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 \(2\) and degree \(1\))

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: \( 3 \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \(- q^{3}\) \(- q^{4}\) \( -3 \beta q^{5} \) \( -\beta q^{6} \) \( -5 \beta q^{7} \) \( + 7 \beta q^{8} \) \( -26 q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \(- q^{3}\) \(- q^{4}\) \( -3 \beta q^{5} \) \( -\beta q^{6} \) \( -5 \beta q^{7} \) \( + 7 \beta q^{8} \) \( -26 q^{9} \) \( + 27 q^{10} \) \( + 16 \beta q^{11} \) \(+ q^{12}\) \( + ( 26 - 13 \beta ) q^{13} \) \( + 45 q^{14} \) \( + 3 \beta q^{15} \) \( -71 q^{16} \) \( -45 q^{17} \) \( -26 \beta q^{18} \) \( + 2 \beta q^{19} \) \( + 3 \beta q^{20} \) \( + 5 \beta q^{21} \) \( -144 q^{22} \) \( + 162 q^{23} \) \( -7 \beta q^{24} \) \( + 44 q^{25} \) \( + ( 117 + 26 \beta ) q^{26} \) \( + 53 q^{27} \) \( + 5 \beta q^{28} \) \( -144 q^{29} \) \( -27 q^{30} \) \( + 88 \beta q^{31} \) \( -15 \beta q^{32} \) \( -16 \beta q^{33} \) \( -45 \beta q^{34} \) \( -135 q^{35} \) \( + 26 q^{36} \) \( -101 \beta q^{37} \) \( -18 q^{38} \) \( + ( -26 + 13 \beta ) q^{39} \) \( + 189 q^{40} \) \( -64 \beta q^{41} \) \( -45 q^{42} \) \( -97 q^{43} \) \( -16 \beta q^{44} \) \( + 78 \beta q^{45} \) \( + 162 \beta q^{46} \) \( -37 \beta q^{47} \) \( + 71 q^{48} \) \( + 118 q^{49} \) \( + 44 \beta q^{50} \) \( + 45 q^{51} \) \( + ( -26 + 13 \beta ) q^{52} \) \( -414 q^{53} \) \( + 53 \beta q^{54} \) \( + 432 q^{55} \) \( + 315 q^{56} \) \( -2 \beta q^{57} \) \( -144 \beta q^{58} \) \( -174 \beta q^{59} \) \( -3 \beta q^{60} \) \( + 376 q^{61} \) \( -792 q^{62} \) \( + 130 \beta q^{63} \) \( -433 q^{64} \) \( + ( -351 - 78 \beta ) q^{65} \) \( + 144 q^{66} \) \( -12 \beta q^{67} \) \( + 45 q^{68} \) \( -162 q^{69} \) \( -135 \beta q^{70} \) \( + 119 \beta q^{71} \) \( -182 \beta q^{72} \) \( + 366 \beta q^{73} \) \( + 909 q^{74} \) \( -44 q^{75} \) \( -2 \beta q^{76} \) \( + 720 q^{77} \) \( + ( -117 - 26 \beta ) q^{78} \) \( -830 q^{79} \) \( + 213 \beta q^{80} \) \( + 649 q^{81} \) \( + 576 q^{82} \) \( -146 \beta q^{83} \) \( -5 \beta q^{84} \) \( + 135 \beta q^{85} \) \( -97 \beta q^{86} \) \( + 144 q^{87} \) \( -1008 q^{88} \) \( + 146 \beta q^{89} \) \( -702 q^{90} \) \( + ( -585 - 130 \beta ) q^{91} \) \( -162 q^{92} \) \( -88 \beta q^{93} \) \( + 333 q^{94} \) \( + 54 q^{95} \) \( + 15 \beta q^{96} \) \( -284 \beta q^{97} \) \( + 118 \beta q^{98} \) \( -416 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 52q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 52q^{9} \) \(\mathstrut +\mathstrut 54q^{10} \) \(\mathstrut +\mathstrut 2q^{12} \) \(\mathstrut +\mathstrut 52q^{13} \) \(\mathstrut +\mathstrut 90q^{14} \) \(\mathstrut -\mathstrut 142q^{16} \) \(\mathstrut -\mathstrut 90q^{17} \) \(\mathstrut -\mathstrut 288q^{22} \) \(\mathstrut +\mathstrut 324q^{23} \) \(\mathstrut +\mathstrut 88q^{25} \) \(\mathstrut +\mathstrut 234q^{26} \) \(\mathstrut +\mathstrut 106q^{27} \) \(\mathstrut -\mathstrut 288q^{29} \) \(\mathstrut -\mathstrut 54q^{30} \) \(\mathstrut -\mathstrut 270q^{35} \) \(\mathstrut +\mathstrut 52q^{36} \) \(\mathstrut -\mathstrut 36q^{38} \) \(\mathstrut -\mathstrut 52q^{39} \) \(\mathstrut +\mathstrut 378q^{40} \) \(\mathstrut -\mathstrut 90q^{42} \) \(\mathstrut -\mathstrut 194q^{43} \) \(\mathstrut +\mathstrut 142q^{48} \) \(\mathstrut +\mathstrut 236q^{49} \) \(\mathstrut +\mathstrut 90q^{51} \) \(\mathstrut -\mathstrut 52q^{52} \) \(\mathstrut -\mathstrut 828q^{53} \) \(\mathstrut +\mathstrut 864q^{55} \) \(\mathstrut +\mathstrut 630q^{56} \) \(\mathstrut +\mathstrut 752q^{61} \) \(\mathstrut -\mathstrut 1584q^{62} \) \(\mathstrut -\mathstrut 866q^{64} \) \(\mathstrut -\mathstrut 702q^{65} \) \(\mathstrut +\mathstrut 288q^{66} \) \(\mathstrut +\mathstrut 90q^{68} \) \(\mathstrut -\mathstrut 324q^{69} \) \(\mathstrut +\mathstrut 1818q^{74} \) \(\mathstrut -\mathstrut 88q^{75} \) \(\mathstrut +\mathstrut 1440q^{77} \) \(\mathstrut -\mathstrut 234q^{78} \) \(\mathstrut -\mathstrut 1660q^{79} \) \(\mathstrut +\mathstrut 1298q^{81} \) \(\mathstrut +\mathstrut 1152q^{82} \) \(\mathstrut +\mathstrut 288q^{87} \) \(\mathstrut -\mathstrut 2016q^{88} \) \(\mathstrut -\mathstrut 1404q^{90} \) \(\mathstrut -\mathstrut 1170q^{91} \) \(\mathstrut -\mathstrut 324q^{92} \) \(\mathstrut +\mathstrut 666q^{94} \) \(\mathstrut +\mathstrut 108q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
13.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(13, [\chi])\).