{2} )^{2} \)
$29$ \( ( 6724 - 82 T + T^{2} )^{2} \)
$31$ \( ( 38416 + T^{2} )^{2} \)
$37$ \( 294499921 - 17161 T^{2} + T^{4} \)
$41$ \( 12745506816 - 112896 T^{2} + T^{4} \)
$43$ \( ( 40401 + 201 T + T^{2} )^{2} \)
$47$ \( ( 11025 + T^{2} )^{2} \)
$53$ \( ( 432 + T )^{4} \)
$59$ \( 7471182096 - 86436 T^{2} + T^{4} \)
$61$ \( ( 3136 - 56 T + T^{2} )^{2} \)
$67$ \( 52204938256 - 228484 T^{2} + T^{4} \)
$71$ \( 6561 - 81 T^{2} + T^{4} \)
$73$ \( ( 9604 + T^{2} )^{2} \)
$79$ \( ( -1304 + T )^{4} \)
$83$ \( ( 94864 + T^{2} )^{2} \)
$89$ \( 2005339210000 - 1416100 T^{2} + T^{4} \)
$97$ \( 24010000 - 4900 T^{2} + T^{4} \)
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