Properties

Label 29.2.d.a
Level 29
Weight 2
Character orbit 29.d
Analytic conductor 0.232
Analytic rank 0
Dimension 6
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.231566165862\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{14} + \zeta_{14}^{3} - \zeta_{14}^{4} + \zeta_{14}^{5} ) q^{2} + ( -1 + \zeta_{14}^{5} ) q^{3} + ( \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{4} + ( 1 - \zeta_{14} - 2 \zeta_{14}^{3} - 2 \zeta_{14}^{5} ) q^{5} + ( -\zeta_{14} - \zeta_{14}^{3} - \zeta_{14}^{5} ) q^{6} + ( 1 - 2 \zeta_{14} + 2 \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{7} + ( -2 + \zeta_{14} - \zeta_{14}^{2} + 2 \zeta_{14}^{3} + \zeta_{14}^{5} ) q^{8} + ( 1 - \zeta_{14}^{3} + \zeta_{14}^{5} ) q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{14} + \zeta_{14}^{3} - \zeta_{14}^{4} + \zeta_{14}^{5} ) q^{2} + ( -1 + \zeta_{14}^{5} ) q^{3} + ( \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{4} + ( 1 - \zeta_{14} - 2 \zeta_{14}^{3} - 2 \zeta_{14}^{5} ) q^{5} + ( -\zeta_{14} - \zeta_{14}^{3} - \zeta_{14}^{5} ) q^{6} + ( 1 - 2 \zeta_{14} + 2 \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{7} + ( -2 + \zeta_{14} - \zeta_{14}^{2} + 2 \zeta_{14}^{3} + \zeta_{14}^{5} ) q^{8} + ( 1 - \zeta_{14}^{3} + \zeta_{14}^{5} ) q^{9} + ( 2 + \zeta_{14} + 2 \zeta_{14}^{2} + \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{10} + ( -1 - 2 \zeta_{14} + 2 \zeta_{14}^{2} - 2 \zeta_{14}^{3} - \zeta_{14}^{4} ) q^{11} + ( -\zeta_{14}^{2} + \zeta_{14}^{3} - \zeta_{14}^{4} + \zeta_{14}^{5} ) q^{12} + ( -3 + 4 \zeta_{14} - 2 \zeta_{14}^{2} + 4 \zeta_{14}^{3} - 3 \zeta_{14}^{4} ) q^{13} + ( 3 \zeta_{14} - 2 \zeta_{14}^{2} - \zeta_{14}^{3} - 2 \zeta_{14}^{4} + 3 \zeta_{14}^{5} ) q^{14} + ( 2 \zeta_{14} + \zeta_{14}^{2} + 3 \zeta_{14}^{3} + \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{15} + ( 2 - 3 \zeta_{14} - 3 \zeta_{14}^{3} + 2 \zeta_{14}^{4} ) q^{16} + ( -2 \zeta_{14}^{2} + 2 \zeta_{14}^{3} - 2 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{17} + ( -2 + \zeta_{14} + \zeta_{14}^{2} + \zeta_{14}^{3} - 2 \zeta_{14}^{4} ) q^{18} + ( 1 + \zeta_{14}^{2} + 2 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{19} + ( 1 - \zeta_{14} + \zeta_{14}^{2} - \zeta_{14}^{3} - \zeta_{14}^{5} ) q^{20} + ( 1 - \zeta_{14}^{3} ) q^{21} + ( 4 - 4 \zeta_{14} + 3 \zeta_{14}^{2} - 3 \zeta_{14}^{3} + 4 \zeta_{14}^{4} - 4 \zeta_{14}^{5} ) q^{22} + ( -\zeta_{14} + 3 \zeta_{14}^{2} + \zeta_{14}^{3} + 3 \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{23} + ( 2 - 2 \zeta_{14} - 2 \zeta_{14}^{3} - \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{24} + ( -7 + 3 \zeta_{14} - 7 \zeta_{14}^{2} - 4 \zeta_{14}^{4} + 4 \zeta_{14}^{5} ) q^{25} + ( 2 - 4 \zeta_{14} - \zeta_{14}^{2} + \zeta_{14}^{3} + 4 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{26} + ( -1 + \zeta_{14} - 3 \zeta_{14}^{3} - 3 \zeta_{14}^{5} ) q^{27} + ( -2 + \zeta_{14}^{2} + \zeta_{14}^{3} - \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{28} + ( 3 + 2 \zeta_{14}^{2} - 2 \zeta_{14}^{3} + 6 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{29} + ( -5 - 4 \zeta_{14}^{2} + 2 \zeta_{14}^{3} - 2 \zeta_{14}^{4} + 4 \zeta_{14}^{5} ) q^{30} + ( -1 + \zeta_{14} + 2 \zeta_{14}^{3} - 6 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{31} + ( -3 + 2 \zeta_{14} + \zeta_{14}^{2} - \zeta_{14}^{3} - 2 \zeta_{14}^{4} + 3 \zeta_{14}^{5} ) q^{32} + ( 1 + 2 \zeta_{14} + \zeta_{14}^{2} + 3 \zeta_{14}^{4} - 3 \zeta_{14}^{5} ) q^{33} -2 \zeta_{14}^{4} q^{34} + ( \zeta_{14}^{2} + \zeta_{14}^{3} + \zeta_{14}^{4} ) q^{35} + ( 1 - \zeta_{14} - \zeta_{14}^{2} + \zeta_{14}^{3} + \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{36} + ( 3 - 2 \zeta_{14} + 2 \zeta_{14}^{2} - 3 \zeta_{14}^{3} ) q^{37} + ( -1 + 2 \zeta_{14} - 2 \zeta_{14}^{2} + \zeta_{14}^{3} + 3 \zeta_{14}^{5} ) q^{38} + ( 1 - 4 \zeta_{14} + \zeta_{14}^{2} - \zeta_{14}^{4} + \zeta_{14}^{5} ) q^{39} + ( 3 + 2 \zeta_{14} + 5 \zeta_{14}^{2} + 2 \zeta_{14}^{3} + 3 \zeta_{14}^{4} ) q^{40} + ( 4 + 2 \zeta_{14}^{2} - 2 \zeta_{14}^{5} ) q^{41} + ( -1 + \zeta_{14} + \zeta_{14}^{2} + \zeta_{14}^{3} - \zeta_{14}^{4} ) q^{42} + ( 3 \zeta_{14} - 3 \zeta_{14}^{2} + \zeta_{14}^{3} - 3 \zeta_{14}^{4} + 3 \zeta_{14}^{5} ) q^{43} + ( 3 \zeta_{14} - 5 \zeta_{14}^{2} + 4 \zeta_{14}^{3} - 5 \zeta_{14}^{4} + 3 \zeta_{14}^{5} ) q^{44} -\zeta_{14}^{2} q^{45} + ( -1 - 4 \zeta_{14}^{2} - \zeta_{14}^{3} + \zeta_{14}^{4} + 4 \zeta_{14}^{5} ) q^{46} + ( 1 - 6 \zeta_{14}^{2} + \zeta_{14}^{4} ) q^{47} + ( 1 + 3 \zeta_{14} + \zeta_{14}^{2} + \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{48} + ( -3 - 4 \zeta_{14} + 4 \zeta_{14}^{2} + 3 \zeta_{14}^{3} + \zeta_{14}^{5} ) q^{49} + ( 4 - 4 \zeta_{14} + 4 \zeta_{14}^{2} - 4 \zeta_{14}^{3} - 11 \zeta_{14}^{5} ) q^{50} + ( 2 - 2 \zeta_{14} + 4 \zeta_{14}^{2} - 4 \zeta_{14}^{3} + 2 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{51} + ( \zeta_{14} + 3 \zeta_{14}^{2} - 6 \zeta_{14}^{3} + 3 \zeta_{14}^{4} + \zeta_{14}^{5} ) q^{52} + ( -1 + \zeta_{14} + 2 \zeta_{14}^{3} - 4 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{53} + ( 3 - \zeta_{14} + 3 \zeta_{14}^{2} + 4 \zeta_{14}^{4} - 4 \zeta_{14}^{5} ) q^{54} + ( -7 + 3 \zeta_{14} - 6 \zeta_{14}^{2} + 6 \zeta_{14}^{3} - 3 \zeta_{14}^{4} + 7 \zeta_{14}^{5} ) q^{55} + ( -2 + 2 \zeta_{14} - 3 \zeta_{14}^{4} ) q^{56} + ( -2 - 3 \zeta_{14}^{2} + 2 \zeta_{14}^{3} - 2 \zeta_{14}^{4} + 3 \zeta_{14}^{5} ) q^{57} + ( -7 + 7 \zeta_{14} - 4 \zeta_{14}^{2} + 3 \zeta_{14}^{3} - 5 \zeta_{14}^{4} + 7 \zeta_{14}^{5} ) q^{58} + ( -8 + 2 \zeta_{14}^{2} - 2 \zeta_{14}^{3} + 2 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{59} + ( -1 + \zeta_{14} + \zeta_{14}^{3} + \zeta_{14}^{4} + \zeta_{14}^{5} ) q^{60} + ( 3 - 4 \zeta_{14} + 2 \zeta_{14}^{2} - 2 \zeta_{14}^{3} + 4 \zeta_{14}^{4} - 3 \zeta_{14}^{5} ) q^{61} + ( 4 - 7 \zeta_{14} + 4 \zeta_{14}^{2} + 5 \zeta_{14}^{4} - 5 \zeta_{14}^{5} ) q^{62} + ( 5 - 5 \zeta_{14} - 2 \zeta_{14}^{3} + 6 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{63} + ( -2 \zeta_{14} - 3 \zeta_{14}^{2} - \zeta_{14}^{3} - 3 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{64} + ( 3 - \zeta_{14} + 4 \zeta_{14}^{2} - 4 \zeta_{14}^{3} + \zeta_{14}^{4} - 3 \zeta_{14}^{5} ) q^{65} + ( -3 + 3 \zeta_{14} - 3 \zeta_{14}^{2} + 3 \zeta_{14}^{3} + 4 \zeta_{14}^{5} ) q^{66} + ( 7 - 4 \zeta_{14} + 4 \zeta_{14}^{2} - 7 \zeta_{14}^{3} - 8 \zeta_{14}^{5} ) q^{67} + ( -2 + 2 \zeta_{14} - 2 \zeta_{14}^{2} + 2 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{68} + ( -2 - \zeta_{14} - 5 \zeta_{14}^{2} - \zeta_{14}^{3} - 2 \zeta_{14}^{4} ) q^{69} + ( -1 - 2 \zeta_{14}^{2} + 2 \zeta_{14}^{5} ) q^{70} + ( 3 - 6 \zeta_{14}^{2} + 3 \zeta_{14}^{4} ) q^{71} + ( -\zeta_{14} + 2 \zeta_{14}^{3} - \zeta_{14}^{5} ) q^{72} + ( -7 \zeta_{14} + 3 \zeta_{14}^{2} - 5 \zeta_{14}^{3} + 3 \zeta_{14}^{4} - 7 \zeta_{14}^{5} ) q^{73} + ( -1 + \zeta_{14} + 3 \zeta_{14}^{2} + \zeta_{14}^{3} - \zeta_{14}^{4} ) q^{74} + ( 11 + 8 \zeta_{14}^{2} - \zeta_{14}^{3} + \zeta_{14}^{4} - 8 \zeta_{14}^{5} ) q^{75} + ( -\zeta_{14} + 3 \zeta_{14}^{2} - \zeta_{14}^{3} ) q^{76} + ( -1 + 6 \zeta_{14} - \zeta_{14}^{2} - 5 \zeta_{14}^{4} + 5 \zeta_{14}^{5} ) q^{77} + ( 3 - \zeta_{14} + \zeta_{14}^{2} - 3 \zeta_{14}^{3} ) q^{78} + ( -3 + 3 \zeta_{14}^{3} + 6 \zeta_{14}^{5} ) q^{79} + ( -6 + \zeta_{14} - 5 \zeta_{14}^{2} + 5 \zeta_{14}^{3} - \zeta_{14}^{4} + 6 \zeta_{14}^{5} ) q^{80} + ( 6 \zeta_{14} - \zeta_{14}^{2} + 4 \zeta_{14}^{3} - \zeta_{14}^{4} + 6 \zeta_{14}^{5} ) q^{81} + ( -2 + 2 \zeta_{14} + 4 \zeta_{14}^{3} + 4 \zeta_{14}^{5} ) q^{82} + ( 1 + 4 \zeta_{14} + \zeta_{14}^{2} - 4 \zeta_{14}^{4} + 4 \zeta_{14}^{5} ) q^{83} + ( 1 - \zeta_{14} + \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{84} + ( -2 + 2 \zeta_{14} + 2 \zeta_{14}^{3} + 2 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{85} + ( -3 + 2 \zeta_{14}^{2} + 3 \zeta_{14}^{3} - 3 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{86} + ( -5 + 2 \zeta_{14} - 8 \zeta_{14}^{2} + 4 \zeta_{14}^{3} - 6 \zeta_{14}^{4} + 5 \zeta_{14}^{5} ) q^{87} + ( 7 + \zeta_{14}^{2} + \zeta_{14}^{3} - \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{88} + ( -1 + \zeta_{14} + 6 \zeta_{14}^{3} + 6 \zeta_{14}^{5} ) q^{89} + ( \zeta_{14} - \zeta_{14}^{4} ) q^{90} + ( -5 + 12 \zeta_{14} - 5 \zeta_{14}^{2} - 9 \zeta_{14}^{4} + 9 \zeta_{14}^{5} ) q^{91} + ( -2 + 2 \zeta_{14} + 3 \zeta_{14}^{3} - 4 \zeta_{14}^{4} + 3 \zeta_{14}^{5} ) q^{92} + ( -2 \zeta_{14} + 5 \zeta_{14}^{2} - 3 \zeta_{14}^{3} + 5 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{93} + ( -2 + 8 \zeta_{14} - \zeta_{14}^{2} + \zeta_{14}^{3} - 8 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{94} + ( 5 - 3 \zeta_{14} + 3 \zeta_{14}^{2} - 5 \zeta_{14}^{3} - 6 \zeta_{14}^{5} ) q^{95} + ( \zeta_{14} - \zeta_{14}^{2} - 4 \zeta_{14}^{5} ) q^{96} + ( 5 - 8 \zeta_{14} + 5 \zeta_{14}^{2} + 8 \zeta_{14}^{4} - 8 \zeta_{14}^{5} ) q^{97} + ( 6 - 7 \zeta_{14} - 3 \zeta_{14}^{2} - 7 \zeta_{14}^{3} + 6 \zeta_{14}^{4} ) q^{98} + ( -4 + 3 \zeta_{14}^{2} - \zeta_{14}^{3} + \zeta_{14}^{4} - 3 \zeta_{14}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{2} - 5q^{3} - 2q^{4} + q^{5} - 3q^{6} + q^{7} - 7q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - 2q^{2} - 5q^{3} - 2q^{4} + q^{5} - 3q^{6} + q^{7} - 7q^{8} + 6q^{9} + 9q^{10} - 11q^{11} + 4q^{12} - 5q^{13} + 9q^{14} + 5q^{15} + 4q^{16} + 8q^{17} - 9q^{18} + q^{19} + 2q^{20} + 5q^{21} + 6q^{22} - 7q^{23} + 7q^{24} - 24q^{25} + 4q^{26} - 11q^{27} - 12q^{28} + 6q^{29} - 18q^{30} + 5q^{31} - 13q^{32} + q^{33} + 2q^{34} - q^{35} + 5q^{36} + 11q^{37} + 2q^{38} + 3q^{39} + 14q^{40} + 20q^{41} - 4q^{42} + 13q^{43} + 20q^{44} + q^{45} + 11q^{47} + 6q^{48} - 22q^{49} + q^{50} - 2q^{51} - 10q^{52} + 3q^{53} + 6q^{54} - 17q^{55} - 7q^{56} - 2q^{57} - 16q^{58} - 56q^{59} - 4q^{60} + 3q^{61} + 3q^{62} + 15q^{63} + q^{64} + 5q^{65} - 5q^{66} + 19q^{67} - 12q^{68} - 7q^{69} - 2q^{70} + 21q^{71} - 25q^{73} - 6q^{74} + 48q^{75} - 5q^{76} + 11q^{77} + 13q^{78} - 9q^{79} - 18q^{80} + 18q^{81} - 2q^{82} + 17q^{83} + 3q^{84} - 8q^{85} - 16q^{86} - 5q^{87} + 42q^{88} + 7q^{89} + 2q^{90} + 5q^{91} - 17q^{93} + 8q^{94} + 13q^{95} - 2q^{96} + q^{97} + 19q^{98} - 32q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{14}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.900969 0.433884i
−0.623490 0.781831i
−0.623490 + 0.781831i
0.222521 + 0.974928i
0.222521 0.974928i
0.900969 + 0.433884i
−0.277479 1.21572i −1.62349 0.781831i 0.400969 0.193096i 0.900969 + 3.94740i −0.500000 + 2.19064i −0.623490 0.300257i −1.90097 2.38374i 0.153989 + 0.193096i 4.54892 2.19064i
16.1 0.400969 + 0.193096i −0.777479 + 0.974928i −1.12349 1.40881i −0.623490 0.300257i −0.500000 + 0.240787i 0.222521 0.279032i −0.376510 1.64960i 0.321552 + 1.40881i −0.192021 0.240787i
20.1 0.400969 0.193096i −0.777479 0.974928i −1.12349 + 1.40881i −0.623490 + 0.300257i −0.500000 0.240787i 0.222521 + 0.279032i −0.376510 + 1.64960i 0.321552 1.40881i −0.192021 + 0.240787i
23.1 −1.12349 + 1.40881i −0.0990311 + 0.433884i −0.277479 1.21572i 0.222521 0.279032i −0.500000 0.626980i 0.900969 3.94740i −1.22252 0.588735i 2.52446 + 1.21572i 0.143104 + 0.626980i
24.1 −1.12349 1.40881i −0.0990311 0.433884i −0.277479 + 1.21572i 0.222521 + 0.279032i −0.500000 + 0.626980i 0.900969 + 3.94740i −1.22252 + 0.588735i 2.52446 1.21572i 0.143104 0.626980i
25.1 −0.277479 + 1.21572i −1.62349 + 0.781831i 0.400969 + 0.193096i 0.900969 3.94740i −0.500000 2.19064i −0.623490 + 0.300257i −1.90097 + 2.38374i 0.153989 0.193096i 4.54892 + 2.19064i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.2.d.a 6
3.b odd 2 1 261.2.k.a 6
4.b odd 2 1 464.2.u.f 6
5.b even 2 1 725.2.l.b 6
5.c odd 4 2 725.2.r.b 12
29.b even 2 1 841.2.d.d 6
29.c odd 4 2 841.2.e.d 12
29.d even 7 1 inner 29.2.d.a 6
29.d even 7 1 841.2.a.e 3
29.d even 7 2 841.2.d.b 6
29.d even 7 2 841.2.d.e 6
29.e even 14 1 841.2.a.f 3
29.e even 14 2 841.2.d.a 6
29.e even 14 2 841.2.d.c 6
29.e even 14 1 841.2.d.d 6
29.f odd 28 2 841.2.b.c 6
29.f odd 28 4 841.2.e.b 12
29.f odd 28 4 841.2.e.c 12
29.f odd 28 2 841.2.e.d 12
87.h odd 14 1 7569.2.a.p 3
87.j odd 14 1 261.2.k.a 6
87.j odd 14 1 7569.2.a.r 3
116.j odd 14 1 464.2.u.f 6
145.n even 14 1 725.2.l.b 6
145.p odd 28 2 725.2.r.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.d.a 6 1.a even 1 1 trivial
29.2.d.a 6 29.d even 7 1 inner
261.2.k.a 6 3.b odd 2 1
261.2.k.a 6 87.j odd 14 1
464.2.u.f 6 4.b odd 2 1
464.2.u.f 6 116.j odd 14 1
725.2.l.b 6 5.b even 2 1
725.2.l.b 6 145.n even 14 1
725.2.r.b 12 5.c odd 4 2
725.2.r.b 12 145.p odd 28 2
841.2.a.e 3 29.d even 7 1
841.2.a.f 3 29.e even 14 1
841.2.b.c 6 29.f odd 28 2
841.2.d.a 6 29.e even 14 2
841.2.d.b 6 29.d even 7 2
841.2.d.c 6 29.e even 14 2
841.2.d.d 6 29.b even 2 1
841.2.d.d 6 29.e even 14 1
841.2.d.e 6 29.d even 7 2
841.2.e.b 12 29.f odd 28 4
841.2.e.c 12 29.f odd 28 4
841.2.e.d 12 29.c odd 4 2
841.2.e.d 12 29.f odd 28 2
7569.2.a.p 3 87.h odd 14 1
7569.2.a.r 3 87.j odd 14 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 7 T^{3} + 10 T^{4} + 13 T^{5} + 27 T^{6} + 26 T^{7} + 40 T^{8} + 56 T^{9} + 32 T^{10} + 64 T^{11} + 64 T^{12} \)
$3$ \( 1 + 5 T + 8 T^{2} + 4 T^{3} + 3 T^{4} + 3 T^{5} - 8 T^{6} + 9 T^{7} + 27 T^{8} + 108 T^{9} + 648 T^{10} + 1215 T^{11} + 729 T^{12} \)
$5$ \( 1 - T + 10 T^{2} - 12 T^{3} + 81 T^{4} - 91 T^{5} + 456 T^{6} - 455 T^{7} + 2025 T^{8} - 1500 T^{9} + 6250 T^{10} - 3125 T^{11} + 15625 T^{12} \)
$7$ \( 1 - T + 8 T^{2} - 22 T^{3} + 127 T^{4} - 169 T^{5} + 848 T^{6} - 1183 T^{7} + 6223 T^{8} - 7546 T^{9} + 19208 T^{10} - 16807 T^{11} + 117649 T^{12} \)
$11$ \( 1 + 11 T + 68 T^{2} + 354 T^{3} + 1683 T^{4} + 6723 T^{5} + 23296 T^{6} + 73953 T^{7} + 203643 T^{8} + 471174 T^{9} + 995588 T^{10} + 1771561 T^{11} + 1771561 T^{12} \)
$13$ \( 1 + 5 T + 12 T^{2} - 40 T^{3} - 111 T^{4} + 875 T^{5} + 6308 T^{6} + 11375 T^{7} - 18759 T^{8} - 87880 T^{9} + 342732 T^{10} + 1856465 T^{11} + 4826809 T^{12} \)
$17$ \( ( 1 - 4 T + 47 T^{2} - 128 T^{3} + 799 T^{4} - 1156 T^{5} + 4913 T^{6} )^{2} \)
$19$ \( 1 - T - 18 T^{2} + 156 T^{3} + 67 T^{4} - 1127 T^{5} + 9612 T^{6} - 21413 T^{7} + 24187 T^{8} + 1070004 T^{9} - 2345778 T^{10} - 2476099 T^{11} + 47045881 T^{12} \)
$23$ \( 1 + 7 T + 26 T^{2} + 84 T^{3} + 431 T^{4} - 175 T^{5} - 12020 T^{6} - 4025 T^{7} + 227999 T^{8} + 1022028 T^{9} + 7275866 T^{10} + 45054401 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 6 T - 13 T^{2} + 316 T^{3} - 377 T^{4} - 5046 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 5 T - 34 T^{2} + 388 T^{3} - 1649 T^{4} - 7283 T^{5} + 109220 T^{6} - 225773 T^{7} - 1584689 T^{8} + 11558908 T^{9} - 31399714 T^{10} - 143145755 T^{11} + 887503681 T^{12} \)
$37$ \( 1 - 11 T + 42 T^{2} - 328 T^{3} + 4049 T^{4} - 22869 T^{5} + 100360 T^{6} - 846153 T^{7} + 5543081 T^{8} - 16614184 T^{9} + 78714762 T^{10} - 762783527 T^{11} + 2565726409 T^{12} \)
$41$ \( ( 1 - 10 T + 147 T^{2} - 828 T^{3} + 6027 T^{4} - 16810 T^{5} + 68921 T^{6} )^{2} \)
$43$ \( 1 - 13 T + 42 T^{2} - 50 T^{3} + 251 T^{4} + 17703 T^{5} - 219260 T^{6} + 761229 T^{7} + 464099 T^{8} - 3975350 T^{9} + 143589642 T^{10} - 1911109759 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 - 11 T + 18 T^{2} + 410 T^{3} - 2409 T^{4} - 12679 T^{5} + 224300 T^{6} - 595913 T^{7} - 5321481 T^{8} + 42567430 T^{9} + 87834258 T^{10} - 2522795077 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 - 3 T - 58 T^{2} + 648 T^{3} - 431 T^{4} - 23629 T^{5} + 179088 T^{6} - 1252337 T^{7} - 1210679 T^{8} + 96472296 T^{9} - 457647898 T^{10} - 1254586479 T^{11} + 22164361129 T^{12} \)
$59$ \( ( 1 + 28 T + 429 T^{2} + 4032 T^{3} + 25311 T^{4} + 97468 T^{5} + 205379 T^{6} )^{2} \)
$61$ \( 1 - 3 T - 24 T^{2} - 74 T^{3} + 4633 T^{4} - 24883 T^{5} - 67236 T^{6} - 1517863 T^{7} + 17239393 T^{8} - 16796594 T^{9} - 332300184 T^{10} - 2533788903 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - 19 T + 168 T^{2} - 904 T^{3} + 1419 T^{4} + 3983 T^{5} + 31592 T^{6} + 266861 T^{7} + 6369891 T^{8} - 271889752 T^{9} + 3385388328 T^{10} - 25652377033 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - 21 T + 118 T^{2} + 546 T^{3} - 4409 T^{4} - 107373 T^{5} + 1690660 T^{6} - 7623483 T^{7} - 22225769 T^{8} + 195419406 T^{9} + 2998578358 T^{10} - 37888816371 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 25 T + 300 T^{2} + 3400 T^{3} + 39069 T^{4} + 348943 T^{5} + 2806700 T^{6} + 25472839 T^{7} + 208198701 T^{8} + 1322657800 T^{9} + 8519472300 T^{10} + 51826789825 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 9 T + 2 T^{2} + 210 T^{3} - 977 T^{4} - 72339 T^{5} - 812260 T^{6} - 5714781 T^{7} - 6097457 T^{8} + 103538190 T^{9} + 77900162 T^{10} + 27693507591 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 17 T + 136 T^{2} - 1776 T^{3} + 27843 T^{4} - 251219 T^{5} + 1969848 T^{6} - 20851177 T^{7} + 191810427 T^{8} - 1015493712 T^{9} + 6454331656 T^{10} - 66963690931 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 7 T + 30 T^{2} + 532 T^{3} + 7389 T^{4} - 23261 T^{5} + 663944 T^{6} - 2070229 T^{7} + 58528269 T^{8} + 375043508 T^{9} + 1882267230 T^{10} - 39088416143 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - T + 114 T^{2} + 382 T^{3} + 20669 T^{4} + 10611 T^{5} + 2196928 T^{6} + 1029267 T^{7} + 194474621 T^{8} + 348641086 T^{9} + 10092338034 T^{10} - 8587340257 T^{11} + 832972004929 T^{12} \)
show more
show less