Properties

Label 735.2.d.a
Level 735
Weight 2
Character orbit 735.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
1.00000i
1.00000i
1.00000i 1.00000i 1.00000 −1.00000 2.00000i 1.00000 0 3.00000i −1.00000 −2.00000 + 1.00000i
589.2 1.00000i 1.00000i 1.00000 −1.00000 + 2.00000i 1.00000 0 3.00000i −1.00000 −2.00000 1.00000i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.d.a 2
3.b odd 2 1 2205.2.d.f 2
5.b even 2 1 inner 735.2.d.a 2
5.c odd 4 1 3675.2.a.d 1
5.c odd 4 1 3675.2.a.l 1
7.b odd 2 1 105.2.d.a 2
7.c even 3 2 735.2.q.b 4
7.d odd 6 2 735.2.q.a 4
15.d odd 2 1 2205.2.d.f 2
21.c even 2 1 315.2.d.c 2
28.d even 2 1 1680.2.t.f 2
35.c odd 2 1 105.2.d.a 2
35.f even 4 1 525.2.a.b 1
35.f even 4 1 525.2.a.c 1
35.i odd 6 2 735.2.q.a 4
35.j even 6 2 735.2.q.b 4
84.h odd 2 1 5040.2.t.e 2
105.g even 2 1 315.2.d.c 2
105.k odd 4 1 1575.2.a.e 1
105.k odd 4 1 1575.2.a.i 1
140.c even 2 1 1680.2.t.f 2
140.j odd 4 1 8400.2.a.bj 1
140.j odd 4 1 8400.2.a.ch 1
420.o odd 2 1 5040.2.t.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 7.b odd 2 1
105.2.d.a 2 35.c odd 2 1
315.2.d.c 2 21.c even 2 1
315.2.d.c 2 105.g even 2 1
525.2.a.b 1 35.f even 4 1
525.2.a.c 1 35.f even 4 1
735.2.d.a 2 1.a even 1 1 trivial
735.2.d.a 2 5.b even 2 1 inner
735.2.q.a 4 7.d odd 6 2
735.2.q.a 4 35.i odd 6 2
735.2.q.b 4 7.c even 3 2
735.2.q.b 4 35.j even 6 2
1575.2.a.e 1 105.k odd 4 1
1575.2.a.i 1 105.k odd 4 1
1680.2.t.f 2 28.d even 2 1
1680.2.t.f 2 140.c even 2 1
2205.2.d.f 2 3.b odd 2 1
2205.2.d.f 2 15.d odd 2 1
3675.2.a.d 1 5.c odd 4 1
3675.2.a.l 1 5.c odd 4 1
5040.2.t.e 2 84.h odd 2 1
5040.2.t.e 2 420.o odd 2 1
8400.2.a.bj 1 140.j odd 4 1
8400.2.a.ch 1 140.j odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 1 \)
\( T_{19} + 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T^{2} + 4 T^{4} \)
$3$ \( 1 + T^{2} \)
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ 1
$11$ \( ( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( 1 - 18 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 6 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 10 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 58 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 47 T^{2} )^{2} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 8 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 122 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 10 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 102 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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