Properties

Label 735.2.i.a
Level $735$
Weight $2$
Character orbit 735.i
Analytic conductor $5.869$
Analytic rank $1$
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.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

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

$q$-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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0 −3.00000 −0.500000 + 0.866025i −0.500000 0.866025i
361.1 −0.500000 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 1.00000 0 −3.00000 −0.500000 0.866025i −0.500000 + 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
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 735.2.i.a 2
7.b odd 2 1 735.2.i.b 2
7.c even 3 1 105.2.a.a 1
7.c even 3 1 inner 735.2.i.a 2
7.d odd 6 1 735.2.a.f 1
7.d odd 6 1 735.2.i.b 2
21.g even 6 1 2205.2.a.b 1
21.h odd 6 1 315.2.a.a 1
28.g odd 6 1 1680.2.a.f 1
35.i odd 6 1 3675.2.a.f 1
35.j even 6 1 525.2.a.a 1
35.l odd 12 2 525.2.d.b 2
56.k odd 6 1 6720.2.a.bk 1
56.p even 6 1 6720.2.a.p 1
84.n even 6 1 5040.2.a.d 1
105.o odd 6 1 1575.2.a.h 1
105.x even 12 2 1575.2.d.b 2
140.p odd 6 1 8400.2.a.co 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 7.c even 3 1
315.2.a.a 1 21.h odd 6 1
525.2.a.a 1 35.j even 6 1
525.2.d.b 2 35.l odd 12 2
735.2.a.f 1 7.d odd 6 1
735.2.i.a 2 1.a even 1 1 trivial
735.2.i.a 2 7.c even 3 1 inner
735.2.i.b 2 7.b odd 2 1
735.2.i.b 2 7.d odd 6 1
1575.2.a.h 1 105.o odd 6 1
1575.2.d.b 2 105.x even 12 2
1680.2.a.f 1 28.g odd 6 1
2205.2.a.b 1 21.g even 6 1
3675.2.a.f 1 35.i odd 6 1
5040.2.a.d 1 84.n even 6 1
6720.2.a.p 1 56.p even 6 1
6720.2.a.bk 1 56.k odd 6 1
8400.2.a.co 1 140.p odd 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \)
\( T_{13} + 6 \)
\( T_{17}^{2} + 2 T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 6 + T )^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 64 - 8 T + T^{2} \)
$23$ \( 64 + 8 T + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 4 - 2 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 64 + 8 T + T^{2} \)
$53$ \( 100 + 10 T + T^{2} \)
$59$ \( 16 + 4 T + T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 16 + 4 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 4 - 2 T + T^{2} \)
$79$ \( 64 + 8 T + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( 18 + T )^{2} \)
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