Properties

Label 105.4.i.c
Level $105$
Weight $4$
Character orbit 105.i
Analytic conductor $6.195$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.646154928.2
Defining polynomial: \(x^{6} + 11 x^{4} - 8 x^{3} + 121 x^{2} - 44 x + 16\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{3} ) q^{2} + ( 3 - 3 \beta_{3} ) q^{3} + ( \beta_{1} + \beta_{2} + \beta_{5} ) q^{4} -5 \beta_{3} q^{5} + ( -3 + 3 \beta_{2} ) q^{6} + ( 2 + 5 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{7} + ( 2 + 5 \beta_{2} - 3 \beta_{4} ) q^{8} -9 \beta_{3} q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{3} ) q^{2} + ( 3 - 3 \beta_{3} ) q^{3} + ( \beta_{1} + \beta_{2} + \beta_{5} ) q^{4} -5 \beta_{3} q^{5} + ( -3 + 3 \beta_{2} ) q^{6} + ( 2 + 5 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{7} + ( 2 + 5 \beta_{2} - 3 \beta_{4} ) q^{8} -9 \beta_{3} q^{9} + ( -5 + 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{10} + ( 7 \beta_{1} + 7 \beta_{2} - \beta_{5} ) q^{11} + ( 3 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} ) q^{12} + ( -25 + 10 \beta_{2} + 4 \beta_{4} ) q^{13} + ( 35 + 7 \beta_{1} - 21 \beta_{3} - 7 \beta_{4} ) q^{14} -15 q^{15} + ( -3 \beta_{1} + 24 \beta_{3} - 7 \beta_{4} + 7 \beta_{5} ) q^{16} + ( 22 - 4 \beta_{1} - 4 \beta_{2} - 22 \beta_{3} - 6 \beta_{5} ) q^{17} + ( -9 + 9 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{18} + ( -4 \beta_{1} + 7 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} ) q^{19} + ( -5 \beta_{2} - 5 \beta_{4} ) q^{20} + ( -3 + 3 \beta_{1} - 12 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} + 9 \beta_{5} ) q^{21} + ( 46 + 3 \beta_{2} - 5 \beta_{4} ) q^{22} + ( -19 \beta_{1} + 18 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{23} + ( 6 + 15 \beta_{1} + 15 \beta_{2} - 6 \beta_{3} - 9 \beta_{5} ) q^{24} + ( -25 + 25 \beta_{3} ) q^{25} + ( 37 \beta_{1} + 107 \beta_{3} - 18 \beta_{4} + 18 \beta_{5} ) q^{26} -27 q^{27} + ( 20 - 27 \beta_{1} - 11 \beta_{2} - 100 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} ) q^{28} + ( 70 + 36 \beta_{2} - 10 \beta_{4} ) q^{29} + ( 15 \beta_{1} + 15 \beta_{3} ) q^{30} + ( -33 - 57 \beta_{1} - 57 \beta_{2} + 33 \beta_{3} + 37 \beta_{5} ) q^{31} + ( 40 - 5 \beta_{1} - 5 \beta_{2} - 40 \beta_{3} - 35 \beta_{5} ) q^{32} + ( 21 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{33} + ( -68 + 40 \beta_{2} + 16 \beta_{4} ) q^{34} + ( -15 - 20 \beta_{1} - 25 \beta_{2} + 5 \beta_{3} - 15 \beta_{4} + 10 \beta_{5} ) q^{35} + ( -9 \beta_{2} - 9 \beta_{4} ) q^{36} + ( -18 \beta_{1} + 3 \beta_{3} + 52 \beta_{4} - 52 \beta_{5} ) q^{37} + ( 3 - 31 \beta_{1} - 31 \beta_{2} - 3 \beta_{3} - 12 \beta_{5} ) q^{38} + ( -75 + 30 \beta_{1} + 30 \beta_{2} + 75 \beta_{3} + 12 \beta_{5} ) q^{39} + ( 25 \beta_{1} - 10 \beta_{3} + 15 \beta_{4} - 15 \beta_{5} ) q^{40} + ( -188 + 11 \beta_{2} + 35 \beta_{4} ) q^{41} + ( 42 - 21 \beta_{2} - 105 \beta_{3} - 21 \beta_{5} ) q^{42} + ( 79 + 11 \beta_{2} + 67 \beta_{4} ) q^{43} + ( -5 \beta_{1} - 40 \beta_{3} + 15 \beta_{4} - 15 \beta_{5} ) q^{44} + ( -45 + 45 \beta_{3} ) q^{45} + ( -88 - 45 \beta_{1} - 45 \beta_{2} + 88 \beta_{3} + \beta_{5} ) q^{46} + ( 87 \beta_{1} + 26 \beta_{3} + 75 \beta_{4} - 75 \beta_{5} ) q^{47} + ( 72 + 9 \beta_{2} - 21 \beta_{4} ) q^{48} + ( 31 - 115 \beta_{1} - 86 \beta_{2} - 99 \beta_{3} + 31 \beta_{4} + 19 \beta_{5} ) q^{49} + ( 25 - 25 \beta_{2} ) q^{50} + ( -12 \beta_{1} - 66 \beta_{3} + 18 \beta_{4} - 18 \beta_{5} ) q^{51} + ( 220 - 81 \beta_{1} - 81 \beta_{2} - 220 \beta_{3} - 41 \beta_{5} ) q^{52} + ( -104 + 41 \beta_{1} + 41 \beta_{2} + 104 \beta_{3} + 19 \beta_{5} ) q^{53} + ( 27 \beta_{1} + 27 \beta_{3} ) q^{54} + ( -35 \beta_{2} + 5 \beta_{4} ) q^{55} + ( -168 + 98 \beta_{1} + 35 \beta_{2} - 70 \beta_{3} - 7 \beta_{4} - 28 \beta_{5} ) q^{56} + ( 21 + 12 \beta_{2} - 24 \beta_{4} ) q^{57} + ( -100 \beta_{1} + 152 \beta_{3} - 16 \beta_{4} + 16 \beta_{5} ) q^{58} + ( 518 - 144 \beta_{1} - 144 \beta_{2} - 518 \beta_{3} + 34 \beta_{5} ) q^{59} + ( -15 \beta_{1} - 15 \beta_{2} - 15 \beta_{5} ) q^{60} + ( 18 \beta_{1} + 378 \beta_{3} - 26 \beta_{4} + 26 \beta_{5} ) q^{61} + ( -255 - 144 \beta_{2} - 17 \beta_{4} ) q^{62} + ( -27 - 36 \beta_{1} - 45 \beta_{2} + 9 \beta_{3} - 27 \beta_{4} + 18 \beta_{5} ) q^{63} + ( 12 + 169 \beta_{2} + 19 \beta_{4} ) q^{64} + ( 50 \beta_{1} + 125 \beta_{3} - 20 \beta_{4} + 20 \beta_{5} ) q^{65} + ( 138 + 9 \beta_{1} + 9 \beta_{2} - 138 \beta_{3} - 15 \beta_{5} ) q^{66} + ( -277 - 101 \beta_{1} - 101 \beta_{2} + 277 \beta_{3} - 25 \beta_{5} ) q^{67} + ( 84 \beta_{1} + 220 \beta_{3} - 24 \beta_{4} + 24 \beta_{5} ) q^{68} + ( 54 + 57 \beta_{2} - 27 \beta_{4} ) q^{69} + ( -105 - 35 \beta_{1} - 35 \beta_{2} - 70 \beta_{3} + 35 \beta_{4} - 35 \beta_{5} ) q^{70} + ( -140 + 264 \beta_{2} - 14 \beta_{4} ) q^{71} + ( 45 \beta_{1} - 18 \beta_{3} + 27 \beta_{4} - 27 \beta_{5} ) q^{72} + ( 405 + 83 \beta_{1} + 83 \beta_{2} - 405 \beta_{3} + 23 \beta_{5} ) q^{73} + ( -279 + 153 \beta_{1} + 153 \beta_{2} + 279 \beta_{3} + 122 \beta_{5} ) q^{74} + 75 \beta_{3} q^{75} + ( -200 + 71 \beta_{2} - 9 \beta_{4} ) q^{76} + ( -196 - 21 \beta_{1} - 77 \beta_{2} - 28 \beta_{3} + 42 \beta_{4} - 35 \beta_{5} ) q^{77} + ( 321 - 111 \beta_{2} - 54 \beta_{4} ) q^{78} + ( -79 \beta_{1} + 829 \beta_{3} - 103 \beta_{4} + 103 \beta_{5} ) q^{79} + ( 120 + 15 \beta_{1} + 15 \beta_{2} - 120 \beta_{3} - 35 \beta_{5} ) q^{80} + ( -81 + 81 \beta_{3} ) q^{81} + ( 293 \beta_{1} + 370 \beta_{3} - 81 \beta_{4} + 81 \beta_{5} ) q^{82} + ( -184 + 102 \beta_{2} - 44 \beta_{4} ) q^{83} + ( -240 - 33 \beta_{1} + 48 \beta_{2} - 60 \beta_{3} - 9 \beta_{4} + 27 \beta_{5} ) q^{84} + ( -110 + 20 \beta_{2} + 30 \beta_{4} ) q^{85} + ( 122 \beta_{1} + 199 \beta_{3} - 145 \beta_{4} + 145 \beta_{5} ) q^{86} + ( 210 + 108 \beta_{1} + 108 \beta_{2} - 210 \beta_{3} - 30 \beta_{5} ) q^{87} + ( 248 + 109 \beta_{1} + 109 \beta_{2} - 248 \beta_{3} - 5 \beta_{5} ) q^{88} + ( -386 \beta_{1} - 34 \beta_{3} - 60 \beta_{4} + 60 \beta_{5} ) q^{89} + ( 45 - 45 \beta_{2} ) q^{90} + ( 446 - 159 \beta_{1} - 169 \beta_{2} - 473 \beta_{3} - 2 \beta_{4} - 85 \beta_{5} ) q^{91} + ( -80 + 61 \beta_{2} - 29 \beta_{4} ) q^{92} + ( -171 \beta_{1} + 99 \beta_{3} - 111 \beta_{4} + 111 \beta_{5} ) q^{93} + ( 410 + 199 \beta_{1} + 199 \beta_{2} - 410 \beta_{3} + 63 \beta_{5} ) q^{94} + ( 35 + 20 \beta_{1} + 20 \beta_{2} - 35 \beta_{3} - 40 \beta_{5} ) q^{95} + ( -15 \beta_{1} - 120 \beta_{3} + 105 \beta_{4} - 105 \beta_{5} ) q^{96} + ( -398 - 242 \beta_{2} - 174 \beta_{4} ) q^{97} + ( -847 + 161 \beta_{1} + 42 \beta_{2} + 364 \beta_{3} - 14 \beta_{4} + 91 \beta_{5} ) q^{98} + ( -63 \beta_{2} + 9 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{2} + 9q^{3} - q^{4} - 15q^{5} - 18q^{6} - 2q^{7} + 18q^{8} - 27q^{9} + O(q^{10}) \) \( 6q - 3q^{2} + 9q^{3} - q^{4} - 15q^{5} - 18q^{6} - 2q^{7} + 18q^{8} - 27q^{9} - 15q^{10} + q^{11} + 3q^{12} - 158q^{13} + 161q^{14} - 90q^{15} + 79q^{16} + 72q^{17} - 27q^{18} + 29q^{19} + 10q^{20} - 39q^{21} + 286q^{22} + 63q^{23} + 27q^{24} - 75q^{25} + 339q^{26} - 162q^{27} - 195q^{28} + 440q^{29} + 45q^{30} - 136q^{31} + 155q^{32} - 3q^{33} - 440q^{34} - 55q^{35} + 18q^{36} - 43q^{37} + 21q^{38} - 237q^{39} - 45q^{40} - 1198q^{41} - 42q^{42} + 340q^{43} - 135q^{44} - 135q^{45} - 265q^{46} + 3q^{47} + 474q^{48} - 192q^{49} + 150q^{50} - 216q^{51} + 701q^{52} - 331q^{53} + 81q^{54} - 10q^{55} - 1176q^{56} + 174q^{57} + 472q^{58} + 1520q^{59} + 15q^{60} + 1160q^{61} - 1496q^{62} - 99q^{63} + 34q^{64} + 395q^{65} + 429q^{66} - 806q^{67} + 684q^{68} + 378q^{69} - 875q^{70} - 812q^{71} - 81q^{72} + 1192q^{73} - 959q^{74} + 225q^{75} - 1182q^{76} - 1309q^{77} + 2034q^{78} + 2590q^{79} + 395q^{80} - 243q^{81} + 1191q^{82} - 1016q^{83} - 1629q^{84} - 720q^{85} + 742q^{86} + 660q^{87} + 749q^{88} - 42q^{89} + 270q^{90} + 1346q^{91} - 422q^{92} + 408q^{93} + 1167q^{94} + 145q^{95} - 465q^{96} - 2040q^{97} - 4053q^{98} - 18q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 11 x^{4} - 8 x^{3} + 121 x^{2} - 44 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 4 \)\()/11\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 11 \nu^{3} - 4 \nu^{2} + 121 \nu \)\()/44\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} + \nu^{3} + 11 \nu^{2} - 4 \nu + 73 \)\()/11\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{5} - 73 \nu^{3} + 72 \nu^{2} - 803 \nu + 292 \)\()/44\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + 7 \beta_{3} - \beta_{2} - \beta_{1} - 7\)
\(\nu^{3}\)\(=\)\(11 \beta_{2} + 4\)
\(\nu^{4}\)\(=\)\(-11 \beta_{5} + 11 \beta_{4} - 77 \beta_{3} + 15 \beta_{1}\)
\(\nu^{5}\)\(=\)\(4 \beta_{5} + 72 \beta_{3} - 125 \beta_{2} - 125 \beta_{1} - 72\)

Character values

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

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(1\)

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
1.55859 2.69955i
0.184087 0.318847i
−1.74267 + 3.01840i
1.55859 + 2.69955i
0.184087 + 0.318847i
−1.74267 3.01840i
−2.05859 + 3.56558i 1.50000 + 2.59808i −4.47557 7.75191i −2.50000 + 4.33013i −12.3515 −9.40914 15.9521i 3.91601 −4.50000 + 7.79423i −10.2929 17.8279i
16.2 −0.684087 + 1.18487i 1.50000 + 2.59808i 3.06405 + 5.30709i −2.50000 + 4.33013i −4.10452 17.2929 + 6.62978i −19.3297 −4.50000 + 7.79423i −3.42043 5.92436i
16.3 1.24267 2.15238i 1.50000 + 2.59808i 0.911519 + 1.57880i −2.50000 + 4.33013i 7.45605 −8.88380 + 16.2505i 24.4137 −4.50000 + 7.79423i 6.21337 + 10.7619i
46.1 −2.05859 3.56558i 1.50000 2.59808i −4.47557 + 7.75191i −2.50000 4.33013i −12.3515 −9.40914 + 15.9521i 3.91601 −4.50000 7.79423i −10.2929 + 17.8279i
46.2 −0.684087 1.18487i 1.50000 2.59808i 3.06405 5.30709i −2.50000 4.33013i −4.10452 17.2929 6.62978i −19.3297 −4.50000 7.79423i −3.42043 + 5.92436i
46.3 1.24267 + 2.15238i 1.50000 2.59808i 0.911519 1.57880i −2.50000 4.33013i 7.45605 −8.88380 16.2505i 24.4137 −4.50000 7.79423i 6.21337 10.7619i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.3
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.c 6
3.b odd 2 1 315.4.j.e 6
7.c even 3 1 inner 105.4.i.c 6
7.c even 3 1 735.4.a.r 3
7.d odd 6 1 735.4.a.s 3
21.g even 6 1 2205.4.a.bj 3
21.h odd 6 1 315.4.j.e 6
21.h odd 6 1 2205.4.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.i.c 6 1.a even 1 1 trivial
105.4.i.c 6 7.c even 3 1 inner
315.4.j.e 6 3.b odd 2 1
315.4.j.e 6 21.h odd 6 1
735.4.a.r 3 7.c even 3 1
735.4.a.s 3 7.d odd 6 1
2205.4.a.bi 3 21.h odd 6 1
2205.4.a.bj 3 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3 T_{2}^{5} + 17 T_{2}^{4} + 4 T_{2}^{3} + 106 T_{2}^{2} + 112 T_{2} + 196 \) acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 196 + 112 T + 106 T^{2} + 4 T^{3} + 17 T^{4} + 3 T^{5} + T^{6} \)
$3$ \( ( 9 - 3 T + T^{2} )^{3} \)
$5$ \( ( 25 + 5 T + T^{2} )^{3} \)
$7$ \( 40353607 + 235298 T + 33614 T^{2} - 10192 T^{3} + 98 T^{4} + 2 T^{5} + T^{6} \)
$11$ \( 14288400 + 1920240 T + 254284 T^{2} + 8068 T^{3} + 509 T^{4} - T^{5} + T^{6} \)
$13$ \( ( -687 - 49 T + 79 T^{2} + T^{3} )^{2} \)
$17$ \( 377291776 - 2020096 T + 1409344 T^{2} - 31360 T^{3} + 5288 T^{4} - 72 T^{5} + T^{6} \)
$19$ \( 4293001441 - 137135453 T + 6280758 T^{2} - 70345 T^{3} + 2934 T^{4} - 29 T^{5} + T^{6} \)
$23$ \( 2819185216 - 218967904 T + 20352424 T^{2} + 153620 T^{3} + 8093 T^{4} - 63 T^{5} + T^{6} \)
$29$ \( ( 28096 + 1544 T - 220 T^{2} + T^{3} )^{2} \)
$31$ \( 22705015340484 + 297005843718 T + 4533190569 T^{2} + 1052940 T^{3} + 80827 T^{4} + 136 T^{5} + T^{6} \)
$37$ \( 6083064833769 + 292673813355 T + 14187436866 T^{2} - 169821 T^{3} + 120514 T^{4} + 43 T^{5} + T^{6} \)
$41$ \( ( -3130740 + 66236 T + 599 T^{2} + T^{3} )^{2} \)
$43$ \( ( 1053316 - 175635 T - 170 T^{2} + T^{3} )^{2} \)
$47$ \( 945954910704400 + 7358648453280 T + 57151164396 T^{2} + 62230528 T^{3} + 239265 T^{4} - 3 T^{5} + T^{6} \)
$53$ \( 118026496 - 43021440 T + 12085616 T^{2} - 1332488 T^{3} + 113521 T^{4} + 331 T^{5} + T^{6} \)
$59$ \( 604774333465600 + 13415711860480 T + 334980881984 T^{2} - 878386880 T^{3} + 1764872 T^{4} - 1520 T^{5} + T^{6} \)
$61$ \( 2052216117137664 - 18741185870400 T + 118598075280 T^{2} - 389289216 T^{3} + 931900 T^{4} - 1160 T^{5} + T^{6} \)
$67$ \( 1883790895075600 - 2365488372660 T + 37952902961 T^{2} + 130733126 T^{3} + 595135 T^{4} + 806 T^{5} + T^{6} \)
$71$ \( ( -232329576 - 682460 T + 406 T^{2} + T^{3} )^{2} \)
$73$ \( 17273582822500 - 1487673111750 T + 123170492225 T^{2} - 418358140 T^{3} + 1062919 T^{4} - 1192 T^{5} + T^{6} \)
$79$ \( 38998197155670544 - 361676997591580 T + 2842791851145 T^{2} - 4348535126 T^{3} + 4876635 T^{4} - 2590 T^{5} + T^{6} \)
$83$ \( ( -30444208 - 59692 T + 508 T^{2} + T^{3} )^{2} \)
$89$ \( 501888344434534656 + 1096927039840512 T + 2427197976352 T^{2} + 1351850112 T^{3} + 1550132 T^{4} + 42 T^{5} + T^{6} \)
$97$ \( ( -1879560768 - 1909340 T + 1020 T^{2} + T^{3} )^{2} \)
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