Properties

Label 735.2.q.b
Level 735
Weight 2
Character orbit 735.q
Analytic conductor 5.869
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −0.866025 + 0.500000i −0.500000 0.866025i −1.23205 1.86603i 1.00000 0 3.00000i 0.500000 0.866025i 0.133975 + 2.23205i
79.2 0.866025 + 0.500000i 0.866025 0.500000i −0.500000 0.866025i 2.23205 + 0.133975i 1.00000 0 3.00000i 0.500000 0.866025i 1.86603 + 1.23205i
214.1 −0.866025 + 0.500000i −0.866025 0.500000i −0.500000 + 0.866025i −1.23205 + 1.86603i 1.00000 0 3.00000i 0.500000 + 0.866025i 0.133975 2.23205i
214.2 0.866025 0.500000i 0.866025 + 0.500000i −0.500000 + 0.866025i 2.23205 0.133975i 1.00000 0 3.00000i 0.500000 + 0.866025i 1.86603 1.23205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j 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.q.b 4
5.b even 2 1 inner 735.2.q.b 4
7.b odd 2 1 735.2.q.a 4
7.c even 3 1 735.2.d.a 2
7.c even 3 1 inner 735.2.q.b 4
7.d odd 6 1 105.2.d.a 2
7.d odd 6 1 735.2.q.a 4
21.g even 6 1 315.2.d.c 2
21.h odd 6 1 2205.2.d.f 2
28.f even 6 1 1680.2.t.f 2
35.c odd 2 1 735.2.q.a 4
35.i odd 6 1 105.2.d.a 2
35.i odd 6 1 735.2.q.a 4
35.j even 6 1 735.2.d.a 2
35.j even 6 1 inner 735.2.q.b 4
35.k even 12 1 525.2.a.b 1
35.k even 12 1 525.2.a.c 1
35.l odd 12 1 3675.2.a.d 1
35.l odd 12 1 3675.2.a.l 1
84.j odd 6 1 5040.2.t.e 2
105.o odd 6 1 2205.2.d.f 2
105.p even 6 1 315.2.d.c 2
105.w odd 12 1 1575.2.a.e 1
105.w odd 12 1 1575.2.a.i 1
140.s even 6 1 1680.2.t.f 2
140.x odd 12 1 8400.2.a.bj 1
140.x odd 12 1 8400.2.a.ch 1
420.be odd 6 1 5040.2.t.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 7.d odd 6 1
105.2.d.a 2 35.i odd 6 1
315.2.d.c 2 21.g even 6 1
315.2.d.c 2 105.p even 6 1
525.2.a.b 1 35.k even 12 1
525.2.a.c 1 35.k even 12 1
735.2.d.a 2 7.c even 3 1
735.2.d.a 2 35.j even 6 1
735.2.q.a 4 7.b odd 2 1
735.2.q.a 4 7.d odd 6 1
735.2.q.a 4 35.c odd 2 1
735.2.q.a 4 35.i odd 6 1
735.2.q.b 4 1.a even 1 1 trivial
735.2.q.b 4 5.b even 2 1 inner
735.2.q.b 4 7.c even 3 1 inner
735.2.q.b 4 35.j even 6 1 inner
1575.2.a.e 1 105.w odd 12 1
1575.2.a.i 1 105.w odd 12 1
1680.2.t.f 2 28.f even 6 1
1680.2.t.f 2 140.s even 6 1
2205.2.d.f 2 21.h odd 6 1
2205.2.d.f 2 105.o odd 6 1
3675.2.a.d 1 35.l odd 12 1
3675.2.a.l 1 35.l odd 12 1
5040.2.t.e 2 84.j odd 6 1
5040.2.t.e 2 420.be odd 6 1
8400.2.a.bj 1 140.x odd 12 1
8400.2.a.ch 1 140.x odd 12 1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{2} + 1 \)
\( T_{19}^{2} - 6 T_{19} + 36 \)
\( T_{73}^{4} - 36 T_{73}^{2} + 1296 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 2 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 8 T + 5 T^{2} - 472 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 - 13 T^{2} + 4489 T^{4} ) \)
$71$ \( ( 1 - 10 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} )( 1 + 16 T + 183 T^{2} + 1168 T^{3} + 5329 T^{4} ) \)
$79$ \( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 102 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 190 T^{2} + 9409 T^{4} )^{2} \)
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