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

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

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\):

\( T_{3} - 8 \) Copy content Toggle raw display
\( T_{5} + 14 \) Copy content Toggle raw display
\( T_{13} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

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