Properties

Label 105.4.a.g
Level 105
Weight 4
Character orbit 105.a
Self dual yes
Analytic conductor 6.195
Analytic rank 0
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + 3 q^{3} + ( 3 + 3 \beta ) q^{4} -5 q^{5} + ( 3 + 3 \beta ) q^{6} + 7 q^{7} + ( 25 + \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + 3 q^{3} + ( 3 + 3 \beta ) q^{4} -5 q^{5} + ( 3 + 3 \beta ) q^{6} + 7 q^{7} + ( 25 + \beta ) q^{8} + 9 q^{9} + ( -5 - 5 \beta ) q^{10} + ( 32 - 2 \beta ) q^{11} + ( 9 + 9 \beta ) q^{12} + ( 2 - 10 \beta ) q^{13} + ( 7 + 7 \beta ) q^{14} -15 q^{15} + ( 11 + 3 \beta ) q^{16} + ( 26 - 12 \beta ) q^{17} + ( 9 + 9 \beta ) q^{18} + ( -60 - 2 \beta ) q^{19} + ( -15 - 15 \beta ) q^{20} + 21 q^{21} + ( 12 + 28 \beta ) q^{22} + ( 32 - 48 \beta ) q^{23} + ( 75 + 3 \beta ) q^{24} + 25 q^{25} + ( -98 - 18 \beta ) q^{26} + 27 q^{27} + ( 21 + 21 \beta ) q^{28} + ( 170 + 12 \beta ) q^{29} + ( -15 - 15 \beta ) q^{30} + ( 52 - 38 \beta ) q^{31} + ( -159 + 9 \beta ) q^{32} + ( 96 - 6 \beta ) q^{33} + ( -94 + 2 \beta ) q^{34} -35 q^{35} + ( 27 + 27 \beta ) q^{36} + ( -134 + 80 \beta ) q^{37} + ( -80 - 64 \beta ) q^{38} + ( 6 - 30 \beta ) q^{39} + ( -125 - 5 \beta ) q^{40} + ( 62 - 108 \beta ) q^{41} + ( 21 + 21 \beta ) q^{42} + ( -248 + 100 \beta ) q^{43} + ( 36 + 84 \beta ) q^{44} -45 q^{45} + ( -448 - 64 \beta ) q^{46} + ( -164 + 140 \beta ) q^{47} + ( 33 + 9 \beta ) q^{48} + 49 q^{49} + ( 25 + 25 \beta ) q^{50} + ( 78 - 36 \beta ) q^{51} + ( -294 - 54 \beta ) q^{52} + ( 462 + 58 \beta ) q^{53} + ( 27 + 27 \beta ) q^{54} + ( -160 + 10 \beta ) q^{55} + ( 175 + 7 \beta ) q^{56} + ( -180 - 6 \beta ) q^{57} + ( 290 + 194 \beta ) q^{58} + ( 220 + 76 \beta ) q^{59} + ( -45 - 45 \beta ) q^{60} + ( -398 - 84 \beta ) q^{61} + ( -328 - 24 \beta ) q^{62} + 63 q^{63} + ( -157 - 165 \beta ) q^{64} + ( -10 + 50 \beta ) q^{65} + ( 36 + 84 \beta ) q^{66} + ( -64 - 228 \beta ) q^{67} + ( -282 + 6 \beta ) q^{68} + ( 96 - 144 \beta ) q^{69} + ( -35 - 35 \beta ) q^{70} + ( 112 + 86 \beta ) q^{71} + ( 225 + 9 \beta ) q^{72} + ( 182 - 38 \beta ) q^{73} + ( 666 + 26 \beta ) q^{74} + 75 q^{75} + ( -240 - 192 \beta ) q^{76} + ( 224 - 14 \beta ) q^{77} + ( -294 - 54 \beta ) q^{78} + ( 920 - 8 \beta ) q^{79} + ( -55 - 15 \beta ) q^{80} + 81 q^{81} + ( -1018 - 154 \beta ) q^{82} + ( -228 - 224 \beta ) q^{83} + ( 63 + 63 \beta ) q^{84} + ( -130 + 60 \beta ) q^{85} + ( 752 - 48 \beta ) q^{86} + ( 510 + 36 \beta ) q^{87} + ( 780 - 20 \beta ) q^{88} + ( 230 + 336 \beta ) q^{89} + ( -45 - 45 \beta ) q^{90} + ( 14 - 70 \beta ) q^{91} + ( -1344 - 192 \beta ) q^{92} + ( 156 - 114 \beta ) q^{93} + ( 1236 + 116 \beta ) q^{94} + ( 300 + 10 \beta ) q^{95} + ( -477 + 27 \beta ) q^{96} + ( -474 + 278 \beta ) q^{97} + ( 49 + 49 \beta ) q^{98} + ( 288 - 18 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} + 6q^{3} + 9q^{4} - 10q^{5} + 9q^{6} + 14q^{7} + 51q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 3q^{2} + 6q^{3} + 9q^{4} - 10q^{5} + 9q^{6} + 14q^{7} + 51q^{8} + 18q^{9} - 15q^{10} + 62q^{11} + 27q^{12} - 6q^{13} + 21q^{14} - 30q^{15} + 25q^{16} + 40q^{17} + 27q^{18} - 122q^{19} - 45q^{20} + 42q^{21} + 52q^{22} + 16q^{23} + 153q^{24} + 50q^{25} - 214q^{26} + 54q^{27} + 63q^{28} + 352q^{29} - 45q^{30} + 66q^{31} - 309q^{32} + 186q^{33} - 186q^{34} - 70q^{35} + 81q^{36} - 188q^{37} - 224q^{38} - 18q^{39} - 255q^{40} + 16q^{41} + 63q^{42} - 396q^{43} + 156q^{44} - 90q^{45} - 960q^{46} - 188q^{47} + 75q^{48} + 98q^{49} + 75q^{50} + 120q^{51} - 642q^{52} + 982q^{53} + 81q^{54} - 310q^{55} + 357q^{56} - 366q^{57} + 774q^{58} + 516q^{59} - 135q^{60} - 880q^{61} - 680q^{62} + 126q^{63} - 479q^{64} + 30q^{65} + 156q^{66} - 356q^{67} - 558q^{68} + 48q^{69} - 105q^{70} + 310q^{71} + 459q^{72} + 326q^{73} + 1358q^{74} + 150q^{75} - 672q^{76} + 434q^{77} - 642q^{78} + 1832q^{79} - 125q^{80} + 162q^{81} - 2190q^{82} - 680q^{83} + 189q^{84} - 200q^{85} + 1456q^{86} + 1056q^{87} + 1540q^{88} + 796q^{89} - 135q^{90} - 42q^{91} - 2880q^{92} + 198q^{93} + 2588q^{94} + 610q^{95} - 927q^{96} - 670q^{97} + 147q^{98} + 558q^{99} + O(q^{100}) \)

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
−2.70156
3.70156
−1.70156 3.00000 −5.10469 −5.00000 −5.10469 7.00000 22.2984 9.00000 8.50781
1.2 4.70156 3.00000 14.1047 −5.00000 14.1047 7.00000 28.7016 9.00000 −23.5078
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.g 2
3.b odd 2 1 315.4.a.g 2
4.b odd 2 1 1680.4.a.y 2
5.b even 2 1 525.4.a.i 2
5.c odd 4 2 525.4.d.j 4
7.b odd 2 1 735.4.a.q 2
15.d odd 2 1 1575.4.a.y 2
21.c even 2 1 2205.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 1.a even 1 1 trivial
315.4.a.g 2 3.b odd 2 1
525.4.a.i 2 5.b even 2 1
525.4.d.j 4 5.c odd 4 2
735.4.a.q 2 7.b odd 2 1
1575.4.a.y 2 15.d odd 2 1
1680.4.a.y 2 4.b odd 2 1
2205.4.a.v 2 21.c even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 T_{2} - 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 8 T^{2} - 24 T^{3} + 64 T^{4} \)
$3$ \( ( 1 - 3 T )^{2} \)
$5$ \( ( 1 + 5 T )^{2} \)
$7$ \( ( 1 - 7 T )^{2} \)
$11$ \( 1 - 62 T + 3582 T^{2} - 82522 T^{3} + 1771561 T^{4} \)
$13$ \( 1 + 6 T + 3378 T^{2} + 13182 T^{3} + 4826809 T^{4} \)
$17$ \( 1 - 40 T + 8750 T^{2} - 196520 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 122 T + 17398 T^{2} + 836798 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 16 T + 782 T^{2} - 194672 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 352 T + 78278 T^{2} - 8584928 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 66 T + 45870 T^{2} - 1966206 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 188 T + 44542 T^{2} + 9522764 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 - 16 T + 18350 T^{2} - 1102736 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 396 T + 95718 T^{2} + 31484772 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 + 188 T + 15582 T^{2} + 19518724 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 982 T + 504354 T^{2} - 146197214 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 516 T + 418118 T^{2} - 105975564 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 880 T + 575238 T^{2} + 199743280 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 356 T + 100374 T^{2} + 107071628 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 310 T + 664038 T^{2} - 110952410 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 - 326 T + 789802 T^{2} - 126819542 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 1832 T + 1824478 T^{2} - 903247448 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 + 680 T + 744870 T^{2} + 388815160 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 - 796 T + 411158 T^{2} - 561155324 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 + 670 T + 1145410 T^{2} + 611490910 T^{3} + 832972004929 T^{4} \)
show more
show less