Properties

Label 7.4.a.a
Level 7
Weight 4
Character orbit 7.a
Self dual yes
Analytic conductor 0.413
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 7.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.413013370040\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - 2q^{3} - 7q^{4} + 16q^{5} + 2q^{6} - 7q^{7} + 15q^{8} - 23q^{9} + O(q^{10}) \) \( q - q^{2} - 2q^{3} - 7q^{4} + 16q^{5} + 2q^{6} - 7q^{7} + 15q^{8} - 23q^{9} - 16q^{10} - 8q^{11} + 14q^{12} + 28q^{13} + 7q^{14} - 32q^{15} + 41q^{16} + 54q^{17} + 23q^{18} - 110q^{19} - 112q^{20} + 14q^{21} + 8q^{22} + 48q^{23} - 30q^{24} + 131q^{25} - 28q^{26} + 100q^{27} + 49q^{28} - 110q^{29} + 32q^{30} + 12q^{31} - 161q^{32} + 16q^{33} - 54q^{34} - 112q^{35} + 161q^{36} - 246q^{37} + 110q^{38} - 56q^{39} + 240q^{40} + 182q^{41} - 14q^{42} + 128q^{43} + 56q^{44} - 368q^{45} - 48q^{46} + 324q^{47} - 82q^{48} + 49q^{49} - 131q^{50} - 108q^{51} - 196q^{52} - 162q^{53} - 100q^{54} - 128q^{55} - 105q^{56} + 220q^{57} + 110q^{58} + 810q^{59} + 224q^{60} - 488q^{61} - 12q^{62} + 161q^{63} - 167q^{64} + 448q^{65} - 16q^{66} + 244q^{67} - 378q^{68} - 96q^{69} + 112q^{70} - 768q^{71} - 345q^{72} - 702q^{73} + 246q^{74} - 262q^{75} + 770q^{76} + 56q^{77} + 56q^{78} + 440q^{79} + 656q^{80} + 421q^{81} - 182q^{82} - 1302q^{83} - 98q^{84} + 864q^{85} - 128q^{86} + 220q^{87} - 120q^{88} + 730q^{89} + 368q^{90} - 196q^{91} - 336q^{92} - 24q^{93} - 324q^{94} - 1760q^{95} + 322q^{96} + 294q^{97} - 49q^{98} + 184q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 −2.00000 −7.00000 16.0000 2.00000 −7.00000 15.0000 −23.0000 −16.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.4.a.a 1
3.b odd 2 1 63.4.a.b 1
4.b odd 2 1 112.4.a.f 1
5.b even 2 1 175.4.a.b 1
5.c odd 4 2 175.4.b.b 2
7.b odd 2 1 49.4.a.b 1
7.c even 3 2 49.4.c.c 2
7.d odd 6 2 49.4.c.b 2
8.b even 2 1 448.4.a.i 1
8.d odd 2 1 448.4.a.e 1
11.b odd 2 1 847.4.a.b 1
12.b even 2 1 1008.4.a.c 1
13.b even 2 1 1183.4.a.b 1
15.d odd 2 1 1575.4.a.e 1
17.b even 2 1 2023.4.a.a 1
21.c even 2 1 441.4.a.i 1
21.g even 6 2 441.4.e.e 2
21.h odd 6 2 441.4.e.h 2
28.d even 2 1 784.4.a.g 1
35.c odd 2 1 1225.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 1.a even 1 1 trivial
49.4.a.b 1 7.b odd 2 1
49.4.c.b 2 7.d odd 6 2
49.4.c.c 2 7.c even 3 2
63.4.a.b 1 3.b odd 2 1
112.4.a.f 1 4.b odd 2 1
175.4.a.b 1 5.b even 2 1
175.4.b.b 2 5.c odd 4 2
441.4.a.i 1 21.c even 2 1
441.4.e.e 2 21.g even 6 2
441.4.e.h 2 21.h odd 6 2
448.4.a.e 1 8.d odd 2 1
448.4.a.i 1 8.b even 2 1
784.4.a.g 1 28.d even 2 1
847.4.a.b 1 11.b odd 2 1
1008.4.a.c 1 12.b even 2 1
1183.4.a.b 1 13.b even 2 1
1225.4.a.j 1 35.c odd 2 1
1575.4.a.e 1 15.d odd 2 1
2023.4.a.a 1 17.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 8 T^{2} \)
$3$ \( 1 + 2 T + 27 T^{2} \)
$5$ \( 1 - 16 T + 125 T^{2} \)
$7$ \( 1 + 7 T \)
$11$ \( 1 + 8 T + 1331 T^{2} \)
$13$ \( 1 - 28 T + 2197 T^{2} \)
$17$ \( 1 - 54 T + 4913 T^{2} \)
$19$ \( 1 + 110 T + 6859 T^{2} \)
$23$ \( 1 - 48 T + 12167 T^{2} \)
$29$ \( 1 + 110 T + 24389 T^{2} \)
$31$ \( 1 - 12 T + 29791 T^{2} \)
$37$ \( 1 + 246 T + 50653 T^{2} \)
$41$ \( 1 - 182 T + 68921 T^{2} \)
$43$ \( 1 - 128 T + 79507 T^{2} \)
$47$ \( 1 - 324 T + 103823 T^{2} \)
$53$ \( 1 + 162 T + 148877 T^{2} \)
$59$ \( 1 - 810 T + 205379 T^{2} \)
$61$ \( 1 + 488 T + 226981 T^{2} \)
$67$ \( 1 - 244 T + 300763 T^{2} \)
$71$ \( 1 + 768 T + 357911 T^{2} \)
$73$ \( 1 + 702 T + 389017 T^{2} \)
$79$ \( 1 - 440 T + 493039 T^{2} \)
$83$ \( 1 + 1302 T + 571787 T^{2} \)
$89$ \( 1 - 730 T + 704969 T^{2} \)
$97$ \( 1 - 294 T + 912673 T^{2} \)
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