Properties

Label 46.4.a.d
Level $46$
Weight $4$
Character orbit 46.a
Self dual yes
Analytic conductor $2.714$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 46 = 2 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 46.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.71408786026\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
Defining polynomial: \( x^{2} - x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta + 2) q^{3} + 4 q^{4} + (2 \beta + 4) q^{5} + ( - 2 \beta + 4) q^{6} + (4 \beta + 4) q^{7} + 8 q^{8} + ( - 3 \beta - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta + 2) q^{3} + 4 q^{4} + (2 \beta + 4) q^{5} + ( - 2 \beta + 4) q^{6} + (4 \beta + 4) q^{7} + 8 q^{8} + ( - 3 \beta - 5) q^{9} + (4 \beta + 8) q^{10} - 6 q^{11} + ( - 4 \beta + 8) q^{12} + ( - 11 \beta - 20) q^{13} + (8 \beta + 8) q^{14} + ( - 2 \beta - 28) q^{15} + 16 q^{16} + ( - 6 \beta - 30) q^{17} + ( - 6 \beta - 10) q^{18} + (20 \beta - 18) q^{19} + (8 \beta + 16) q^{20} - 64 q^{21} - 12 q^{22} + 23 q^{23} + ( - 8 \beta + 16) q^{24} + (20 \beta - 37) q^{25} + ( - 22 \beta - 40) q^{26} + (29 \beta - 10) q^{27} + (16 \beta + 16) q^{28} + ( - 33 \beta - 12) q^{29} + ( - 4 \beta - 56) q^{30} + ( - 69 \beta + 26) q^{31} + 32 q^{32} + (6 \beta - 12) q^{33} + ( - 12 \beta - 60) q^{34} + (32 \beta + 160) q^{35} + ( - 12 \beta - 20) q^{36} + (38 \beta + 84) q^{37} + (40 \beta - 36) q^{38} + (9 \beta + 158) q^{39} + (16 \beta + 32) q^{40} + (29 \beta + 172) q^{41} - 128 q^{42} + ( - 82 \beta + 198) q^{43} - 24 q^{44} + ( - 28 \beta - 128) q^{45} + 46 q^{46} + ( - 25 \beta + 442) q^{47} + ( - 16 \beta + 32) q^{48} + (48 \beta - 39) q^{49} + (40 \beta - 74) q^{50} + (24 \beta + 48) q^{51} + ( - 44 \beta - 80) q^{52} + ( - 38 \beta + 44) q^{53} + (58 \beta - 20) q^{54} + ( - 12 \beta - 24) q^{55} + (32 \beta + 32) q^{56} + (38 \beta - 396) q^{57} + ( - 66 \beta - 24) q^{58} + (60 \beta + 276) q^{59} + ( - 8 \beta - 112) q^{60} + (58 \beta - 560) q^{61} + ( - 138 \beta + 52) q^{62} + ( - 44 \beta - 236) q^{63} + 64 q^{64} + ( - 106 \beta - 476) q^{65} + (12 \beta - 24) q^{66} + (56 \beta - 450) q^{67} + ( - 24 \beta - 120) q^{68} + ( - 23 \beta + 46) q^{69} + (64 \beta + 320) q^{70} + ( - 21 \beta + 210) q^{71} + ( - 24 \beta - 40) q^{72} + (3 \beta - 640) q^{73} + (76 \beta + 168) q^{74} + (57 \beta - 434) q^{75} + (80 \beta - 72) q^{76} + ( - 24 \beta - 24) q^{77} + (18 \beta + 316) q^{78} + (26 \beta + 48) q^{79} + (32 \beta + 64) q^{80} + (120 \beta - 407) q^{81} + (58 \beta + 344) q^{82} + (22 \beta + 890) q^{83} - 256 q^{84} + ( - 96 \beta - 336) q^{85} + ( - 164 \beta + 396) q^{86} + ( - 21 \beta + 570) q^{87} - 48 q^{88} + (18 \beta + 1014) q^{89} + ( - 56 \beta - 256) q^{90} + ( - 168 \beta - 872) q^{91} + 92 q^{92} + ( - 95 \beta + 1294) q^{93} + ( - 50 \beta + 884) q^{94} + (84 \beta + 648) q^{95} + ( - 32 \beta + 64) q^{96} + ( - 274 \beta - 318) q^{97} + (96 \beta - 78) q^{98} + (18 \beta + 30) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 3 q^{3} + 8 q^{4} + 10 q^{5} + 6 q^{6} + 12 q^{7} + 16 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 3 q^{3} + 8 q^{4} + 10 q^{5} + 6 q^{6} + 12 q^{7} + 16 q^{8} - 13 q^{9} + 20 q^{10} - 12 q^{11} + 12 q^{12} - 51 q^{13} + 24 q^{14} - 58 q^{15} + 32 q^{16} - 66 q^{17} - 26 q^{18} - 16 q^{19} + 40 q^{20} - 128 q^{21} - 24 q^{22} + 46 q^{23} + 24 q^{24} - 54 q^{25} - 102 q^{26} + 9 q^{27} + 48 q^{28} - 57 q^{29} - 116 q^{30} - 17 q^{31} + 64 q^{32} - 18 q^{33} - 132 q^{34} + 352 q^{35} - 52 q^{36} + 206 q^{37} - 32 q^{38} + 325 q^{39} + 80 q^{40} + 373 q^{41} - 256 q^{42} + 314 q^{43} - 48 q^{44} - 284 q^{45} + 92 q^{46} + 859 q^{47} + 48 q^{48} - 30 q^{49} - 108 q^{50} + 120 q^{51} - 204 q^{52} + 50 q^{53} + 18 q^{54} - 60 q^{55} + 96 q^{56} - 754 q^{57} - 114 q^{58} + 612 q^{59} - 232 q^{60} - 1062 q^{61} - 34 q^{62} - 516 q^{63} + 128 q^{64} - 1058 q^{65} - 36 q^{66} - 844 q^{67} - 264 q^{68} + 69 q^{69} + 704 q^{70} + 399 q^{71} - 104 q^{72} - 1277 q^{73} + 412 q^{74} - 811 q^{75} - 64 q^{76} - 72 q^{77} + 650 q^{78} + 122 q^{79} + 160 q^{80} - 694 q^{81} + 746 q^{82} + 1802 q^{83} - 512 q^{84} - 768 q^{85} + 628 q^{86} + 1119 q^{87} - 96 q^{88} + 2046 q^{89} - 568 q^{90} - 1912 q^{91} + 184 q^{92} + 2493 q^{93} + 1718 q^{94} + 1380 q^{95} + 96 q^{96} - 910 q^{97} - 60 q^{98} + 78 q^{99}+O(q^{100}) \) 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
4.77200
−3.77200
2.00000 −2.77200 4.00000 13.5440 −5.54400 23.0880 8.00000 −19.3160 27.0880
1.2 2.00000 5.77200 4.00000 −3.54400 11.5440 −11.0880 8.00000 6.31601 −7.08801
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 46.4.a.d 2
3.b odd 2 1 414.4.a.f 2
4.b odd 2 1 368.4.a.f 2
5.b even 2 1 1150.4.a.j 2
5.c odd 4 2 1150.4.b.j 4
7.b odd 2 1 2254.4.a.f 2
8.b even 2 1 1472.4.a.k 2
8.d odd 2 1 1472.4.a.n 2
23.b odd 2 1 1058.4.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.4.a.d 2 1.a even 1 1 trivial
368.4.a.f 2 4.b odd 2 1
414.4.a.f 2 3.b odd 2 1
1058.4.a.j 2 23.b odd 2 1
1150.4.a.j 2 5.b even 2 1
1150.4.b.j 4 5.c odd 4 2
1472.4.a.k 2 8.b even 2 1
1472.4.a.n 2 8.d odd 2 1
2254.4.a.f 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3T_{3} - 16 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(46))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T - 16 \) Copy content Toggle raw display
$5$ \( T^{2} - 10T - 48 \) Copy content Toggle raw display
$7$ \( T^{2} - 12T - 256 \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 51T - 1558 \) Copy content Toggle raw display
$17$ \( T^{2} + 66T + 432 \) Copy content Toggle raw display
$19$ \( T^{2} + 16T - 7236 \) Copy content Toggle raw display
$23$ \( (T - 23)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 57T - 19062 \) Copy content Toggle raw display
$31$ \( T^{2} + 17T - 86816 \) Copy content Toggle raw display
$37$ \( T^{2} - 206T - 15744 \) Copy content Toggle raw display
$41$ \( T^{2} - 373T + 19434 \) Copy content Toggle raw display
$43$ \( T^{2} - 314T - 98064 \) Copy content Toggle raw display
$47$ \( T^{2} - 859T + 173064 \) Copy content Toggle raw display
$53$ \( T^{2} - 50T - 25728 \) Copy content Toggle raw display
$59$ \( T^{2} - 612T + 27936 \) Copy content Toggle raw display
$61$ \( T^{2} + 1062 T + 220568 \) Copy content Toggle raw display
$67$ \( T^{2} + 844T + 120852 \) Copy content Toggle raw display
$71$ \( T^{2} - 399T + 31752 \) Copy content Toggle raw display
$73$ \( T^{2} + 1277 T + 407518 \) Copy content Toggle raw display
$79$ \( T^{2} - 122T - 8616 \) Copy content Toggle raw display
$83$ \( T^{2} - 1802 T + 802968 \) Copy content Toggle raw display
$89$ \( T^{2} - 2046 T + 1040616 \) Copy content Toggle raw display
$97$ \( T^{2} + 910 T - 1163112 \) Copy content Toggle raw display
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