Properties

Label 105.2.i.d
Level 105
Weight 2
Character orbit 105.i
Analytic conductor 0.838
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: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 105.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.838429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{3} + ( -2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{4} -\zeta_{12}^{2} q^{5} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{8} -\zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{3} + ( -2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{4} -\zeta_{12}^{2} q^{5} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{8} -\zeta_{12}^{2} q^{9} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{10} + ( 1 + \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{12} + ( 4 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{13} + ( 5 - 3 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{14} - q^{15} + ( 4 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{16} + ( -5 - \zeta_{12} + 5 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{17} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{18} + ( 2 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{19} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{20} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{21} -2 q^{22} + ( -\zeta_{12} + 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{23} + ( -6 + 2 \zeta_{12} + 6 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{24} + ( -1 + \zeta_{12}^{2} ) q^{25} + ( -3 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{26} - q^{27} + ( 2 - 4 \zeta_{12} + 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{28} + ( 1 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{29} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{30} + ( -3 - 2 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{31} + ( 8 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 16 \zeta_{12}^{3} ) q^{32} + ( -\zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{33} + ( -2 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{34} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{35} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{36} + ( -3 \zeta_{12} - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{37} + ( 7 - 3 \zeta_{12} - 7 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{38} + ( 4 + \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{39} + ( -2 \zeta_{12} + 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{40} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{41} + ( 4 - \zeta_{12} - 5 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{42} + ( -2 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{43} -4 \zeta_{12}^{2} q^{44} + ( -1 + \zeta_{12}^{2} ) q^{45} + ( -6 + 4 \zeta_{12} + 6 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{46} -2 \zeta_{12}^{2} q^{47} + ( -8 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{48} + ( -3 - 5 \zeta_{12}^{2} ) q^{49} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{50} + ( \zeta_{12} + 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{51} + ( -2 + 6 \zeta_{12} + 2 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{52} + ( -2 + 6 \zeta_{12} + 2 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{53} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{54} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{55} + ( -8 + 6 \zeta_{12} + 10 \zeta_{12}^{2} - 18 \zeta_{12}^{3} ) q^{56} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{57} + ( 2 \zeta_{12} - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{58} + ( -5 - 3 \zeta_{12} + 5 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{59} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{60} -4 \zeta_{12}^{2} q^{61} + ( 3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{62} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{63} + ( 16 - 16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{64} + ( -\zeta_{12} - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{65} + ( -2 + 2 \zeta_{12}^{2} ) q^{66} + ( 6 - 5 \zeta_{12} - 6 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{67} + ( 8 \zeta_{12} - 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{68} + ( 3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{69} + ( -1 + 2 \zeta_{12} - 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{70} + ( 1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{71} + ( -2 \zeta_{12} + 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{72} + ( -4 + 5 \zeta_{12} + 4 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{73} + ( -7 + \zeta_{12} + 7 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{74} + \zeta_{12}^{2} q^{75} + ( 14 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{76} + ( 1 + 2 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{77} + ( 1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{78} + ( -6 \zeta_{12} - 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{79} + ( -8 + 4 \zeta_{12} + 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{80} + ( -1 + \zeta_{12}^{2} ) q^{81} -2 \zeta_{12}^{2} q^{82} + ( 3 - 14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{83} + ( 10 - 6 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{84} + ( 5 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{85} + ( -\zeta_{12} + 7 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{86} + ( 1 + 3 \zeta_{12} - \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{87} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{88} + ( 7 \zeta_{12} - 3 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{89} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{90} + ( -4 - 4 \zeta_{12} + 5 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{91} + ( -12 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{92} + ( 2 \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{93} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{94} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{95} + ( 8 \zeta_{12} - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{96} + ( 8 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{97} + ( 5 - 2 \zeta_{12} - 8 \zeta_{12}^{2} + 13 \zeta_{12}^{3} ) q^{98} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 2q^{3} - 4q^{4} - 2q^{5} + 4q^{6} - 24q^{8} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{2} + 2q^{3} - 4q^{4} - 2q^{5} + 4q^{6} - 24q^{8} - 2q^{9} + 2q^{10} + 2q^{11} + 4q^{12} + 16q^{13} + 18q^{14} - 4q^{15} - 16q^{16} - 10q^{17} + 2q^{18} - 2q^{19} + 8q^{20} - 8q^{22} + 6q^{23} - 12q^{24} - 2q^{25} + 2q^{26} - 4q^{27} + 24q^{28} + 4q^{29} - 2q^{30} - 6q^{31} + 16q^{32} - 2q^{33} - 8q^{34} + 8q^{36} - 4q^{37} + 14q^{38} + 8q^{39} + 12q^{40} + 4q^{41} + 6q^{42} - 8q^{43} - 8q^{44} - 2q^{45} - 12q^{46} - 4q^{47} - 32q^{48} - 22q^{49} - 4q^{50} + 10q^{51} - 4q^{52} - 4q^{53} - 2q^{54} - 4q^{55} - 12q^{56} - 4q^{57} - 16q^{58} - 10q^{59} + 4q^{60} - 8q^{61} + 12q^{62} + 64q^{64} - 8q^{65} - 4q^{66} + 12q^{67} - 8q^{68} + 12q^{69} - 12q^{70} + 4q^{71} + 12q^{72} - 8q^{73} - 14q^{74} + 2q^{75} + 56q^{76} + 12q^{77} + 4q^{78} - 6q^{79} - 16q^{80} - 2q^{81} - 4q^{82} + 12q^{83} + 36q^{84} + 20q^{85} + 14q^{86} + 2q^{87} - 6q^{89} - 4q^{90} - 6q^{91} - 48q^{92} + 6q^{93} + 4q^{94} - 2q^{95} - 16q^{96} + 32q^{97} + 4q^{98} - 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(-\zeta_{12}^{2}\) \(1\)

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
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.366025 + 0.633975i 0.500000 + 0.866025i 0.732051 + 1.26795i −0.500000 + 0.866025i −0.732051 −0.866025 2.50000i −2.53590 −0.500000 + 0.866025i −0.366025 0.633975i
16.2 1.36603 2.36603i 0.500000 + 0.866025i −2.73205 4.73205i −0.500000 + 0.866025i 2.73205 0.866025 + 2.50000i −9.46410 −0.500000 + 0.866025i 1.36603 + 2.36603i
46.1 −0.366025 0.633975i 0.500000 0.866025i 0.732051 1.26795i −0.500000 0.866025i −0.732051 −0.866025 + 2.50000i −2.53590 −0.500000 0.866025i −0.366025 + 0.633975i
46.2 1.36603 + 2.36603i 0.500000 0.866025i −2.73205 + 4.73205i −0.500000 0.866025i 2.73205 0.866025 2.50000i −9.46410 −0.500000 0.866025i 1.36603 2.36603i
\(n\): 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 105.2.i.d 4
3.b odd 2 1 315.2.j.c 4
4.b odd 2 1 1680.2.bg.o 4
5.b even 2 1 525.2.i.f 4
5.c odd 4 1 525.2.r.a 4
5.c odd 4 1 525.2.r.f 4
7.b odd 2 1 735.2.i.l 4
7.c even 3 1 inner 105.2.i.d 4
7.c even 3 1 735.2.a.g 2
7.d odd 6 1 735.2.a.h 2
7.d odd 6 1 735.2.i.l 4
21.g even 6 1 2205.2.a.ba 2
21.h odd 6 1 315.2.j.c 4
21.h odd 6 1 2205.2.a.z 2
28.g odd 6 1 1680.2.bg.o 4
35.i odd 6 1 3675.2.a.be 2
35.j even 6 1 525.2.i.f 4
35.j even 6 1 3675.2.a.bg 2
35.l odd 12 1 525.2.r.a 4
35.l odd 12 1 525.2.r.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 1.a even 1 1 trivial
105.2.i.d 4 7.c even 3 1 inner
315.2.j.c 4 3.b odd 2 1
315.2.j.c 4 21.h odd 6 1
525.2.i.f 4 5.b even 2 1
525.2.i.f 4 35.j even 6 1
525.2.r.a 4 5.c odd 4 1
525.2.r.a 4 35.l odd 12 1
525.2.r.f 4 5.c odd 4 1
525.2.r.f 4 35.l odd 12 1
735.2.a.g 2 7.c even 3 1
735.2.a.h 2 7.d odd 6 1
735.2.i.l 4 7.b odd 2 1
735.2.i.l 4 7.d odd 6 1
1680.2.bg.o 4 4.b odd 2 1
1680.2.bg.o 4 28.g odd 6 1
2205.2.a.z 2 21.h odd 6 1
2205.2.a.ba 2 21.g even 6 1
3675.2.a.be 2 35.i odd 6 1
3675.2.a.bg 2 35.j even 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 2 T^{2} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} ) \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( 1 + 11 T^{2} + 49 T^{4} \)
$11$ \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 - 8 T + 39 T^{2} - 104 T^{3} + 169 T^{4} )^{2} \)
$17$ \( 1 + 10 T + 44 T^{2} + 220 T^{3} + 1147 T^{4} + 3740 T^{5} + 12716 T^{6} + 49130 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 2 T - 23 T^{2} - 22 T^{3} + 292 T^{4} - 418 T^{5} - 8303 T^{6} + 13718 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 6 T - 16 T^{2} - 36 T^{3} + 1347 T^{4} - 828 T^{5} - 8464 T^{6} - 73002 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 - 2 T + 32 T^{2} - 58 T^{3} + 841 T^{4} )^{2} \)
$31$ \( 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 558 T^{5} - 22103 T^{6} + 178746 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 4 T - 35 T^{2} - 92 T^{3} + 640 T^{4} - 3404 T^{5} - 47915 T^{6} + 202612 T^{7} + 1874161 T^{8} \)
$41$ \( ( 1 - 2 T + 80 T^{2} - 82 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 4 T + 63 T^{2} + 172 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( 1 + 4 T + 14 T^{2} - 416 T^{3} - 3653 T^{4} - 22048 T^{5} + 39326 T^{6} + 595508 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 10 T - 16 T^{2} - 20 T^{3} + 4075 T^{4} - 1180 T^{5} - 55696 T^{6} + 2053790 T^{7} + 12117361 T^{8} \)
$61$ \( ( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 - 12 T + 49 T^{2} + 468 T^{3} - 5112 T^{4} + 31356 T^{5} + 219961 T^{6} - 3609156 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 - 2 T + 116 T^{2} - 142 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 + 8 T - 23 T^{2} - 472 T^{3} - 2432 T^{4} - 34456 T^{5} - 122567 T^{6} + 3112136 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 6 T - 23 T^{2} - 594 T^{3} - 5604 T^{4} - 46926 T^{5} - 143543 T^{6} + 2958234 T^{7} + 38950081 T^{8} \)
$83$ \( ( 1 - 6 T + 28 T^{2} - 498 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( 1 + 6 T - 4 T^{2} - 828 T^{3} - 9525 T^{4} - 73692 T^{5} - 31684 T^{6} + 4229814 T^{7} + 62742241 T^{8} \)
$97$ \( ( 1 - 16 T + 210 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \)
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