Properties

Label 1035.2.b.a
Level $1035$
Weight $2$
Character orbit 1035.b
Analytic conductor $8.265$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1035.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.26451660920\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1035\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(622\) \(856\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
1.00000i
1.00000i
2.00000i 0 −2.00000 −2.00000 1.00000i 0 1.00000i 0 0 −2.00000 + 4.00000i
829.2 2.00000i 0 −2.00000 −2.00000 + 1.00000i 0 1.00000i 0 0 −2.00000 4.00000i
\(n\): 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 1035.2.b.a 2
3.b odd 2 1 115.2.b.a 2
5.b even 2 1 inner 1035.2.b.a 2
5.c odd 4 1 5175.2.a.a 1
5.c odd 4 1 5175.2.a.z 1
12.b even 2 1 1840.2.e.b 2
15.d odd 2 1 115.2.b.a 2
15.e even 4 1 575.2.a.a 1
15.e even 4 1 575.2.a.e 1
60.h even 2 1 1840.2.e.b 2
60.l odd 4 1 9200.2.a.g 1
60.l odd 4 1 9200.2.a.bg 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.a 2 3.b odd 2 1
115.2.b.a 2 15.d odd 2 1
575.2.a.a 1 15.e even 4 1
575.2.a.e 1 15.e even 4 1
1035.2.b.a 2 1.a even 1 1 trivial
1035.2.b.a 2 5.b even 2 1 inner
1840.2.e.b 2 12.b even 2 1
1840.2.e.b 2 60.h even 2 1
5175.2.a.a 1 5.c odd 4 1
5175.2.a.z 1 5.c odd 4 1
9200.2.a.g 1 60.l odd 4 1
9200.2.a.bg 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1035, [\chi])\):

\( T_{2}^{2} + 4 \)
\( T_{7}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 25 + T^{2} \)
$19$ \( ( 8 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( ( 5 + T )^{2} \)
$37$ \( 49 + T^{2} \)
$41$ \( ( -7 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 4 + T^{2} \)
$53$ \( 1 + T^{2} \)
$59$ \( ( -3 + T )^{2} \)
$61$ \( ( 6 + T )^{2} \)
$67$ \( 169 + T^{2} \)
$71$ \( ( 13 + T )^{2} \)
$73$ \( 64 + T^{2} \)
$79$ \( ( -14 + T )^{2} \)
$83$ \( 9 + T^{2} \)
$89$ \( ( 14 + T )^{2} \)
$97$ \( 196 + T^{2} \)
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