Properties

Label 1682.4.a.c
Level $1682$
Weight $4$
Character orbit 1682.a
Self dual yes
Analytic conductor $99.241$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1682.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(99.2412126297\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + (\beta + 1) q^{3} + 4 q^{4} + (6 \beta - 5) q^{5} + (2 \beta + 2) q^{6} + ( - 8 \beta - 8) q^{7} + 8 q^{8} + (2 \beta - 20) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + (\beta + 1) q^{3} + 4 q^{4} + (6 \beta - 5) q^{5} + (2 \beta + 2) q^{6} + ( - 8 \beta - 8) q^{7} + 8 q^{8} + (2 \beta - 20) q^{9} + (12 \beta - 10) q^{10} + (3 \beta + 45) q^{11} + (4 \beta + 4) q^{12} + (8 \beta - 25) q^{13} + ( - 16 \beta - 16) q^{14} + (\beta + 31) q^{15} + 16 q^{16} + ( - 16 \beta + 22) q^{17} + (4 \beta - 40) q^{18} + (4 \beta - 54) q^{19} + (24 \beta - 20) q^{20} + ( - 16 \beta - 56) q^{21} + (6 \beta + 90) q^{22} + ( - 78 \beta - 14) q^{23} + (8 \beta + 8) q^{24} + ( - 60 \beta + 116) q^{25} + (16 \beta - 50) q^{26} + ( - 45 \beta - 35) q^{27} + ( - 32 \beta - 32) q^{28} + (2 \beta + 62) q^{30} + ( - 109 \beta - 33) q^{31} + 32 q^{32} + (48 \beta + 63) q^{33} + ( - 32 \beta + 44) q^{34} + ( - 8 \beta - 248) q^{35} + (8 \beta - 80) q^{36} + ( - 4 \beta - 20) q^{37} + (8 \beta - 108) q^{38} + ( - 17 \beta + 23) q^{39} + (48 \beta - 40) q^{40} + (80 \beta - 152) q^{41} + ( - 32 \beta - 112) q^{42} + ( - 53 \beta + 65) q^{43} + (12 \beta + 180) q^{44} + ( - 130 \beta + 172) q^{45} + ( - 156 \beta - 28) q^{46} + (99 \beta + 257) q^{47} + (16 \beta + 16) q^{48} + (128 \beta + 105) q^{49} + ( - 120 \beta + 232) q^{50} + (6 \beta - 74) q^{51} + (32 \beta - 100) q^{52} + (52 \beta - 479) q^{53} + ( - 90 \beta - 70) q^{54} + (255 \beta - 117) q^{55} + ( - 64 \beta - 64) q^{56} + ( - 50 \beta - 30) q^{57} + (250 \beta - 90) q^{59} + (4 \beta + 124) q^{60} + (12 \beta - 514) q^{61} + ( - 218 \beta - 66) q^{62} + (144 \beta + 64) q^{63} + 64 q^{64} + ( - 190 \beta + 413) q^{65} + (96 \beta + 126) q^{66} + ( - 20 \beta - 456) q^{67} + ( - 64 \beta + 88) q^{68} + ( - 92 \beta - 482) q^{69} + ( - 16 \beta - 496) q^{70} + (34 \beta + 398) q^{71} + (16 \beta - 160) q^{72} + ( - 176 \beta + 428) q^{73} + ( - 8 \beta - 40) q^{74} + (56 \beta - 244) q^{75} + (16 \beta - 216) q^{76} + ( - 384 \beta - 504) q^{77} + ( - 34 \beta + 46) q^{78} + (361 \beta + 159) q^{79} + (96 \beta - 80) q^{80} + ( - 134 \beta + 235) q^{81} + (160 \beta - 304) q^{82} + ( - 38 \beta - 914) q^{83} + ( - 64 \beta - 224) q^{84} + (212 \beta - 686) q^{85} + ( - 106 \beta + 130) q^{86} + (24 \beta + 360) q^{88} + ( - 72 \beta + 472) q^{89} + ( - 260 \beta + 344) q^{90} + (136 \beta - 184) q^{91} + ( - 312 \beta - 56) q^{92} + ( - 142 \beta - 687) q^{93} + (198 \beta + 514) q^{94} + ( - 344 \beta + 414) q^{95} + (32 \beta + 32) q^{96} + ( - 100 \beta - 184) q^{97} + (256 \beta + 210) q^{98} + (30 \beta - 864) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} + 16 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} + 16 q^{8} - 40 q^{9} - 20 q^{10} + 90 q^{11} + 8 q^{12} - 50 q^{13} - 32 q^{14} + 62 q^{15} + 32 q^{16} + 44 q^{17} - 80 q^{18} - 108 q^{19} - 40 q^{20} - 112 q^{21} + 180 q^{22} - 28 q^{23} + 16 q^{24} + 232 q^{25} - 100 q^{26} - 70 q^{27} - 64 q^{28} + 124 q^{30} - 66 q^{31} + 64 q^{32} + 126 q^{33} + 88 q^{34} - 496 q^{35} - 160 q^{36} - 40 q^{37} - 216 q^{38} + 46 q^{39} - 80 q^{40} - 304 q^{41} - 224 q^{42} + 130 q^{43} + 360 q^{44} + 344 q^{45} - 56 q^{46} + 514 q^{47} + 32 q^{48} + 210 q^{49} + 464 q^{50} - 148 q^{51} - 200 q^{52} - 958 q^{53} - 140 q^{54} - 234 q^{55} - 128 q^{56} - 60 q^{57} - 180 q^{59} + 248 q^{60} - 1028 q^{61} - 132 q^{62} + 128 q^{63} + 128 q^{64} + 826 q^{65} + 252 q^{66} - 912 q^{67} + 176 q^{68} - 964 q^{69} - 992 q^{70} + 796 q^{71} - 320 q^{72} + 856 q^{73} - 80 q^{74} - 488 q^{75} - 432 q^{76} - 1008 q^{77} + 92 q^{78} + 318 q^{79} - 160 q^{80} + 470 q^{81} - 608 q^{82} - 1828 q^{83} - 448 q^{84} - 1372 q^{85} + 260 q^{86} + 720 q^{88} + 944 q^{89} + 688 q^{90} - 368 q^{91} - 112 q^{92} - 1374 q^{93} + 1028 q^{94} + 828 q^{95} + 64 q^{96} - 368 q^{97} + 420 q^{98} - 1728 q^{99}+O(q^{100}) \) 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
−2.44949
2.44949
2.00000 −1.44949 4.00000 −19.6969 −2.89898 11.5959 8.00000 −24.8990 −39.3939
1.2 2.00000 3.44949 4.00000 9.69694 6.89898 −27.5959 8.00000 −15.1010 19.3939
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(1\)

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 1682.4.a.c 2
29.b even 2 1 58.4.a.c 2
87.d odd 2 1 522.4.a.j 2
116.d odd 2 1 464.4.a.e 2
145.d even 2 1 1450.4.a.g 2
232.b odd 2 1 1856.4.a.i 2
232.g even 2 1 1856.4.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.c 2 29.b even 2 1
464.4.a.e 2 116.d odd 2 1
522.4.a.j 2 87.d odd 2 1
1450.4.a.g 2 145.d even 2 1
1682.4.a.c 2 1.a even 1 1 trivial
1856.4.a.i 2 232.b odd 2 1
1856.4.a.l 2 232.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1682))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 5 \) Copy content Toggle raw display
$5$ \( T^{2} + 10T - 191 \) Copy content Toggle raw display
$7$ \( T^{2} + 16T - 320 \) Copy content Toggle raw display
$11$ \( T^{2} - 90T + 1971 \) Copy content Toggle raw display
$13$ \( T^{2} + 50T + 241 \) Copy content Toggle raw display
$17$ \( T^{2} - 44T - 1052 \) Copy content Toggle raw display
$19$ \( T^{2} + 108T + 2820 \) Copy content Toggle raw display
$23$ \( T^{2} + 28T - 36308 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 66T - 70197 \) Copy content Toggle raw display
$37$ \( T^{2} + 40T + 304 \) Copy content Toggle raw display
$41$ \( T^{2} + 304T - 15296 \) Copy content Toggle raw display
$43$ \( T^{2} - 130T - 12629 \) Copy content Toggle raw display
$47$ \( T^{2} - 514T + 7243 \) Copy content Toggle raw display
$53$ \( T^{2} + 958T + 213217 \) Copy content Toggle raw display
$59$ \( T^{2} + 180T - 366900 \) Copy content Toggle raw display
$61$ \( T^{2} + 1028 T + 263332 \) Copy content Toggle raw display
$67$ \( T^{2} + 912T + 205536 \) Copy content Toggle raw display
$71$ \( T^{2} - 796T + 151468 \) Copy content Toggle raw display
$73$ \( T^{2} - 856T - 2672 \) Copy content Toggle raw display
$79$ \( T^{2} - 318T - 756645 \) Copy content Toggle raw display
$83$ \( T^{2} + 1828 T + 826732 \) Copy content Toggle raw display
$89$ \( T^{2} - 944T + 191680 \) Copy content Toggle raw display
$97$ \( T^{2} + 368T - 26144 \) Copy content Toggle raw display
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