Properties

Label 105.4.i.b
Level $105$
Weight $4$
Character orbit 105.i
Analytic conductor $6.195$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{2} + ( 3 + 3 \beta_{2} ) q^{3} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{4} -5 \beta_{2} q^{5} + ( 6 - 3 \beta_{3} ) q^{6} + ( 10 + 10 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{7} + ( 12 - 2 \beta_{3} ) q^{8} + 9 \beta_{2} q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{2} + ( 3 + 3 \beta_{2} ) q^{3} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{4} -5 \beta_{2} q^{5} + ( 6 - 3 \beta_{3} ) q^{6} + ( 10 + 10 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{7} + ( 12 - 2 \beta_{3} ) q^{8} + 9 \beta_{2} q^{9} + ( -10 - 5 \beta_{1} - 10 \beta_{2} ) q^{10} + ( 16 - 38 \beta_{1} + 16 \beta_{2} ) q^{11} + ( -12 \beta_{1} + 6 \beta_{2} - 12 \beta_{3} ) q^{12} + ( 7 - 26 \beta_{3} ) q^{13} + ( 10 - 11 \beta_{1} - 26 \beta_{2} - 26 \beta_{3} ) q^{14} + 15 q^{15} + ( -48 \beta_{1} - 12 \beta_{2} - 48 \beta_{3} ) q^{16} + ( 10 - 34 \beta_{1} + 10 \beta_{2} ) q^{17} + ( 18 + 9 \beta_{1} + 18 \beta_{2} ) q^{18} + ( 64 \beta_{1} + 9 \beta_{2} + 64 \beta_{3} ) q^{19} + ( 10 + 20 \beta_{3} ) q^{20} + ( 45 + 24 \beta_{1} + 30 \beta_{2} + 30 \beta_{3} ) q^{21} + ( -44 + 60 \beta_{3} ) q^{22} + ( 26 \beta_{1} - 34 \beta_{2} + 26 \beta_{3} ) q^{23} + ( 36 + 6 \beta_{1} + 36 \beta_{2} ) q^{24} + ( -25 - 25 \beta_{2} ) q^{25} + ( -59 \beta_{1} - 66 \beta_{2} - 59 \beta_{3} ) q^{26} -27 q^{27} + ( 46 - 24 \beta_{1} - 44 \beta_{2} + 40 \beta_{3} ) q^{28} + ( -166 + 6 \beta_{3} ) q^{29} + ( -15 \beta_{1} - 30 \beta_{2} - 15 \beta_{3} ) q^{30} + ( -33 + 36 \beta_{1} - 33 \beta_{2} ) q^{31} + ( -24 - 92 \beta_{1} - 24 \beta_{2} ) q^{32} + ( -114 \beta_{1} + 48 \beta_{2} - 114 \beta_{3} ) q^{33} + ( -48 + 58 \beta_{3} ) q^{34} + ( -25 + 10 \beta_{1} - 75 \beta_{2} - 40 \beta_{3} ) q^{35} + ( -18 - 36 \beta_{3} ) q^{36} + ( 222 \beta_{1} + 9 \beta_{2} + 222 \beta_{3} ) q^{37} + ( 146 + 137 \beta_{1} + 146 \beta_{2} ) q^{38} + ( 21 + 78 \beta_{1} + 21 \beta_{2} ) q^{39} + ( -10 \beta_{1} - 60 \beta_{2} - 10 \beta_{3} ) q^{40} + ( 76 - 122 \beta_{3} ) q^{41} + ( 108 + 45 \beta_{1} + 30 \beta_{2} - 33 \beta_{3} ) q^{42} + ( -421 + 38 \beta_{3} ) q^{43} + ( -140 \beta_{1} + 336 \beta_{2} - 140 \beta_{3} ) q^{44} + ( 45 + 45 \beta_{2} ) q^{45} + ( -16 + 18 \beta_{1} - 16 \beta_{2} ) q^{46} + ( 120 \beta_{1} - 106 \beta_{2} + 120 \beta_{3} ) q^{47} + ( 36 - 144 \beta_{3} ) q^{48} + ( 3 + 220 \beta_{1} - 5 \beta_{2} - 40 \beta_{3} ) q^{49} + ( -50 + 25 \beta_{3} ) q^{50} + ( -102 \beta_{1} + 30 \beta_{2} - 102 \beta_{3} ) q^{51} + ( -194 + 24 \beta_{1} - 194 \beta_{2} ) q^{52} + ( 184 + 56 \beta_{1} + 184 \beta_{2} ) q^{53} + ( 27 \beta_{1} + 54 \beta_{2} + 27 \beta_{3} ) q^{54} + ( 80 + 190 \beta_{3} ) q^{55} + ( 152 + 110 \beta_{1} - 20 \beta_{2} - 6 \beta_{3} ) q^{56} + ( -27 + 192 \beta_{3} ) q^{57} + ( 178 \beta_{1} + 344 \beta_{2} + 178 \beta_{3} ) q^{58} + ( 70 + 186 \beta_{1} + 70 \beta_{2} ) q^{59} + ( 30 - 60 \beta_{1} + 30 \beta_{2} ) q^{60} + ( 60 \beta_{1} - 366 \beta_{2} + 60 \beta_{3} ) q^{61} + ( 6 - 39 \beta_{3} ) q^{62} + ( 45 - 18 \beta_{1} + 135 \beta_{2} + 72 \beta_{3} ) q^{63} + ( -136 - 176 \beta_{3} ) q^{64} + ( -130 \beta_{1} - 35 \beta_{2} - 130 \beta_{3} ) q^{65} + ( -132 - 180 \beta_{1} - 132 \beta_{2} ) q^{66} + ( -533 - 250 \beta_{1} - 533 \beta_{2} ) q^{67} + ( -108 \beta_{1} + 292 \beta_{2} - 108 \beta_{3} ) q^{68} + ( 102 + 78 \beta_{3} ) q^{69} + ( -130 - 130 \beta_{1} - 180 \beta_{2} - 75 \beta_{3} ) q^{70} + ( -604 + 246 \beta_{3} ) q^{71} + ( 18 \beta_{1} + 108 \beta_{2} + 18 \beta_{3} ) q^{72} + ( -827 + 10 \beta_{1} - 827 \beta_{2} ) q^{73} + ( 462 + 453 \beta_{1} + 462 \beta_{2} ) q^{74} -75 \beta_{2} q^{75} + ( 494 + 92 \beta_{3} ) q^{76} + ( 392 - 252 \beta_{1} - 448 \beta_{2} + 350 \beta_{3} ) q^{77} + ( 198 - 177 \beta_{3} ) q^{78} + ( -32 \beta_{1} + 567 \beta_{2} - 32 \beta_{3} ) q^{79} + ( -60 - 240 \beta_{1} - 60 \beta_{2} ) q^{80} + ( -81 - 81 \beta_{2} ) q^{81} + ( -320 \beta_{1} - 396 \beta_{2} - 320 \beta_{3} ) q^{82} + ( 484 + 102 \beta_{3} ) q^{83} + ( 270 - 192 \beta_{1} + 138 \beta_{2} - 72 \beta_{3} ) q^{84} + ( 50 + 170 \beta_{3} ) q^{85} + ( 497 \beta_{1} + 918 \beta_{2} + 497 \beta_{3} ) q^{86} + ( -498 - 18 \beta_{1} - 498 \beta_{2} ) q^{87} + ( 40 - 424 \beta_{1} + 40 \beta_{2} ) q^{88} + ( -1046 \beta_{1} + 102 \beta_{2} - 1046 \beta_{3} ) q^{89} + ( 90 - 45 \beta_{3} ) q^{90} + ( 486 - 60 \beta_{1} + 485 \beta_{2} - 376 \beta_{3} ) q^{91} + ( 276 + 188 \beta_{3} ) q^{92} + ( 108 \beta_{1} - 99 \beta_{2} + 108 \beta_{3} ) q^{93} + ( 28 + 134 \beta_{1} + 28 \beta_{2} ) q^{94} + ( 45 + 320 \beta_{1} + 45 \beta_{2} ) q^{95} + ( -276 \beta_{1} - 72 \beta_{2} - 276 \beta_{3} ) q^{96} + ( 846 + 736 \beta_{3} ) q^{97} + ( 430 - 88 \beta_{1} - 96 \beta_{2} - 523 \beta_{3} ) q^{98} + ( -144 - 342 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 6q^{3} + 4q^{4} + 10q^{5} + 24q^{6} + 50q^{7} + 48q^{8} - 18q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 6q^{3} + 4q^{4} + 10q^{5} + 24q^{6} + 50q^{7} + 48q^{8} - 18q^{9} - 20q^{10} + 32q^{11} - 12q^{12} + 28q^{13} + 92q^{14} + 60q^{15} + 24q^{16} + 20q^{17} + 36q^{18} - 18q^{19} + 40q^{20} + 120q^{21} - 176q^{22} + 68q^{23} + 72q^{24} - 50q^{25} + 132q^{26} - 108q^{27} + 272q^{28} - 664q^{29} + 60q^{30} - 66q^{31} - 48q^{32} - 96q^{33} - 192q^{34} + 50q^{35} - 72q^{36} - 18q^{37} + 292q^{38} + 42q^{39} + 120q^{40} + 304q^{41} + 372q^{42} - 1684q^{43} - 672q^{44} + 90q^{45} - 32q^{46} + 212q^{47} + 144q^{48} + 22q^{49} - 200q^{50} - 60q^{51} - 388q^{52} + 368q^{53} - 108q^{54} + 320q^{55} + 648q^{56} - 108q^{57} - 688q^{58} + 140q^{59} + 60q^{60} + 732q^{61} + 24q^{62} - 90q^{63} - 544q^{64} + 70q^{65} - 264q^{66} - 1066q^{67} - 584q^{68} + 408q^{69} - 160q^{70} - 2416q^{71} - 216q^{72} - 1654q^{73} + 924q^{74} + 150q^{75} + 1976q^{76} + 2464q^{77} + 792q^{78} - 1134q^{79} - 120q^{80} - 162q^{81} + 792q^{82} + 1936q^{83} + 804q^{84} + 200q^{85} - 1836q^{86} - 996q^{87} + 80q^{88} - 204q^{89} + 360q^{90} + 974q^{91} + 1104q^{92} + 198q^{93} + 56q^{94} + 90q^{95} + 144q^{96} + 3384q^{97} + 1912q^{98} - 576q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0.292893 0.507306i 1.50000 + 2.59808i 3.82843 + 6.63103i 2.50000 4.33013i 1.75736 8.25736 16.5776i 9.17157 −4.50000 + 7.79423i −1.46447 2.53653i
16.2 1.70711 2.95680i 1.50000 + 2.59808i −1.82843 3.16693i 2.50000 4.33013i 10.2426 16.7426 + 7.91732i 14.8284 −4.50000 + 7.79423i −8.53553 14.7840i
46.1 0.292893 + 0.507306i 1.50000 2.59808i 3.82843 6.63103i 2.50000 + 4.33013i 1.75736 8.25736 + 16.5776i 9.17157 −4.50000 7.79423i −1.46447 + 2.53653i
46.2 1.70711 + 2.95680i 1.50000 2.59808i −1.82843 + 3.16693i 2.50000 + 4.33013i 10.2426 16.7426 7.91732i 14.8284 −4.50000 7.79423i −8.53553 + 14.7840i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.i.b 4
3.b odd 2 1 315.4.j.d 4
7.c even 3 1 inner 105.4.i.b 4
7.c even 3 1 735.4.a.l 2
7.d odd 6 1 735.4.a.n 2
21.g even 6 1 2205.4.a.bc 2
21.h odd 6 1 315.4.j.d 4
21.h odd 6 1 2205.4.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.i.b 4 1.a even 1 1 trivial
105.4.i.b 4 7.c even 3 1 inner
315.4.j.d 4 3.b odd 2 1
315.4.j.d 4 21.h odd 6 1
735.4.a.l 2 7.c even 3 1
735.4.a.n 2 7.d odd 6 1
2205.4.a.bc 2 21.g even 6 1
2205.4.a.bd 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4 T_{2}^{3} + 14 T_{2}^{2} - 8 T_{2} + 4 \) acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 8 T + 14 T^{2} - 4 T^{3} + T^{4} \)
$3$ \( ( 9 - 3 T + T^{2} )^{2} \)
$5$ \( ( 25 - 5 T + T^{2} )^{2} \)
$7$ \( 117649 - 17150 T + 1239 T^{2} - 50 T^{3} + T^{4} \)
$11$ \( 6927424 + 84224 T + 3656 T^{2} - 32 T^{3} + T^{4} \)
$13$ \( ( -1303 - 14 T + T^{2} )^{2} \)
$17$ \( 4892944 + 44240 T + 2612 T^{2} - 20 T^{3} + T^{4} \)
$19$ \( 65788321 - 145998 T + 8435 T^{2} + 18 T^{3} + T^{4} \)
$23$ \( 38416 + 13328 T + 4820 T^{2} - 68 T^{3} + T^{4} \)
$29$ \( ( 27484 + 332 T + T^{2} )^{2} \)
$31$ \( 2259009 - 99198 T + 5859 T^{2} + 66 T^{3} + T^{4} \)
$37$ \( 9699689169 - 1772766 T + 98811 T^{2} + 18 T^{3} + T^{4} \)
$41$ \( ( -23992 - 152 T + T^{2} )^{2} \)
$43$ \( ( 174353 + 842 T + T^{2} )^{2} \)
$47$ \( 308494096 + 3723568 T + 62508 T^{2} - 212 T^{3} + T^{4} \)
$53$ \( 760877056 - 10150912 T + 107840 T^{2} - 368 T^{3} + T^{4} \)
$59$ \( 4133461264 + 9000880 T + 83892 T^{2} - 140 T^{3} + T^{4} \)
$61$ \( 16067083536 - 92785392 T + 409068 T^{2} - 732 T^{3} + T^{4} \)
$67$ \( 25309309921 + 169588874 T + 977267 T^{2} + 1066 T^{3} + T^{4} \)
$71$ \( ( 243784 + 1208 T + T^{2} )^{2} \)
$73$ \( 467485345441 + 1130887766 T + 2051987 T^{2} + 1654 T^{3} + T^{4} \)
$79$ \( 102042552481 + 362246094 T + 966515 T^{2} + 1134 T^{3} + T^{4} \)
$83$ \( ( 213448 - 968 T + T^{2} )^{2} \)
$89$ \( 4742934797584 - 444276912 T + 2219444 T^{2} + 204 T^{3} + T^{4} \)
$97$ \( ( -367676 - 1692 T + T^{2} )^{2} \)
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