Properties

Label 735.2.i.e
Level $735$
Weight $2$
Character orbit 735.i
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.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + ( - \zeta_{6} + 1) q^{4} - \zeta_{6} q^{5} + q^{6} + 3 q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + ( - \zeta_{6} + 1) q^{4} - \zeta_{6} q^{5} + q^{6} + 3 q^{8} - \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} + ( - 4 \zeta_{6} + 4) q^{11} - \zeta_{6} q^{12} - 2 q^{13} - q^{15} + \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + ( - \zeta_{6} + 1) q^{18} - 4 \zeta_{6} q^{19} - q^{20} + 4 q^{22} + ( - 3 \zeta_{6} + 3) q^{24} + (\zeta_{6} - 1) q^{25} - 2 \zeta_{6} q^{26} - q^{27} - 2 q^{29} - \zeta_{6} q^{30} + ( - 5 \zeta_{6} + 5) q^{32} - 4 \zeta_{6} q^{33} - 2 q^{34} - q^{36} + 10 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + (2 \zeta_{6} - 2) q^{39} - 3 \zeta_{6} q^{40} + 10 q^{41} + 4 q^{43} - 4 \zeta_{6} q^{44} + (\zeta_{6} - 1) q^{45} - 8 \zeta_{6} q^{47} + q^{48} - q^{50} + 2 \zeta_{6} q^{51} + (2 \zeta_{6} - 2) q^{52} + ( - 10 \zeta_{6} + 10) q^{53} - \zeta_{6} q^{54} - 4 q^{55} - 4 q^{57} - 2 \zeta_{6} q^{58} + ( - 4 \zeta_{6} + 4) q^{59} + (\zeta_{6} - 1) q^{60} + 2 \zeta_{6} q^{61} + 7 q^{64} + 2 \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{66} + (12 \zeta_{6} - 12) q^{67} + 2 \zeta_{6} q^{68} - 8 q^{71} - 3 \zeta_{6} q^{72} + (10 \zeta_{6} - 10) q^{73} + (10 \zeta_{6} - 10) q^{74} + \zeta_{6} q^{75} - 4 q^{76} - 2 q^{78} + ( - \zeta_{6} + 1) q^{80} + (\zeta_{6} - 1) q^{81} + 10 \zeta_{6} q^{82} + 12 q^{83} + 2 q^{85} + 4 \zeta_{6} q^{86} + (2 \zeta_{6} - 2) q^{87} + ( - 12 \zeta_{6} + 12) q^{88} + 6 \zeta_{6} q^{89} - q^{90} + ( - 8 \zeta_{6} + 8) q^{94} + (4 \zeta_{6} - 4) q^{95} - 5 \zeta_{6} q^{96} + 2 q^{97} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} + q^{4} - q^{5} + 2 q^{6} + 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{3} + q^{4} - q^{5} + 2 q^{6} + 6 q^{8} - q^{9} + q^{10} + 4 q^{11} - q^{12} - 4 q^{13} - 2 q^{15} + q^{16} - 2 q^{17} + q^{18} - 4 q^{19} - 2 q^{20} + 8 q^{22} + 3 q^{24} - q^{25} - 2 q^{26} - 2 q^{27} - 4 q^{29} - q^{30} + 5 q^{32} - 4 q^{33} - 4 q^{34} - 2 q^{36} + 10 q^{37} + 4 q^{38} - 2 q^{39} - 3 q^{40} + 20 q^{41} + 8 q^{43} - 4 q^{44} - q^{45} - 8 q^{47} + 2 q^{48} - 2 q^{50} + 2 q^{51} - 2 q^{52} + 10 q^{53} - q^{54} - 8 q^{55} - 8 q^{57} - 2 q^{58} + 4 q^{59} - q^{60} + 2 q^{61} + 14 q^{64} + 2 q^{65} + 4 q^{66} - 12 q^{67} + 2 q^{68} - 16 q^{71} - 3 q^{72} - 10 q^{73} - 10 q^{74} + q^{75} - 8 q^{76} - 4 q^{78} + q^{80} - q^{81} + 10 q^{82} + 24 q^{83} + 4 q^{85} + 4 q^{86} - 2 q^{87} + 12 q^{88} + 6 q^{89} - 2 q^{90} + 8 q^{94} - 4 q^{95} - 5 q^{96} + 4 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.e 2
7.b odd 2 1 735.2.i.d 2
7.c even 3 1 15.2.a.a 1
7.c even 3 1 inner 735.2.i.e 2
7.d odd 6 1 735.2.a.c 1
7.d odd 6 1 735.2.i.d 2
21.g even 6 1 2205.2.a.i 1
21.h odd 6 1 45.2.a.a 1
28.g odd 6 1 240.2.a.d 1
35.i odd 6 1 3675.2.a.j 1
35.j even 6 1 75.2.a.b 1
35.l odd 12 2 75.2.b.b 2
56.k odd 6 1 960.2.a.a 1
56.p even 6 1 960.2.a.l 1
63.g even 3 1 405.2.e.f 2
63.h even 3 1 405.2.e.f 2
63.j odd 6 1 405.2.e.c 2
63.n odd 6 1 405.2.e.c 2
77.h odd 6 1 1815.2.a.d 1
84.n even 6 1 720.2.a.c 1
91.r even 6 1 2535.2.a.j 1
105.o odd 6 1 225.2.a.b 1
105.x even 12 2 225.2.b.b 2
112.u odd 12 2 3840.2.k.r 2
112.w even 12 2 3840.2.k.m 2
119.j even 6 1 4335.2.a.c 1
133.r odd 6 1 5415.2.a.j 1
140.p odd 6 1 1200.2.a.e 1
140.w even 12 2 1200.2.f.h 2
161.f odd 6 1 7935.2.a.d 1
168.s odd 6 1 2880.2.a.y 1
168.v even 6 1 2880.2.a.bc 1
231.l even 6 1 5445.2.a.c 1
273.w odd 6 1 7605.2.a.g 1
280.bf even 6 1 4800.2.a.t 1
280.bi odd 6 1 4800.2.a.bz 1
280.br even 12 2 4800.2.f.c 2
280.bt odd 12 2 4800.2.f.bf 2
385.q odd 6 1 9075.2.a.g 1
420.ba even 6 1 3600.2.a.u 1
420.bp odd 12 2 3600.2.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 7.c even 3 1
45.2.a.a 1 21.h odd 6 1
75.2.a.b 1 35.j even 6 1
75.2.b.b 2 35.l odd 12 2
225.2.a.b 1 105.o odd 6 1
225.2.b.b 2 105.x even 12 2
240.2.a.d 1 28.g odd 6 1
405.2.e.c 2 63.j odd 6 1
405.2.e.c 2 63.n odd 6 1
405.2.e.f 2 63.g even 3 1
405.2.e.f 2 63.h even 3 1
720.2.a.c 1 84.n even 6 1
735.2.a.c 1 7.d odd 6 1
735.2.i.d 2 7.b odd 2 1
735.2.i.d 2 7.d odd 6 1
735.2.i.e 2 1.a even 1 1 trivial
735.2.i.e 2 7.c even 3 1 inner
960.2.a.a 1 56.k odd 6 1
960.2.a.l 1 56.p even 6 1
1200.2.a.e 1 140.p odd 6 1
1200.2.f.h 2 140.w even 12 2
1815.2.a.d 1 77.h odd 6 1
2205.2.a.i 1 21.g even 6 1
2535.2.a.j 1 91.r even 6 1
2880.2.a.y 1 168.s odd 6 1
2880.2.a.bc 1 168.v even 6 1
3600.2.a.u 1 420.ba even 6 1
3600.2.f.e 2 420.bp odd 12 2
3675.2.a.j 1 35.i odd 6 1
3840.2.k.m 2 112.w even 12 2
3840.2.k.r 2 112.u odd 12 2
4335.2.a.c 1 119.j even 6 1
4800.2.a.t 1 280.bf even 6 1
4800.2.a.bz 1 280.bi odd 6 1
4800.2.f.c 2 280.br even 12 2
4800.2.f.bf 2 280.bt odd 12 2
5415.2.a.j 1 133.r odd 6 1
5445.2.a.c 1 231.l even 6 1
7605.2.a.g 1 273.w odd 6 1
7935.2.a.d 1 161.f odd 6 1
9075.2.a.g 1 385.q odd 6 1

Hecke kernels

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

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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