Properties

Label 966.4.a.c
Level $966$
Weight $4$
Character orbit 966.a
Self dual yes
Analytic conductor $56.996$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.9958450655\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta - 7) q^{5} + 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta - 7) q^{5} + 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + ( - 2 \beta - 14) q^{10} + ( - 8 \beta - 11) q^{11} + 12 q^{12} + (9 \beta - 56) q^{13} + 14 q^{14} + ( - 3 \beta - 21) q^{15} + 16 q^{16} + (4 \beta - 39) q^{17} + 18 q^{18} + (42 \beta - 53) q^{19} + ( - 4 \beta - 28) q^{20} + 21 q^{21} + ( - 16 \beta - 22) q^{22} - 23 q^{23} + 24 q^{24} + (15 \beta - 73) q^{25} + (18 \beta - 112) q^{26} + 27 q^{27} + 28 q^{28} + (96 \beta - 121) q^{29} + ( - 6 \beta - 42) q^{30} + ( - 70 \beta - 29) q^{31} + 32 q^{32} + ( - 24 \beta - 33) q^{33} + (8 \beta - 78) q^{34} + ( - 7 \beta - 49) q^{35} + 36 q^{36} + ( - 142 \beta + 15) q^{37} + (84 \beta - 106) q^{38} + (27 \beta - 168) q^{39} + ( - 8 \beta - 56) q^{40} + ( - 132 \beta - 33) q^{41} + 42 q^{42} + ( - 91 \beta + 86) q^{43} + ( - 32 \beta - 44) q^{44} + ( - 9 \beta - 63) q^{45} - 46 q^{46} + (106 \beta - 124) q^{47} + 48 q^{48} + 49 q^{49} + (30 \beta - 146) q^{50} + (12 \beta - 117) q^{51} + (36 \beta - 224) q^{52} + (137 \beta + 15) q^{53} + 54 q^{54} + (75 \beta + 101) q^{55} + 56 q^{56} + (126 \beta - 159) q^{57} + (192 \beta - 242) q^{58} + ( - 71 \beta - 7) q^{59} + ( - 12 \beta - 84) q^{60} + (41 \beta - 370) q^{61} + ( - 140 \beta - 58) q^{62} + 63 q^{63} + 64 q^{64} + ( - 16 \beta + 365) q^{65} + ( - 48 \beta - 66) q^{66} + ( - 347 \beta + 127) q^{67} + (16 \beta - 156) q^{68} - 69 q^{69} + ( - 14 \beta - 98) q^{70} + ( - 509 \beta + 512) q^{71} + 72 q^{72} + (12 \beta - 645) q^{73} + ( - 284 \beta + 30) q^{74} + (45 \beta - 219) q^{75} + (168 \beta - 212) q^{76} + ( - 56 \beta - 77) q^{77} + (54 \beta - 336) q^{78} + (564 \beta - 129) q^{79} + ( - 16 \beta - 112) q^{80} + 81 q^{81} + ( - 264 \beta - 66) q^{82} + (698 \beta - 369) q^{83} + 84 q^{84} + (7 \beta + 261) q^{85} + ( - 182 \beta + 172) q^{86} + (288 \beta - 363) q^{87} + ( - 64 \beta - 88) q^{88} + (647 \beta - 638) q^{89} + ( - 18 \beta - 126) q^{90} + (63 \beta - 392) q^{91} - 92 q^{92} + ( - 210 \beta - 87) q^{93} + (212 \beta - 248) q^{94} + ( - 283 \beta + 245) q^{95} + 96 q^{96} + ( - 280 \beta - 617) q^{97} + 98 q^{98} + ( - 72 \beta - 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 15 q^{5} + 12 q^{6} + 14 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 15 q^{5} + 12 q^{6} + 14 q^{7} + 16 q^{8} + 18 q^{9} - 30 q^{10} - 30 q^{11} + 24 q^{12} - 103 q^{13} + 28 q^{14} - 45 q^{15} + 32 q^{16} - 74 q^{17} + 36 q^{18} - 64 q^{19} - 60 q^{20} + 42 q^{21} - 60 q^{22} - 46 q^{23} + 48 q^{24} - 131 q^{25} - 206 q^{26} + 54 q^{27} + 56 q^{28} - 146 q^{29} - 90 q^{30} - 128 q^{31} + 64 q^{32} - 90 q^{33} - 148 q^{34} - 105 q^{35} + 72 q^{36} - 112 q^{37} - 128 q^{38} - 309 q^{39} - 120 q^{40} - 198 q^{41} + 84 q^{42} + 81 q^{43} - 120 q^{44} - 135 q^{45} - 92 q^{46} - 142 q^{47} + 96 q^{48} + 98 q^{49} - 262 q^{50} - 222 q^{51} - 412 q^{52} + 167 q^{53} + 108 q^{54} + 277 q^{55} + 112 q^{56} - 192 q^{57} - 292 q^{58} - 85 q^{59} - 180 q^{60} - 699 q^{61} - 256 q^{62} + 126 q^{63} + 128 q^{64} + 714 q^{65} - 180 q^{66} - 93 q^{67} - 296 q^{68} - 138 q^{69} - 210 q^{70} + 515 q^{71} + 144 q^{72} - 1278 q^{73} - 224 q^{74} - 393 q^{75} - 256 q^{76} - 210 q^{77} - 618 q^{78} + 306 q^{79} - 240 q^{80} + 162 q^{81} - 396 q^{82} - 40 q^{83} + 168 q^{84} + 529 q^{85} + 162 q^{86} - 438 q^{87} - 240 q^{88} - 629 q^{89} - 270 q^{90} - 721 q^{91} - 184 q^{92} - 384 q^{93} - 284 q^{94} + 207 q^{95} + 192 q^{96} - 1514 q^{97} + 196 q^{98} - 270 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
2.00000 3.00000 4.00000 −9.30278 6.00000 7.00000 8.00000 9.00000 −18.6056
1.2 2.00000 3.00000 4.00000 −5.69722 6.00000 7.00000 8.00000 9.00000 −11.3944
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(23\) \(1\)

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 966.4.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.c 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 15T_{5} + 53 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(966))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 15T + 53 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 30T + 17 \) Copy content Toggle raw display
$13$ \( T^{2} + 103T + 2389 \) Copy content Toggle raw display
$17$ \( T^{2} + 74T + 1317 \) Copy content Toggle raw display
$19$ \( T^{2} + 64T - 4709 \) Copy content Toggle raw display
$23$ \( (T + 23)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 146T - 24623 \) Copy content Toggle raw display
$31$ \( T^{2} + 128T - 11829 \) Copy content Toggle raw display
$37$ \( T^{2} + 112T - 62397 \) Copy content Toggle raw display
$41$ \( T^{2} + 198T - 46827 \) Copy content Toggle raw display
$43$ \( T^{2} - 81T - 25273 \) Copy content Toggle raw display
$47$ \( T^{2} + 142T - 31476 \) Copy content Toggle raw display
$53$ \( T^{2} - 167T - 54027 \) Copy content Toggle raw display
$59$ \( T^{2} + 85T - 14577 \) Copy content Toggle raw display
$61$ \( T^{2} + 699T + 116687 \) Copy content Toggle raw display
$67$ \( T^{2} + 93T - 389167 \) Copy content Toggle raw display
$71$ \( T^{2} - 515T - 775707 \) Copy content Toggle raw display
$73$ \( T^{2} + 1278 T + 407853 \) Copy content Toggle raw display
$79$ \( T^{2} - 306 T - 1010403 \) Copy content Toggle raw display
$83$ \( T^{2} + 40T - 1583013 \) Copy content Toggle raw display
$89$ \( T^{2} + 629 T - 1261569 \) Copy content Toggle raw display
$97$ \( T^{2} + 1514 T + 318249 \) Copy content Toggle raw display
show more
show less