Properties

Label 1530.2.f.b
Level 1530
Weight 2
Character orbit 1530.f
Analytic conductor 12.217
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1530.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2171115093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 170)
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + ( -2 - i ) q^{5} + 2 q^{7} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} + ( -2 - i ) q^{5} + 2 q^{7} -i q^{8} + ( 1 - 2 i ) q^{10} -i q^{13} + 2 i q^{14} + q^{16} + ( -1 + 4 i ) q^{17} -5 q^{19} + ( 2 + i ) q^{20} + 4 q^{23} + ( 3 + 4 i ) q^{25} + q^{26} -2 q^{28} -9 i q^{29} -5 i q^{31} + i q^{32} + ( -4 - i ) q^{34} + ( -4 - 2 i ) q^{35} + 2 q^{37} -5 i q^{38} + ( -1 + 2 i ) q^{40} -10 i q^{41} -6 i q^{43} + 4 i q^{46} -7 i q^{47} -3 q^{49} + ( -4 + 3 i ) q^{50} + i q^{52} + i q^{53} -2 i q^{56} + 9 q^{58} + 5 q^{59} -5 i q^{61} + 5 q^{62} - q^{64} + ( -1 + 2 i ) q^{65} + 2 i q^{67} + ( 1 - 4 i ) q^{68} + ( 2 - 4 i ) q^{70} + 5 i q^{71} + 11 q^{73} + 2 i q^{74} + 5 q^{76} -16 i q^{79} + ( -2 - i ) q^{80} + 10 q^{82} + 6 i q^{83} + ( 6 - 7 i ) q^{85} + 6 q^{86} -5 q^{89} -2 i q^{91} -4 q^{92} + 7 q^{94} + ( 10 + 5 i ) q^{95} + 7 q^{97} -3 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 4q^{5} + 4q^{7} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{5} + 4q^{7} + 2q^{10} + 2q^{16} - 2q^{17} - 10q^{19} + 4q^{20} + 8q^{23} + 6q^{25} + 2q^{26} - 4q^{28} - 8q^{34} - 8q^{35} + 4q^{37} - 2q^{40} - 6q^{49} - 8q^{50} + 18q^{58} + 10q^{59} + 10q^{62} - 2q^{64} - 2q^{65} + 2q^{68} + 4q^{70} + 22q^{73} + 10q^{76} - 4q^{80} + 20q^{82} + 12q^{85} + 12q^{86} - 10q^{89} - 8q^{92} + 14q^{94} + 20q^{95} + 14q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times\).

\(n\) \(307\) \(1261\) \(1361\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1189.1
1.00000i
1.00000i
1.00000i 0 −1.00000 −2.00000 + 1.00000i 0 2.00000 1.00000i 0 1.00000 + 2.00000i
1189.2 1.00000i 0 −1.00000 −2.00000 1.00000i 0 2.00000 1.00000i 0 1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1530.2.f.b 2
3.b odd 2 1 170.2.d.b yes 2
5.b even 2 1 1530.2.f.e 2
12.b even 2 1 1360.2.o.a 2
15.d odd 2 1 170.2.d.a 2
15.e even 4 1 850.2.b.c 2
15.e even 4 1 850.2.b.i 2
17.b even 2 1 1530.2.f.e 2
51.c odd 2 1 170.2.d.a 2
60.h even 2 1 1360.2.o.b 2
85.c even 2 1 inner 1530.2.f.b 2
204.h even 2 1 1360.2.o.b 2
255.h odd 2 1 170.2.d.b yes 2
255.o even 4 1 850.2.b.c 2
255.o even 4 1 850.2.b.i 2
1020.b even 2 1 1360.2.o.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.d.a 2 15.d odd 2 1
170.2.d.a 2 51.c odd 2 1
170.2.d.b yes 2 3.b odd 2 1
170.2.d.b yes 2 255.h odd 2 1
850.2.b.c 2 15.e even 4 1
850.2.b.c 2 255.o even 4 1
850.2.b.i 2 15.e even 4 1
850.2.b.i 2 255.o even 4 1
1360.2.o.a 2 12.b even 2 1
1360.2.o.a 2 1020.b even 2 1
1360.2.o.b 2 60.h even 2 1
1360.2.o.b 2 204.h even 2 1
1530.2.f.b 2 1.a even 1 1 trivial
1530.2.f.b 2 85.c even 2 1 inner
1530.2.f.e 2 5.b even 2 1
1530.2.f.e 2 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1530, [\chi])\):

\( T_{7} - 2 \)
\( T_{23} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ 1
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( 1 - 25 T^{2} + 169 T^{4} \)
$17$ \( 1 + 2 T + 17 T^{2} \)
$19$ \( ( 1 + 5 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 4 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 23 T^{2} + 841 T^{4} \)
$31$ \( 1 - 37 T^{2} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 8 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} ) \)
$43$ \( 1 - 50 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 45 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 105 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 - 5 T + 59 T^{2} )^{2} \)
$61$ \( 1 - 97 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 130 T^{2} + 4489 T^{4} \)
$71$ \( 1 - 117 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 - 11 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 98 T^{2} + 6241 T^{4} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 5 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 7 T + 97 T^{2} )^{2} \)
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