Properties

Label 169.4.a.i
Level $169$
Weight $4$
Character orbit 169.a
Self dual yes
Analytic conductor $9.971$
Analytic rank $1$
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.97132279097\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 2 q^{3} - 5 q^{4} + \beta q^{5} + 2 \beta q^{6} - 8 \beta q^{7} - 13 \beta q^{8} - 23 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 2 q^{3} - 5 q^{4} + \beta q^{5} + 2 \beta q^{6} - 8 \beta q^{7} - 13 \beta q^{8} - 23 q^{9} + 3 q^{10} - 8 \beta q^{11} - 10 q^{12} - 24 q^{14} + 2 \beta q^{15} + q^{16} - 117 q^{17} - 23 \beta q^{18} + 66 \beta q^{19} - 5 \beta q^{20} - 16 \beta q^{21} - 24 q^{22} + 78 q^{23} - 26 \beta q^{24} - 122 q^{25} - 100 q^{27} + 40 \beta q^{28} - 141 q^{29} + 6 q^{30} - 90 \beta q^{31} + 105 \beta q^{32} - 16 \beta q^{33} - 117 \beta q^{34} - 24 q^{35} + 115 q^{36} + 83 \beta q^{37} + 198 q^{38} - 39 q^{40} + 157 \beta q^{41} - 48 q^{42} - 104 q^{43} + 40 \beta q^{44} - 23 \beta q^{45} + 78 \beta q^{46} + 174 \beta q^{47} + 2 q^{48} - 151 q^{49} - 122 \beta q^{50} - 234 q^{51} + 93 q^{53} - 100 \beta q^{54} - 24 q^{55} + 312 q^{56} + 132 \beta q^{57} - 141 \beta q^{58} - 164 \beta q^{59} - 10 \beta q^{60} + 145 q^{61} - 270 q^{62} + 184 \beta q^{63} + 307 q^{64} - 48 q^{66} - 454 \beta q^{67} + 585 q^{68} + 156 q^{69} - 24 \beta q^{70} - 610 \beta q^{71} + 299 \beta q^{72} + 265 \beta q^{73} + 249 q^{74} - 244 q^{75} - 330 \beta q^{76} + 192 q^{77} + 1276 q^{79} + \beta q^{80} + 421 q^{81} + 471 q^{82} + 456 \beta q^{83} + 80 \beta q^{84} - 117 \beta q^{85} - 104 \beta q^{86} - 282 q^{87} + 312 q^{88} + 564 \beta q^{89} - 69 q^{90} - 390 q^{92} - 180 \beta q^{93} + 522 q^{94} + 198 q^{95} + 210 \beta q^{96} - 116 \beta q^{97} - 151 \beta q^{98} + 184 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} - 10 q^{4} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} - 10 q^{4} - 46 q^{9} + 6 q^{10} - 20 q^{12} - 48 q^{14} + 2 q^{16} - 234 q^{17} - 48 q^{22} + 156 q^{23} - 244 q^{25} - 200 q^{27} - 282 q^{29} + 12 q^{30} - 48 q^{35} + 230 q^{36} + 396 q^{38} - 78 q^{40} - 96 q^{42} - 208 q^{43} + 4 q^{48} - 302 q^{49} - 468 q^{51} + 186 q^{53} - 48 q^{55} + 624 q^{56} + 290 q^{61} - 540 q^{62} + 614 q^{64} - 96 q^{66} + 1170 q^{68} + 312 q^{69} + 498 q^{74} - 488 q^{75} + 384 q^{77} + 2552 q^{79} + 842 q^{81} + 942 q^{82} - 564 q^{87} + 624 q^{88} - 138 q^{90} - 780 q^{92} + 1044 q^{94} + 396 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
−1.73205 2.00000 −5.00000 −1.73205 −3.46410 13.8564 22.5167 −23.0000 3.00000
1.2 1.73205 2.00000 −5.00000 1.73205 3.46410 −13.8564 −22.5167 −23.0000 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3 \) Copy content Toggle raw display
$7$ \( T^{2} - 192 \) Copy content Toggle raw display
$11$ \( T^{2} - 192 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 117)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 13068 \) Copy content Toggle raw display
$23$ \( (T - 78)^{2} \) Copy content Toggle raw display
$29$ \( (T + 141)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 24300 \) Copy content Toggle raw display
$37$ \( T^{2} - 20667 \) Copy content Toggle raw display
$41$ \( T^{2} - 73947 \) Copy content Toggle raw display
$43$ \( (T + 104)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 90828 \) Copy content Toggle raw display
$53$ \( (T - 93)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 80688 \) Copy content Toggle raw display
$61$ \( (T - 145)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 618348 \) Copy content Toggle raw display
$71$ \( T^{2} - 1116300 \) Copy content Toggle raw display
$73$ \( T^{2} - 210675 \) Copy content Toggle raw display
$79$ \( (T - 1276)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 623808 \) Copy content Toggle raw display
$89$ \( T^{2} - 954288 \) Copy content Toggle raw display
$97$ \( T^{2} - 40368 \) Copy content Toggle raw display
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