Properties

Label 105.4.a.e
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} - 3 q^{3} + ( - 2 \beta + 1) q^{4} + 5 q^{5} + ( - 3 \beta + 3) q^{6} - 7 q^{7} + ( - 5 \beta - 9) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} - 3 q^{3} + ( - 2 \beta + 1) q^{4} + 5 q^{5} + ( - 3 \beta + 3) q^{6} - 7 q^{7} + ( - 5 \beta - 9) q^{8} + 9 q^{9} + (5 \beta - 5) q^{10} + ( - 20 \beta - 8) q^{11} + (6 \beta - 3) q^{12} + (2 \beta - 38) q^{13} + ( - 7 \beta + 7) q^{14} - 15 q^{15} + (12 \beta - 39) q^{16} + (2 \beta - 62) q^{17} + (9 \beta - 9) q^{18} + (16 \beta - 48) q^{19} + ( - 10 \beta + 5) q^{20} + 21 q^{21} + (12 \beta - 152) q^{22} + (34 \beta - 8) q^{23} + (15 \beta + 27) q^{24} + 25 q^{25} + ( - 40 \beta + 54) q^{26} - 27 q^{27} + (14 \beta - 7) q^{28} + (54 \beta + 94) q^{29} + ( - 15 \beta + 15) q^{30} + ( - 18 \beta - 60) q^{31} + ( - 11 \beta + 207) q^{32} + (60 \beta + 24) q^{33} + ( - 64 \beta + 78) q^{34} - 35 q^{35} + ( - 18 \beta + 9) q^{36} + (66 \beta - 66) q^{37} + ( - 64 \beta + 176) q^{38} + ( - 6 \beta + 114) q^{39} + ( - 25 \beta - 45) q^{40} + ( - 80 \beta + 50) q^{41} + (21 \beta - 21) q^{42} + ( - 62 \beta - 268) q^{43} + ( - 4 \beta + 312) q^{44} + 45 q^{45} + ( - 42 \beta + 280) q^{46} + (42 \beta - 464) q^{47} + ( - 36 \beta + 117) q^{48} + 49 q^{49} + (25 \beta - 25) q^{50} + ( - 6 \beta + 186) q^{51} + (78 \beta - 70) q^{52} + ( - 64 \beta + 442) q^{53} + ( - 27 \beta + 27) q^{54} + ( - 100 \beta - 40) q^{55} + (35 \beta + 63) q^{56} + ( - 48 \beta + 144) q^{57} + (40 \beta + 338) q^{58} + (204 \beta + 52) q^{59} + (30 \beta - 15) q^{60} + (42 \beta - 234) q^{61} + ( - 42 \beta - 84) q^{62} - 63 q^{63} + (122 \beta + 17) q^{64} + (10 \beta - 190) q^{65} + ( - 36 \beta + 456) q^{66} + ( - 38 \beta - 844) q^{67} + (126 \beta - 94) q^{68} + ( - 102 \beta + 24) q^{69} + ( - 35 \beta + 35) q^{70} + (150 \beta - 68) q^{71} + ( - 45 \beta - 81) q^{72} + ( - 322 \beta + 254) q^{73} + ( - 132 \beta + 594) q^{74} - 75 q^{75} + (112 \beta - 304) q^{76} + (140 \beta + 56) q^{77} + (120 \beta - 162) q^{78} + (232 \beta - 216) q^{79} + (60 \beta - 195) q^{80} + 81 q^{81} + (130 \beta - 690) q^{82} + (84 \beta - 292) q^{83} + ( - 42 \beta + 21) q^{84} + (10 \beta - 310) q^{85} + ( - 206 \beta - 228) q^{86} + ( - 162 \beta - 282) q^{87} + (220 \beta + 872) q^{88} + ( - 112 \beta - 702) q^{89} + (45 \beta - 45) q^{90} + ( - 14 \beta + 266) q^{91} + (50 \beta - 552) q^{92} + (54 \beta + 180) q^{93} + ( - 506 \beta + 800) q^{94} + (80 \beta - 240) q^{95} + (33 \beta - 621) q^{96} + ( - 46 \beta - 594) q^{97} + (49 \beta - 49) q^{98} + ( - 180 \beta - 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 6 q^{3} + 2 q^{4} + 10 q^{5} + 6 q^{6} - 14 q^{7} - 18 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 6 q^{3} + 2 q^{4} + 10 q^{5} + 6 q^{6} - 14 q^{7} - 18 q^{8} + 18 q^{9} - 10 q^{10} - 16 q^{11} - 6 q^{12} - 76 q^{13} + 14 q^{14} - 30 q^{15} - 78 q^{16} - 124 q^{17} - 18 q^{18} - 96 q^{19} + 10 q^{20} + 42 q^{21} - 304 q^{22} - 16 q^{23} + 54 q^{24} + 50 q^{25} + 108 q^{26} - 54 q^{27} - 14 q^{28} + 188 q^{29} + 30 q^{30} - 120 q^{31} + 414 q^{32} + 48 q^{33} + 156 q^{34} - 70 q^{35} + 18 q^{36} - 132 q^{37} + 352 q^{38} + 228 q^{39} - 90 q^{40} + 100 q^{41} - 42 q^{42} - 536 q^{43} + 624 q^{44} + 90 q^{45} + 560 q^{46} - 928 q^{47} + 234 q^{48} + 98 q^{49} - 50 q^{50} + 372 q^{51} - 140 q^{52} + 884 q^{53} + 54 q^{54} - 80 q^{55} + 126 q^{56} + 288 q^{57} + 676 q^{58} + 104 q^{59} - 30 q^{60} - 468 q^{61} - 168 q^{62} - 126 q^{63} + 34 q^{64} - 380 q^{65} + 912 q^{66} - 1688 q^{67} - 188 q^{68} + 48 q^{69} + 70 q^{70} - 136 q^{71} - 162 q^{72} + 508 q^{73} + 1188 q^{74} - 150 q^{75} - 608 q^{76} + 112 q^{77} - 324 q^{78} - 432 q^{79} - 390 q^{80} + 162 q^{81} - 1380 q^{82} - 584 q^{83} + 42 q^{84} - 620 q^{85} - 456 q^{86} - 564 q^{87} + 1744 q^{88} - 1404 q^{89} - 90 q^{90} + 532 q^{91} - 1104 q^{92} + 360 q^{93} + 1600 q^{94} - 480 q^{95} - 1242 q^{96} - 1188 q^{97} - 98 q^{98} - 144 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−3.82843 −3.00000 6.65685 5.00000 11.4853 −7.00000 5.14214 9.00000 −19.1421
1.2 1.82843 −3.00000 −4.65685 5.00000 −5.48528 −7.00000 −23.1421 9.00000 9.14214
\(n\): 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.e 2
3.b odd 2 1 315.4.a.k 2
4.b odd 2 1 1680.4.a.bo 2
5.b even 2 1 525.4.a.l 2
5.c odd 4 2 525.4.d.l 4
7.b odd 2 1 735.4.a.o 2
15.d odd 2 1 1575.4.a.q 2
21.c even 2 1 2205.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 1.a even 1 1 trivial
315.4.a.k 2 3.b odd 2 1
525.4.a.l 2 5.b even 2 1
525.4.d.l 4 5.c odd 4 2
735.4.a.o 2 7.b odd 2 1
1575.4.a.q 2 15.d odd 2 1
1680.4.a.bo 2 4.b odd 2 1
2205.4.a.bb 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 7 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16T - 3136 \) Copy content Toggle raw display
$13$ \( T^{2} + 76T + 1412 \) Copy content Toggle raw display
$17$ \( T^{2} + 124T + 3812 \) Copy content Toggle raw display
$19$ \( T^{2} + 96T + 256 \) Copy content Toggle raw display
$23$ \( T^{2} + 16T - 9184 \) Copy content Toggle raw display
$29$ \( T^{2} - 188T - 14492 \) Copy content Toggle raw display
$31$ \( T^{2} + 120T + 1008 \) Copy content Toggle raw display
$37$ \( T^{2} + 132T - 30492 \) Copy content Toggle raw display
$41$ \( T^{2} - 100T - 48700 \) Copy content Toggle raw display
$43$ \( T^{2} + 536T + 41072 \) Copy content Toggle raw display
$47$ \( T^{2} + 928T + 201184 \) Copy content Toggle raw display
$53$ \( T^{2} - 884T + 162596 \) Copy content Toggle raw display
$59$ \( T^{2} - 104T - 330224 \) Copy content Toggle raw display
$61$ \( T^{2} + 468T + 40644 \) Copy content Toggle raw display
$67$ \( T^{2} + 1688 T + 700784 \) Copy content Toggle raw display
$71$ \( T^{2} + 136T - 175376 \) Copy content Toggle raw display
$73$ \( T^{2} - 508T - 764956 \) Copy content Toggle raw display
$79$ \( T^{2} + 432T - 383936 \) Copy content Toggle raw display
$83$ \( T^{2} + 584T + 28816 \) Copy content Toggle raw display
$89$ \( T^{2} + 1404 T + 392452 \) Copy content Toggle raw display
$97$ \( T^{2} + 1188 T + 335908 \) Copy content Toggle raw display
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