Properties

Label 105.4.i.a
Level $105$
Weight $4$
Character orbit 105.i
Analytic conductor $6.195$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 5 \zeta_{6} q^{5} -9 q^{6} + ( -7 - 14 \zeta_{6} ) q^{7} -21 q^{8} -9 \zeta_{6} q^{9} +O(q^{10})\) \( q -3 \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 5 \zeta_{6} q^{5} -9 q^{6} + ( -7 - 14 \zeta_{6} ) q^{7} -21 q^{8} -9 \zeta_{6} q^{9} + ( 15 - 15 \zeta_{6} ) q^{10} + ( 45 - 45 \zeta_{6} ) q^{11} + 3 \zeta_{6} q^{12} -31 q^{13} + ( -42 + 63 \zeta_{6} ) q^{14} + 15 q^{15} + 71 \zeta_{6} q^{16} + ( -96 + 96 \zeta_{6} ) q^{17} + ( -27 + 27 \zeta_{6} ) q^{18} -149 \zeta_{6} q^{19} -5 q^{20} + ( -63 + 21 \zeta_{6} ) q^{21} -135 q^{22} + 141 \zeta_{6} q^{23} + ( -63 + 63 \zeta_{6} ) q^{24} + ( -25 + 25 \zeta_{6} ) q^{25} + 93 \zeta_{6} q^{26} -27 q^{27} + ( 21 - 7 \zeta_{6} ) q^{28} + 48 q^{29} -45 \zeta_{6} q^{30} + ( 178 - 178 \zeta_{6} ) q^{31} + ( 45 - 45 \zeta_{6} ) q^{32} -135 \zeta_{6} q^{33} + 288 q^{34} + ( 70 - 105 \zeta_{6} ) q^{35} + 9 q^{36} -371 \zeta_{6} q^{37} + ( -447 + 447 \zeta_{6} ) q^{38} + ( -93 + 93 \zeta_{6} ) q^{39} -105 \zeta_{6} q^{40} + 225 q^{41} + ( 63 + 126 \zeta_{6} ) q^{42} + 344 q^{43} + 45 \zeta_{6} q^{44} + ( 45 - 45 \zeta_{6} ) q^{45} + ( 423 - 423 \zeta_{6} ) q^{46} -375 \zeta_{6} q^{47} + 213 q^{48} + ( -147 + 392 \zeta_{6} ) q^{49} + 75 q^{50} + 288 \zeta_{6} q^{51} + ( 31 - 31 \zeta_{6} ) q^{52} + ( 663 - 663 \zeta_{6} ) q^{53} + 81 \zeta_{6} q^{54} + 225 q^{55} + ( 147 + 294 \zeta_{6} ) q^{56} -447 q^{57} -144 \zeta_{6} q^{58} + ( 60 - 60 \zeta_{6} ) q^{59} + ( -15 + 15 \zeta_{6} ) q^{60} -392 \zeta_{6} q^{61} -534 q^{62} + ( -126 + 189 \zeta_{6} ) q^{63} + 433 q^{64} -155 \zeta_{6} q^{65} + ( -405 + 405 \zeta_{6} ) q^{66} + ( 280 - 280 \zeta_{6} ) q^{67} -96 \zeta_{6} q^{68} + 423 q^{69} + ( -315 + 105 \zeta_{6} ) q^{70} + 258 q^{71} + 189 \zeta_{6} q^{72} + ( -578 + 578 \zeta_{6} ) q^{73} + ( -1113 + 1113 \zeta_{6} ) q^{74} + 75 \zeta_{6} q^{75} + 149 q^{76} + ( -945 + 315 \zeta_{6} ) q^{77} + 279 q^{78} -152 \zeta_{6} q^{79} + ( -355 + 355 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} -675 \zeta_{6} q^{82} -432 q^{83} + ( 42 - 63 \zeta_{6} ) q^{84} -480 q^{85} -1032 \zeta_{6} q^{86} + ( 144 - 144 \zeta_{6} ) q^{87} + ( -945 + 945 \zeta_{6} ) q^{88} + 234 \zeta_{6} q^{89} -135 q^{90} + ( 217 + 434 \zeta_{6} ) q^{91} -141 q^{92} -534 \zeta_{6} q^{93} + ( -1125 + 1125 \zeta_{6} ) q^{94} + ( 745 - 745 \zeta_{6} ) q^{95} -135 \zeta_{6} q^{96} + 1352 q^{97} + ( 1176 - 735 \zeta_{6} ) q^{98} -405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + 3q^{3} - q^{4} + 5q^{5} - 18q^{6} - 28q^{7} - 42q^{8} - 9q^{9} + O(q^{10}) \) \( 2q - 3q^{2} + 3q^{3} - q^{4} + 5q^{5} - 18q^{6} - 28q^{7} - 42q^{8} - 9q^{9} + 15q^{10} + 45q^{11} + 3q^{12} - 62q^{13} - 21q^{14} + 30q^{15} + 71q^{16} - 96q^{17} - 27q^{18} - 149q^{19} - 10q^{20} - 105q^{21} - 270q^{22} + 141q^{23} - 63q^{24} - 25q^{25} + 93q^{26} - 54q^{27} + 35q^{28} + 96q^{29} - 45q^{30} + 178q^{31} + 45q^{32} - 135q^{33} + 576q^{34} + 35q^{35} + 18q^{36} - 371q^{37} - 447q^{38} - 93q^{39} - 105q^{40} + 450q^{41} + 252q^{42} + 688q^{43} + 45q^{44} + 45q^{45} + 423q^{46} - 375q^{47} + 426q^{48} + 98q^{49} + 150q^{50} + 288q^{51} + 31q^{52} + 663q^{53} + 81q^{54} + 450q^{55} + 588q^{56} - 894q^{57} - 144q^{58} + 60q^{59} - 15q^{60} - 392q^{61} - 1068q^{62} - 63q^{63} + 866q^{64} - 155q^{65} - 405q^{66} + 280q^{67} - 96q^{68} + 846q^{69} - 525q^{70} + 516q^{71} + 189q^{72} - 578q^{73} - 1113q^{74} + 75q^{75} + 298q^{76} - 1575q^{77} + 558q^{78} - 152q^{79} - 355q^{80} - 81q^{81} - 675q^{82} - 864q^{83} + 21q^{84} - 960q^{85} - 1032q^{86} + 144q^{87} - 945q^{88} + 234q^{89} - 270q^{90} + 868q^{91} - 282q^{92} - 534q^{93} - 1125q^{94} + 745q^{95} - 135q^{96} + 2704q^{97} + 1617q^{98} - 810q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
−1.50000 + 2.59808i 1.50000 + 2.59808i −0.500000 0.866025i 2.50000 4.33013i −9.00000 −14.0000 + 12.1244i −21.0000 −4.50000 + 7.79423i 7.50000 + 12.9904i
46.1 −1.50000 2.59808i 1.50000 2.59808i −0.500000 + 0.866025i 2.50000 + 4.33013i −9.00000 −14.0000 12.1244i −21.0000 −4.50000 7.79423i 7.50000 12.9904i
\(n\): 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.a 2
3.b odd 2 1 315.4.j.a 2
7.c even 3 1 inner 105.4.i.a 2
7.c even 3 1 735.4.a.g 1
7.d odd 6 1 735.4.a.h 1
21.g even 6 1 2205.4.a.d 1
21.h odd 6 1 315.4.j.a 2
21.h odd 6 1 2205.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.i.a 2 1.a even 1 1 trivial
105.4.i.a 2 7.c even 3 1 inner
315.4.j.a 2 3.b odd 2 1
315.4.j.a 2 21.h odd 6 1
735.4.a.g 1 7.c even 3 1
735.4.a.h 1 7.d odd 6 1
2205.4.a.d 1 21.g even 6 1
2205.4.a.h 1 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 3 T + T^{2} \)
$3$ \( 9 - 3 T + T^{2} \)
$5$ \( 25 - 5 T + T^{2} \)
$7$ \( 343 + 28 T + T^{2} \)
$11$ \( 2025 - 45 T + T^{2} \)
$13$ \( ( 31 + T )^{2} \)
$17$ \( 9216 + 96 T + T^{2} \)
$19$ \( 22201 + 149 T + T^{2} \)
$23$ \( 19881 - 141 T + T^{2} \)
$29$ \( ( -48 + T )^{2} \)
$31$ \( 31684 - 178 T + T^{2} \)
$37$ \( 137641 + 371 T + T^{2} \)
$41$ \( ( -225 + T )^{2} \)
$43$ \( ( -344 + T )^{2} \)
$47$ \( 140625 + 375 T + T^{2} \)
$53$ \( 439569 - 663 T + T^{2} \)
$59$ \( 3600 - 60 T + T^{2} \)
$61$ \( 153664 + 392 T + T^{2} \)
$67$ \( 78400 - 280 T + T^{2} \)
$71$ \( ( -258 + T )^{2} \)
$73$ \( 334084 + 578 T + T^{2} \)
$79$ \( 23104 + 152 T + T^{2} \)
$83$ \( ( 432 + T )^{2} \)
$89$ \( 54756 - 234 T + T^{2} \)
$97$ \( ( -1352 + T )^{2} \)
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