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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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

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

Display 100 coefficients 10 coefficients

Character values

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

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

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 \( T_{2}^{4} - 25T_{2}^{2} + 625 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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