Properties

Label 105.4.a.c
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

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 - \beta ) q^{2} -3 q^{3} + ( 5 + 7 \beta ) q^{4} -5 q^{5} + ( 9 + 3 \beta ) q^{6} -7 q^{7} + ( -19 - 25 \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( -3 - \beta ) q^{2} -3 q^{3} + ( 5 + 7 \beta ) q^{4} -5 q^{5} + ( 9 + 3 \beta ) q^{6} -7 q^{7} + ( -19 - 25 \beta ) q^{8} + 9 q^{9} + ( 15 + 5 \beta ) q^{10} + ( -8 - 10 \beta ) q^{11} + ( -15 - 21 \beta ) q^{12} + ( 18 - 22 \beta ) q^{13} + ( 21 + 7 \beta ) q^{14} + 15 q^{15} + ( 117 + 63 \beta ) q^{16} + ( -6 + 28 \beta ) q^{17} + ( -27 - 9 \beta ) q^{18} + ( 100 - 26 \beta ) q^{19} + ( -25 - 35 \beta ) q^{20} + 21 q^{21} + ( 64 + 48 \beta ) q^{22} + ( 64 + 56 \beta ) q^{23} + ( 57 + 75 \beta ) q^{24} + 25 q^{25} + ( 34 + 70 \beta ) q^{26} -27 q^{27} + ( -35 - 49 \beta ) q^{28} + ( 26 - 84 \beta ) q^{29} + ( -45 - 15 \beta ) q^{30} + ( 156 + 18 \beta ) q^{31} + ( -451 - 169 \beta ) q^{32} + ( 24 + 30 \beta ) q^{33} + ( -94 - 106 \beta ) q^{34} + 35 q^{35} + ( 45 + 63 \beta ) q^{36} + ( -78 + 24 \beta ) q^{37} + ( -196 + 4 \beta ) q^{38} + ( -54 + 66 \beta ) q^{39} + ( 95 + 125 \beta ) q^{40} + ( 30 + 140 \beta ) q^{41} + ( -63 - 21 \beta ) q^{42} + ( 216 - 68 \beta ) q^{43} + ( -320 - 176 \beta ) q^{44} -45 q^{45} + ( -416 - 288 \beta ) q^{46} + ( 92 + 108 \beta ) q^{47} + ( -351 - 189 \beta ) q^{48} + 49 q^{49} + ( -75 - 25 \beta ) q^{50} + ( 18 - 84 \beta ) q^{51} + ( -526 - 138 \beta ) q^{52} + ( -90 + 214 \beta ) q^{53} + ( 81 + 27 \beta ) q^{54} + ( 40 + 50 \beta ) q^{55} + ( 133 + 175 \beta ) q^{56} + ( -300 + 78 \beta ) q^{57} + ( 258 + 310 \beta ) q^{58} + ( 164 + 36 \beta ) q^{59} + ( 75 + 105 \beta ) q^{60} + ( 522 - 252 \beta ) q^{61} + ( -540 - 228 \beta ) q^{62} -63 q^{63} + ( 1093 + 623 \beta ) q^{64} + ( -90 + 110 \beta ) q^{65} + ( -192 - 144 \beta ) q^{66} + ( -408 + 28 \beta ) q^{67} + ( 754 + 294 \beta ) q^{68} + ( -192 - 168 \beta ) q^{69} + ( -105 - 35 \beta ) q^{70} + ( 392 - 330 \beta ) q^{71} + ( -171 - 225 \beta ) q^{72} + ( 478 - 178 \beta ) q^{73} + ( 138 - 18 \beta ) q^{74} -75 q^{75} + ( -228 + 388 \beta ) q^{76} + ( 56 + 70 \beta ) q^{77} + ( -102 - 210 \beta ) q^{78} + ( 160 + 88 \beta ) q^{79} + ( -585 - 315 \beta ) q^{80} + 81 q^{81} + ( -650 - 590 \beta ) q^{82} + ( 700 - 264 \beta ) q^{83} + ( 105 + 147 \beta ) q^{84} + ( 30 - 140 \beta ) q^{85} + ( -376 + 56 \beta ) q^{86} + ( -78 + 252 \beta ) q^{87} + ( 1152 + 640 \beta ) q^{88} + ( 382 - 728 \beta ) q^{89} + ( 135 + 45 \beta ) q^{90} + ( -126 + 154 \beta ) q^{91} + ( 1888 + 1120 \beta ) q^{92} + ( -468 - 54 \beta ) q^{93} + ( -708 - 524 \beta ) q^{94} + ( -500 + 130 \beta ) q^{95} + ( 1353 + 507 \beta ) q^{96} + ( -322 + 146 \beta ) q^{97} + ( -147 - 49 \beta ) q^{98} + ( -72 - 90 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 7q^{2} - 6q^{3} + 17q^{4} - 10q^{5} + 21q^{6} - 14q^{7} - 63q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 7q^{2} - 6q^{3} + 17q^{4} - 10q^{5} + 21q^{6} - 14q^{7} - 63q^{8} + 18q^{9} + 35q^{10} - 26q^{11} - 51q^{12} + 14q^{13} + 49q^{14} + 30q^{15} + 297q^{16} + 16q^{17} - 63q^{18} + 174q^{19} - 85q^{20} + 42q^{21} + 176q^{22} + 184q^{23} + 189q^{24} + 50q^{25} + 138q^{26} - 54q^{27} - 119q^{28} - 32q^{29} - 105q^{30} + 330q^{31} - 1071q^{32} + 78q^{33} - 294q^{34} + 70q^{35} + 153q^{36} - 132q^{37} - 388q^{38} - 42q^{39} + 315q^{40} + 200q^{41} - 147q^{42} + 364q^{43} - 816q^{44} - 90q^{45} - 1120q^{46} + 292q^{47} - 891q^{48} + 98q^{49} - 175q^{50} - 48q^{51} - 1190q^{52} + 34q^{53} + 189q^{54} + 130q^{55} + 441q^{56} - 522q^{57} + 826q^{58} + 364q^{59} + 255q^{60} + 792q^{61} - 1308q^{62} - 126q^{63} + 2809q^{64} - 70q^{65} - 528q^{66} - 788q^{67} + 1802q^{68} - 552q^{69} - 245q^{70} + 454q^{71} - 567q^{72} + 778q^{73} + 258q^{74} - 150q^{75} - 68q^{76} + 182q^{77} - 414q^{78} + 408q^{79} - 1485q^{80} + 162q^{81} - 1890q^{82} + 1136q^{83} + 357q^{84} - 80q^{85} - 696q^{86} + 96q^{87} + 2944q^{88} + 36q^{89} + 315q^{90} - 98q^{91} + 4896q^{92} - 990q^{93} - 1940q^{94} - 870q^{95} + 3213q^{96} - 498q^{97} - 343q^{98} - 234q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
−5.56155 −3.00000 22.9309 −5.00000 16.6847 −7.00000 −83.0388 9.00000 27.8078
1.2 −1.43845 −3.00000 −5.93087 −5.00000 4.31534 −7.00000 20.0388 9.00000 7.19224
\(n\): 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.c 2
3.b odd 2 1 315.4.a.m 2
4.b odd 2 1 1680.4.a.bk 2
5.b even 2 1 525.4.a.p 2
5.c odd 4 2 525.4.d.i 4
7.b odd 2 1 735.4.a.k 2
15.d odd 2 1 1575.4.a.m 2
21.c even 2 1 2205.4.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.c 2 1.a even 1 1 trivial
315.4.a.m 2 3.b odd 2 1
525.4.a.p 2 5.b even 2 1
525.4.d.i 4 5.c odd 4 2
735.4.a.k 2 7.b odd 2 1
1575.4.a.m 2 15.d odd 2 1
1680.4.a.bk 2 4.b odd 2 1
2205.4.a.bh 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 \( T_{2}^{2} + 7 T_{2} + 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 7 T + 24 T^{2} + 56 T^{3} + 64 T^{4} \)
$3$ \( ( 1 + 3 T )^{2} \)
$5$ \( ( 1 + 5 T )^{2} \)
$7$ \( ( 1 + 7 T )^{2} \)
$11$ \( 1 + 26 T + 2406 T^{2} + 34606 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 14 T + 2386 T^{2} - 30758 T^{3} + 4826809 T^{4} \)
$17$ \( 1 - 16 T + 6558 T^{2} - 78608 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 174 T + 18414 T^{2} - 1193466 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 184 T + 19470 T^{2} - 2238728 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 32 T + 19046 T^{2} + 780448 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 330 T + 85430 T^{2} - 9831030 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 132 T + 103214 T^{2} + 6686196 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 - 200 T + 64542 T^{2} - 13784200 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 - 364 T + 172486 T^{2} - 28940548 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 292 T + 179390 T^{2} - 30316316 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 34 T + 103410 T^{2} - 5061818 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 364 T + 438374 T^{2} - 74757956 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 792 T + 340886 T^{2} - 179768952 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 788 T + 753430 T^{2} + 237001244 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 454 T + 304526 T^{2} - 162491594 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 - 778 T + 794698 T^{2} - 302655226 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 408 T + 994782 T^{2} - 201159912 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 1136 T + 1169990 T^{2} - 649550032 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 - 36 T - 842170 T^{2} - 25378884 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 + 498 T + 1796754 T^{2} + 454511154 T^{3} + 832972004929 T^{4} \)
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