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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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

$p$ $F_p(T)$
$2$ \( T^{2} + 9 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 81 \) Copy content Toggle raw display
$7$ \( T^{2} + 225 \) Copy content Toggle raw display
$11$ \( T^{2} + 2304 \) Copy content Toggle raw display
$13$ \( T^{2} - 52T + 2197 \) Copy content Toggle raw display
$17$ \( (T + 45)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T - 162)^{2} \) Copy content Toggle raw display
$29$ \( (T + 144)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 69696 \) Copy content Toggle raw display
$37$ \( T^{2} + 91809 \) Copy content Toggle raw display
$41$ \( T^{2} + 36864 \) Copy content Toggle raw display
$43$ \( (T + 97)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12321 \) Copy content Toggle raw display
$53$ \( (T + 414)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 272484 \) Copy content Toggle raw display
$61$ \( (T - 376)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1296 \) Copy content Toggle raw display
$71$ \( T^{2} + 127449 \) Copy content Toggle raw display
$73$ \( T^{2} + 1205604 \) Copy content Toggle raw display
$79$ \( (T + 830)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 191844 \) Copy content Toggle raw display
$89$ \( T^{2} + 191844 \) Copy content Toggle raw display
$97$ \( T^{2} + 725904 \) Copy content Toggle raw display
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