Properties

Label 1694.4.a.g
Level $1694$
Weight $4$
Character orbit 1694.a
Self dual yes
Analytic conductor $99.949$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1694.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(99.9492355497\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 8q^{3} + 4q^{4} - 14q^{5} + 16q^{6} + 7q^{7} + 8q^{8} + 37q^{9} + O(q^{10}) \) \( q + 2q^{2} + 8q^{3} + 4q^{4} - 14q^{5} + 16q^{6} + 7q^{7} + 8q^{8} + 37q^{9} - 28q^{10} + 32q^{12} - 18q^{13} + 14q^{14} - 112q^{15} + 16q^{16} - 74q^{17} + 74q^{18} - 80q^{19} - 56q^{20} + 56q^{21} - 112q^{23} + 64q^{24} + 71q^{25} - 36q^{26} + 80q^{27} + 28q^{28} - 190q^{29} - 224q^{30} + 72q^{31} + 32q^{32} - 148q^{34} - 98q^{35} + 148q^{36} - 346q^{37} - 160q^{38} - 144q^{39} - 112q^{40} - 162q^{41} + 112q^{42} + 412q^{43} - 518q^{45} - 224q^{46} + 24q^{47} + 128q^{48} + 49q^{49} + 142q^{50} - 592q^{51} - 72q^{52} + 318q^{53} + 160q^{54} + 56q^{56} - 640q^{57} - 380q^{58} - 200q^{59} - 448q^{60} + 198q^{61} + 144q^{62} + 259q^{63} + 64q^{64} + 252q^{65} - 716q^{67} - 296q^{68} - 896q^{69} - 196q^{70} + 392q^{71} + 296q^{72} - 538q^{73} - 692q^{74} + 568q^{75} - 320q^{76} - 288q^{78} - 240q^{79} - 224q^{80} - 359q^{81} - 324q^{82} + 1072q^{83} + 224q^{84} + 1036q^{85} + 824q^{86} - 1520q^{87} + 810q^{89} - 1036q^{90} - 126q^{91} - 448q^{92} + 576q^{93} + 48q^{94} + 1120q^{95} + 256q^{96} + 1354q^{97} + 98q^{98} + O(q^{100}) \)

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
2.00000 8.00000 4.00000 −14.0000 16.0000 7.00000 8.00000 37.0000 −28.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 1694.4.a.g 1
11.b odd 2 1 14.4.a.a 1
33.d even 2 1 126.4.a.h 1
44.c even 2 1 112.4.a.a 1
55.d odd 2 1 350.4.a.l 1
55.e even 4 2 350.4.c.b 2
77.b even 2 1 98.4.a.a 1
77.h odd 6 2 98.4.c.d 2
77.i even 6 2 98.4.c.f 2
88.b odd 2 1 448.4.a.b 1
88.g even 2 1 448.4.a.o 1
132.d odd 2 1 1008.4.a.s 1
143.d odd 2 1 2366.4.a.h 1
231.h odd 2 1 882.4.a.i 1
231.k odd 6 2 882.4.g.k 2
231.l even 6 2 882.4.g.b 2
308.g odd 2 1 784.4.a.s 1
385.h even 2 1 2450.4.a.bo 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 11.b odd 2 1
98.4.a.a 1 77.b even 2 1
98.4.c.d 2 77.h odd 6 2
98.4.c.f 2 77.i even 6 2
112.4.a.a 1 44.c even 2 1
126.4.a.h 1 33.d even 2 1
350.4.a.l 1 55.d odd 2 1
350.4.c.b 2 55.e even 4 2
448.4.a.b 1 88.b odd 2 1
448.4.a.o 1 88.g even 2 1
784.4.a.s 1 308.g odd 2 1
882.4.a.i 1 231.h odd 2 1
882.4.g.b 2 231.l even 6 2
882.4.g.k 2 231.k odd 6 2
1008.4.a.s 1 132.d odd 2 1
1694.4.a.g 1 1.a even 1 1 trivial
2366.4.a.h 1 143.d odd 2 1
2450.4.a.bo 1 385.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\):

\( T_{3} - 8 \)
\( T_{5} + 14 \)
\( T_{13} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( -8 + T \)
$5$ \( 14 + T \)
$7$ \( -7 + T \)
$11$ \( T \)
$13$ \( 18 + T \)
$17$ \( 74 + T \)
$19$ \( 80 + T \)
$23$ \( 112 + T \)
$29$ \( 190 + T \)
$31$ \( -72 + T \)
$37$ \( 346 + T \)
$41$ \( 162 + T \)
$43$ \( -412 + T \)
$47$ \( -24 + T \)
$53$ \( -318 + T \)
$59$ \( 200 + T \)
$61$ \( -198 + T \)
$67$ \( 716 + T \)
$71$ \( -392 + T \)
$73$ \( 538 + T \)
$79$ \( 240 + T \)
$83$ \( -1072 + T \)
$89$ \( -810 + T \)
$97$ \( -1354 + T \)
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