Properties

Label 29.2.b.a
Level 29
Weight 2
Character orbit 29.b
Analytic conductor 0.232
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 29.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.231566165862\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \(x^{2} + 5\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} -\beta q^{3} -3 q^{4} -3 q^{5} + 5 q^{6} + 2 q^{7} -\beta q^{8} -2 q^{9} +O(q^{10})\) \( q + \beta q^{2} -\beta q^{3} -3 q^{4} -3 q^{5} + 5 q^{6} + 2 q^{7} -\beta q^{8} -2 q^{9} -3 \beta q^{10} + \beta q^{11} + 3 \beta q^{12} - q^{13} + 2 \beta q^{14} + 3 \beta q^{15} - q^{16} -2 \beta q^{17} -2 \beta q^{18} + 9 q^{20} -2 \beta q^{21} -5 q^{22} + 6 q^{23} -5 q^{24} + 4 q^{25} -\beta q^{26} -\beta q^{27} -6 q^{28} + ( -3 + 2 \beta ) q^{29} -15 q^{30} + 3 \beta q^{31} -3 \beta q^{32} + 5 q^{33} + 10 q^{34} -6 q^{35} + 6 q^{36} + \beta q^{39} + 3 \beta q^{40} -2 \beta q^{41} + 10 q^{42} -3 \beta q^{43} -3 \beta q^{44} + 6 q^{45} + 6 \beta q^{46} + \beta q^{47} + \beta q^{48} -3 q^{49} + 4 \beta q^{50} -10 q^{51} + 3 q^{52} -9 q^{53} + 5 q^{54} -3 \beta q^{55} -2 \beta q^{56} + ( -10 - 3 \beta ) q^{58} + 6 q^{59} -9 \beta q^{60} + 6 \beta q^{61} -15 q^{62} -4 q^{63} + 13 q^{64} + 3 q^{65} + 5 \beta q^{66} + 8 q^{67} + 6 \beta q^{68} -6 \beta q^{69} -6 \beta q^{70} + 2 \beta q^{72} -4 \beta q^{75} + 2 \beta q^{77} -5 q^{78} -3 \beta q^{79} + 3 q^{80} -11 q^{81} + 10 q^{82} -6 q^{83} + 6 \beta q^{84} + 6 \beta q^{85} + 15 q^{86} + ( 10 + 3 \beta ) q^{87} + 5 q^{88} -2 \beta q^{89} + 6 \beta q^{90} -2 q^{91} -18 q^{92} + 15 q^{93} -5 q^{94} -15 q^{96} + 6 \beta q^{97} -3 \beta q^{98} -2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{4} - 6q^{5} + 10q^{6} + 4q^{7} - 4q^{9} + O(q^{10}) \) \( 2q - 6q^{4} - 6q^{5} + 10q^{6} + 4q^{7} - 4q^{9} - 2q^{13} - 2q^{16} + 18q^{20} - 10q^{22} + 12q^{23} - 10q^{24} + 8q^{25} - 12q^{28} - 6q^{29} - 30q^{30} + 10q^{33} + 20q^{34} - 12q^{35} + 12q^{36} + 20q^{42} + 12q^{45} - 6q^{49} - 20q^{51} + 6q^{52} - 18q^{53} + 10q^{54} - 20q^{58} + 12q^{59} - 30q^{62} - 8q^{63} + 26q^{64} + 6q^{65} + 16q^{67} - 10q^{78} + 6q^{80} - 22q^{81} + 20q^{82} - 12q^{83} + 30q^{86} + 20q^{87} + 10q^{88} - 4q^{91} - 36q^{92} + 30q^{93} - 10q^{94} - 30q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
2.23607i
2.23607i
2.23607i 2.23607i −3.00000 −3.00000 5.00000 2.00000 2.23607i −2.00000 6.70820i
28.2 2.23607i 2.23607i −3.00000 −3.00000 5.00000 2.00000 2.23607i −2.00000 6.70820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.2.b.a 2
3.b odd 2 1 261.2.c.a 2
4.b odd 2 1 464.2.e.a 2
5.b even 2 1 725.2.c.c 2
5.c odd 4 2 725.2.d.a 4
7.b odd 2 1 1421.2.b.b 2
8.b even 2 1 1856.2.e.g 2
8.d odd 2 1 1856.2.e.f 2
12.b even 2 1 4176.2.o.k 2
29.b even 2 1 inner 29.2.b.a 2
29.c odd 4 2 841.2.a.b 2
29.d even 7 6 841.2.e.g 12
29.e even 14 6 841.2.e.g 12
29.f odd 28 12 841.2.d.h 12
87.d odd 2 1 261.2.c.a 2
87.f even 4 2 7569.2.a.i 2
116.d odd 2 1 464.2.e.a 2
145.d even 2 1 725.2.c.c 2
145.h odd 4 2 725.2.d.a 4
203.c odd 2 1 1421.2.b.b 2
232.b odd 2 1 1856.2.e.f 2
232.g even 2 1 1856.2.e.g 2
348.b even 2 1 4176.2.o.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.b.a 2 1.a even 1 1 trivial
29.2.b.a 2 29.b even 2 1 inner
261.2.c.a 2 3.b odd 2 1
261.2.c.a 2 87.d odd 2 1
464.2.e.a 2 4.b odd 2 1
464.2.e.a 2 116.d odd 2 1
725.2.c.c 2 5.b even 2 1
725.2.c.c 2 145.d even 2 1
725.2.d.a 4 5.c odd 4 2
725.2.d.a 4 145.h odd 4 2
841.2.a.b 2 29.c odd 4 2
841.2.d.h 12 29.f odd 28 12
841.2.e.g 12 29.d even 7 6
841.2.e.g 12 29.e even 14 6
1421.2.b.b 2 7.b odd 2 1
1421.2.b.b 2 203.c odd 2 1
1856.2.e.f 2 8.d odd 2 1
1856.2.e.f 2 232.b odd 2 1
1856.2.e.g 2 8.b even 2 1
1856.2.e.g 2 232.g even 2 1
4176.2.o.k 2 12.b even 2 1
4176.2.o.k 2 348.b even 2 1
7569.2.a.i 2 87.f even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(29, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} + 4 T^{4} \)
$3$ \( 1 - T^{2} + 9 T^{4} \)
$5$ \( ( 1 + 3 T + 5 T^{2} )^{2} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( 1 - 17 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + T + 13 T^{2} )^{2} \)
$17$ \( 1 - 14 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 19 T^{2} )^{2} \)
$23$ \( ( 1 - 6 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 6 T + 29 T^{2} \)
$31$ \( 1 - 17 T^{2} + 961 T^{4} \)
$37$ \( ( 1 - 37 T^{2} )^{2} \)
$41$ \( ( 1 - 12 T + 41 T^{2} )( 1 + 12 T + 41 T^{2} ) \)
$43$ \( 1 - 41 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 89 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 + 9 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 8 T + 61 T^{2} )( 1 + 8 T + 61 T^{2} ) \)
$67$ \( ( 1 - 8 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{2} \)
$79$ \( 1 - 113 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 158 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 14 T^{2} + 9409 T^{4} \)
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