Properties

Label 105.4.a.d
Level $105$
Weight $4$
Character orbit 105.a
Self dual yes
Analytic conductor $6.195$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.19520055060\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - \beta ) q^{2} + 3 q^{3} + ( 1 + 4 \beta ) q^{4} -5 q^{5} + ( -6 - 3 \beta ) q^{6} -7 q^{7} + ( -6 - \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( -2 - \beta ) q^{2} + 3 q^{3} + ( 1 + 4 \beta ) q^{4} -5 q^{5} + ( -6 - 3 \beta ) q^{6} -7 q^{7} + ( -6 - \beta ) q^{8} + 9 q^{9} + ( 10 + 5 \beta ) q^{10} + ( -46 + 2 \beta ) q^{11} + ( 3 + 12 \beta ) q^{12} + ( 4 + 38 \beta ) q^{13} + ( 14 + 7 \beta ) q^{14} -15 q^{15} + ( 9 - 24 \beta ) q^{16} + ( -22 - 44 \beta ) q^{17} + ( -18 - 9 \beta ) q^{18} + ( -54 - 26 \beta ) q^{19} + ( -5 - 20 \beta ) q^{20} -21 q^{21} + ( 82 + 42 \beta ) q^{22} + ( -160 + 20 \beta ) q^{23} + ( -18 - 3 \beta ) q^{24} + 25 q^{25} + ( -198 - 80 \beta ) q^{26} + 27 q^{27} + ( -7 - 28 \beta ) q^{28} + ( -118 - 12 \beta ) q^{29} + ( 30 + 15 \beta ) q^{30} + ( -30 - 102 \beta ) q^{31} + ( 150 + 47 \beta ) q^{32} + ( -138 + 6 \beta ) q^{33} + ( 264 + 110 \beta ) q^{34} + 35 q^{35} + ( 9 + 36 \beta ) q^{36} + ( 102 - 24 \beta ) q^{37} + ( 238 + 106 \beta ) q^{38} + ( 12 + 114 \beta ) q^{39} + ( 30 + 5 \beta ) q^{40} + ( 22 + 80 \beta ) q^{41} + ( 42 + 21 \beta ) q^{42} + ( 68 - 128 \beta ) q^{43} + ( -6 - 182 \beta ) q^{44} -45 q^{45} + ( 220 + 120 \beta ) q^{46} + ( 200 + 168 \beta ) q^{47} + ( 27 - 72 \beta ) q^{48} + 49 q^{49} + ( -50 - 25 \beta ) q^{50} + ( -66 - 132 \beta ) q^{51} + ( 764 + 54 \beta ) q^{52} + ( 8 - 86 \beta ) q^{53} + ( -54 - 27 \beta ) q^{54} + ( 230 - 10 \beta ) q^{55} + ( 42 + 7 \beta ) q^{56} + ( -162 - 78 \beta ) q^{57} + ( 296 + 142 \beta ) q^{58} + ( -232 + 36 \beta ) q^{59} + ( -15 - 60 \beta ) q^{60} + ( -342 - 84 \beta ) q^{61} + ( 570 + 234 \beta ) q^{62} -63 q^{63} + ( -607 - 52 \beta ) q^{64} + ( -20 - 190 \beta ) q^{65} + ( 246 + 126 \beta ) q^{66} + ( 368 - 164 \beta ) q^{67} + ( -902 - 132 \beta ) q^{68} + ( -480 + 60 \beta ) q^{69} + ( -70 - 35 \beta ) q^{70} + ( -370 + 138 \beta ) q^{71} + ( -54 - 9 \beta ) q^{72} + ( 212 + 122 \beta ) q^{73} + ( -84 - 54 \beta ) q^{74} + 75 q^{75} + ( -574 - 242 \beta ) q^{76} + ( 322 - 14 \beta ) q^{77} + ( -594 - 240 \beta ) q^{78} + ( -204 + 484 \beta ) q^{79} + ( -45 + 120 \beta ) q^{80} + 81 q^{81} + ( -444 - 182 \beta ) q^{82} + ( 304 + 84 \beta ) q^{83} + ( -21 - 84 \beta ) q^{84} + ( 110 + 220 \beta ) q^{85} + ( 504 + 188 \beta ) q^{86} + ( -354 - 36 \beta ) q^{87} + ( 266 + 34 \beta ) q^{88} + ( -666 + 112 \beta ) q^{89} + ( 90 + 45 \beta ) q^{90} + ( -28 - 266 \beta ) q^{91} + ( 240 - 620 \beta ) q^{92} + ( -90 - 306 \beta ) q^{93} + ( -1240 - 536 \beta ) q^{94} + ( 270 + 130 \beta ) q^{95} + ( 450 + 141 \beta ) q^{96} + ( -1224 + 86 \beta ) q^{97} + ( -98 - 49 \beta ) q^{98} + ( -414 + 18 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 6q^{3} + 2q^{4} - 10q^{5} - 12q^{6} - 14q^{7} - 12q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 4q^{2} + 6q^{3} + 2q^{4} - 10q^{5} - 12q^{6} - 14q^{7} - 12q^{8} + 18q^{9} + 20q^{10} - 92q^{11} + 6q^{12} + 8q^{13} + 28q^{14} - 30q^{15} + 18q^{16} - 44q^{17} - 36q^{18} - 108q^{19} - 10q^{20} - 42q^{21} + 164q^{22} - 320q^{23} - 36q^{24} + 50q^{25} - 396q^{26} + 54q^{27} - 14q^{28} - 236q^{29} + 60q^{30} - 60q^{31} + 300q^{32} - 276q^{33} + 528q^{34} + 70q^{35} + 18q^{36} + 204q^{37} + 476q^{38} + 24q^{39} + 60q^{40} + 44q^{41} + 84q^{42} + 136q^{43} - 12q^{44} - 90q^{45} + 440q^{46} + 400q^{47} + 54q^{48} + 98q^{49} - 100q^{50} - 132q^{51} + 1528q^{52} + 16q^{53} - 108q^{54} + 460q^{55} + 84q^{56} - 324q^{57} + 592q^{58} - 464q^{59} - 30q^{60} - 684q^{61} + 1140q^{62} - 126q^{63} - 1214q^{64} - 40q^{65} + 492q^{66} + 736q^{67} - 1804q^{68} - 960q^{69} - 140q^{70} - 740q^{71} - 108q^{72} + 424q^{73} - 168q^{74} + 150q^{75} - 1148q^{76} + 644q^{77} - 1188q^{78} - 408q^{79} - 90q^{80} + 162q^{81} - 888q^{82} + 608q^{83} - 42q^{84} + 220q^{85} + 1008q^{86} - 708q^{87} + 532q^{88} - 1332q^{89} + 180q^{90} - 56q^{91} + 480q^{92} - 180q^{93} - 2480q^{94} + 540q^{95} + 900q^{96} - 2448q^{97} - 196q^{98} - 828q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
−4.23607 3.00000 9.94427 −5.00000 −12.7082 −7.00000 −8.23607 9.00000 21.1803
1.2 0.236068 3.00000 −7.94427 −5.00000 0.708204 −7.00000 −3.76393 9.00000 −1.18034
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

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 105.4.a.d 2
3.b odd 2 1 315.4.a.l 2
4.b odd 2 1 1680.4.a.bd 2
5.b even 2 1 525.4.a.o 2
5.c odd 4 2 525.4.d.k 4
7.b odd 2 1 735.4.a.m 2
15.d odd 2 1 1575.4.a.n 2
21.c even 2 1 2205.4.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.d 2 1.a even 1 1 trivial
315.4.a.l 2 3.b odd 2 1
525.4.a.o 2 5.b even 2 1
525.4.d.k 4 5.c odd 4 2
735.4.a.m 2 7.b odd 2 1
1575.4.a.n 2 15.d odd 2 1
1680.4.a.bd 2 4.b odd 2 1
2205.4.a.be 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4 T_{2} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 4 T + T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( ( 5 + T )^{2} \)
$7$ \( ( 7 + T )^{2} \)
$11$ \( 2096 + 92 T + T^{2} \)
$13$ \( -7204 - 8 T + T^{2} \)
$17$ \( -9196 + 44 T + T^{2} \)
$19$ \( -464 + 108 T + T^{2} \)
$23$ \( 23600 + 320 T + T^{2} \)
$29$ \( 13204 + 236 T + T^{2} \)
$31$ \( -51120 + 60 T + T^{2} \)
$37$ \( 7524 - 204 T + T^{2} \)
$41$ \( -31516 - 44 T + T^{2} \)
$43$ \( -77296 - 136 T + T^{2} \)
$47$ \( -101120 - 400 T + T^{2} \)
$53$ \( -36916 - 16 T + T^{2} \)
$59$ \( 47344 + 464 T + T^{2} \)
$61$ \( 81684 + 684 T + T^{2} \)
$67$ \( 944 - 736 T + T^{2} \)
$71$ \( 41680 + 740 T + T^{2} \)
$73$ \( -29476 - 424 T + T^{2} \)
$79$ \( -1129664 + 408 T + T^{2} \)
$83$ \( 57136 - 608 T + T^{2} \)
$89$ \( 380836 + 1332 T + T^{2} \)
$97$ \( 1461196 + 2448 T + T^{2} \)
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