Properties

Label 13.4.b.a
Level 13
Weight 4
Character orbit 13.b
Analytic conductor 0.767
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.767024830075\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{2} - q^{3} - q^{4} -9 i q^{5} -3 i q^{6} -15 i q^{7} + 21 i q^{8} -26 q^{9} +O(q^{10})\) \( q + 3 i q^{2} - q^{3} - q^{4} -9 i q^{5} -3 i q^{6} -15 i q^{7} + 21 i q^{8} -26 q^{9} + 27 q^{10} + 48 i q^{11} + q^{12} + ( 26 - 39 i ) q^{13} + 45 q^{14} + 9 i q^{15} -71 q^{16} -45 q^{17} -78 i q^{18} + 6 i q^{19} + 9 i q^{20} + 15 i q^{21} -144 q^{22} + 162 q^{23} -21 i q^{24} + 44 q^{25} + ( 117 + 78 i ) q^{26} + 53 q^{27} + 15 i q^{28} -144 q^{29} -27 q^{30} + 264 i q^{31} -45 i q^{32} -48 i q^{33} -135 i q^{34} -135 q^{35} + 26 q^{36} -303 i q^{37} -18 q^{38} + ( -26 + 39 i ) q^{39} + 189 q^{40} -192 i q^{41} -45 q^{42} -97 q^{43} -48 i q^{44} + 234 i q^{45} + 486 i q^{46} -111 i q^{47} + 71 q^{48} + 118 q^{49} + 132 i q^{50} + 45 q^{51} + ( -26 + 39 i ) q^{52} -414 q^{53} + 159 i q^{54} + 432 q^{55} + 315 q^{56} -6 i q^{57} -432 i q^{58} -522 i q^{59} -9 i q^{60} + 376 q^{61} -792 q^{62} + 390 i q^{63} -433 q^{64} + ( -351 - 234 i ) q^{65} + 144 q^{66} -36 i q^{67} + 45 q^{68} -162 q^{69} -405 i q^{70} + 357 i q^{71} -546 i q^{72} + 1098 i q^{73} + 909 q^{74} -44 q^{75} -6 i q^{76} + 720 q^{77} + ( -117 - 78 i ) q^{78} -830 q^{79} + 639 i q^{80} + 649 q^{81} + 576 q^{82} -438 i q^{83} -15 i q^{84} + 405 i q^{85} -291 i q^{86} + 144 q^{87} -1008 q^{88} + 438 i q^{89} -702 q^{90} + ( -585 - 390 i ) q^{91} -162 q^{92} -264 i q^{93} + 333 q^{94} + 54 q^{95} + 45 i q^{96} -852 i q^{97} + 354 i q^{98} -1248 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} - 52q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} - 52q^{9} + 54q^{10} + 2q^{12} + 52q^{13} + 90q^{14} - 142q^{16} - 90q^{17} - 288q^{22} + 324q^{23} + 88q^{25} + 234q^{26} + 106q^{27} - 288q^{29} - 54q^{30} - 270q^{35} + 52q^{36} - 36q^{38} - 52q^{39} + 378q^{40} - 90q^{42} - 194q^{43} + 142q^{48} + 236q^{49} + 90q^{51} - 52q^{52} - 828q^{53} + 864q^{55} + 630q^{56} + 752q^{61} - 1584q^{62} - 866q^{64} - 702q^{65} + 288q^{66} + 90q^{68} - 324q^{69} + 1818q^{74} - 88q^{75} + 1440q^{77} - 234q^{78} - 1660q^{79} + 1298q^{81} + 1152q^{82} + 288q^{87} - 2016q^{88} - 1404q^{90} - 1170q^{91} - 324q^{92} + 666q^{94} + 108q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1
1.00000i
1.00000i
3.00000i −1.00000 −1.00000 9.00000i 3.00000i 15.0000i 21.0000i −26.0000 27.0000
12.2 3.00000i −1.00000 −1.00000 9.00000i 3.00000i 15.0000i 21.0000i −26.0000 27.0000
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.4.b.a 2
3.b odd 2 1 117.4.b.a 2
4.b odd 2 1 208.4.f.b 2
5.b even 2 1 325.4.c.b 2
5.c odd 4 1 325.4.d.a 2
5.c odd 4 1 325.4.d.b 2
8.b even 2 1 832.4.f.e 2
8.d odd 2 1 832.4.f.c 2
13.b even 2 1 inner 13.4.b.a 2
13.c even 3 2 169.4.e.d 4
13.d odd 4 1 169.4.a.b 1
13.d odd 4 1 169.4.a.c 1
13.e even 6 2 169.4.e.d 4
13.f odd 12 2 169.4.c.b 2
13.f odd 12 2 169.4.c.c 2
39.d odd 2 1 117.4.b.a 2
39.f even 4 1 1521.4.a.d 1
39.f even 4 1 1521.4.a.i 1
52.b odd 2 1 208.4.f.b 2
65.d even 2 1 325.4.c.b 2
65.h odd 4 1 325.4.d.a 2
65.h odd 4 1 325.4.d.b 2
104.e even 2 1 832.4.f.e 2
104.h odd 2 1 832.4.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 1.a even 1 1 trivial
13.4.b.a 2 13.b even 2 1 inner
117.4.b.a 2 3.b odd 2 1
117.4.b.a 2 39.d odd 2 1
169.4.a.b 1 13.d odd 4 1
169.4.a.c 1 13.d odd 4 1
169.4.c.b 2 13.f odd 12 2
169.4.c.c 2 13.f odd 12 2
169.4.e.d 4 13.c even 3 2
169.4.e.d 4 13.e even 6 2
208.4.f.b 2 4.b odd 2 1
208.4.f.b 2 52.b odd 2 1
325.4.c.b 2 5.b even 2 1
325.4.c.b 2 65.d even 2 1
325.4.d.a 2 5.c odd 4 1
325.4.d.a 2 65.h odd 4 1
325.4.d.b 2 5.c odd 4 1
325.4.d.b 2 65.h odd 4 1
832.4.f.c 2 8.d odd 2 1
832.4.f.c 2 104.h odd 2 1
832.4.f.e 2 8.b even 2 1
832.4.f.e 2 104.e even 2 1
1521.4.a.d 1 39.f even 4 1
1521.4.a.i 1 39.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 7 T^{2} + 64 T^{4} \)
$3$ \( ( 1 + T + 27 T^{2} )^{2} \)
$5$ \( 1 - 169 T^{2} + 15625 T^{4} \)
$7$ \( 1 - 461 T^{2} + 117649 T^{4} \)
$11$ \( 1 - 358 T^{2} + 1771561 T^{4} \)
$13$ \( 1 - 52 T + 2197 T^{2} \)
$17$ \( ( 1 + 45 T + 4913 T^{2} )^{2} \)
$19$ \( 1 - 13682 T^{2} + 47045881 T^{4} \)
$23$ \( ( 1 - 162 T + 12167 T^{2} )^{2} \)
$29$ \( ( 1 + 144 T + 24389 T^{2} )^{2} \)
$31$ \( 1 + 10114 T^{2} + 887503681 T^{4} \)
$37$ \( 1 - 9497 T^{2} + 2565726409 T^{4} \)
$41$ \( 1 - 100978 T^{2} + 4750104241 T^{4} \)
$43$ \( ( 1 + 97 T + 79507 T^{2} )^{2} \)
$47$ \( 1 - 195325 T^{2} + 10779215329 T^{4} \)
$53$ \( ( 1 + 414 T + 148877 T^{2} )^{2} \)
$59$ \( 1 - 138274 T^{2} + 42180533641 T^{4} \)
$61$ \( ( 1 - 376 T + 226981 T^{2} )^{2} \)
$67$ \( 1 - 600230 T^{2} + 90458382169 T^{4} \)
$71$ \( 1 - 588373 T^{2} + 128100283921 T^{4} \)
$73$ \( ( 1 - 592 T + 389017 T^{2} )( 1 + 592 T + 389017 T^{2} ) \)
$79$ \( ( 1 + 830 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 951730 T^{2} + 326940373369 T^{4} \)
$89$ \( 1 - 1218094 T^{2} + 496981290961 T^{4} \)
$97$ \( 1 - 1099442 T^{2} + 832972004929 T^{4} \)
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