Properties

Label 735.2.s.f
Level 735
Weight 2
Character orbit 735.s
Analytic conductor 5.869
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 735.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(\zeta_{6}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
0.500000 + 0.866025i
0.500000 0.866025i
1.50000 0.866025i 1.50000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 3.00000 0 1.73205i 1.50000 + 2.59808i −1.50000 0.866025i
656.1 1.50000 + 0.866025i 1.50000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 3.00000 0 1.73205i 1.50000 2.59808i −1.50000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.s.f 2
3.b odd 2 1 735.2.s.b 2
7.b odd 2 1 735.2.s.d 2
7.c even 3 1 105.2.b.b yes 2
7.c even 3 1 735.2.s.a 2
7.d odd 6 1 105.2.b.a 2
7.d odd 6 1 735.2.s.b 2
21.c even 2 1 735.2.s.a 2
21.g even 6 1 105.2.b.b yes 2
21.g even 6 1 inner 735.2.s.f 2
21.h odd 6 1 105.2.b.a 2
21.h odd 6 1 735.2.s.d 2
28.f even 6 1 1680.2.f.b 2
28.g odd 6 1 1680.2.f.c 2
35.i odd 6 1 525.2.b.a 2
35.j even 6 1 525.2.b.b 2
35.k even 12 2 525.2.g.b 4
35.l odd 12 2 525.2.g.c 4
84.j odd 6 1 1680.2.f.c 2
84.n even 6 1 1680.2.f.b 2
105.o odd 6 1 525.2.b.a 2
105.p even 6 1 525.2.b.b 2
105.w odd 12 2 525.2.g.c 4
105.x even 12 2 525.2.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.a 2 7.d odd 6 1
105.2.b.a 2 21.h odd 6 1
105.2.b.b yes 2 7.c even 3 1
105.2.b.b yes 2 21.g even 6 1
525.2.b.a 2 35.i odd 6 1
525.2.b.a 2 105.o odd 6 1
525.2.b.b 2 35.j even 6 1
525.2.b.b 2 105.p even 6 1
525.2.g.b 4 35.k even 12 2
525.2.g.b 4 105.x even 12 2
525.2.g.c 4 35.l odd 12 2
525.2.g.c 4 105.w odd 12 2
735.2.s.a 2 7.c even 3 1
735.2.s.a 2 21.c even 2 1
735.2.s.b 2 3.b odd 2 1
735.2.s.b 2 7.d odd 6 1
735.2.s.d 2 7.b odd 2 1
735.2.s.d 2 21.h odd 6 1
735.2.s.f 2 1.a even 1 1 trivial
735.2.s.f 2 21.g even 6 1 inner
1680.2.f.b 2 28.f even 6 1
1680.2.f.b 2 84.n even 6 1
1680.2.f.c 2 28.g odd 6 1
1680.2.f.c 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{2} - 3 T_{2} + 3 \)
\( T_{13} \)
\( T_{17}^{2} - 6 T_{17} + 36 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \)
$3$ \( 1 - 3 T + 3 T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ 1
$11$ \( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 - 6 T + 35 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( 1 - 10 T^{2} + 841 T^{4} \)
$31$ \( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} \)
$37$ \( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 53 T^{2} + 2809 T^{4} \)
$59$ \( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \)
$67$ \( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 130 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 146 T^{2} + 9409 T^{4} \)
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