Properties

Label 900.3.l.a
Level $900$
Weight $3$
Character orbit 900.l
Analytic conductor $24.523$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 7 + 7 i ) q^{7} +O(q^{10})\) \( q + ( 7 + 7 i ) q^{7} -10 q^{11} + ( -9 + 9 i ) q^{13} + ( 1 + i ) q^{17} -8 i q^{19} + ( -23 + 23 i ) q^{23} + 8 i q^{29} -14 q^{31} + ( -33 - 33 i ) q^{37} + 14 q^{41} + ( 15 - 15 i ) q^{43} + ( -39 - 39 i ) q^{47} + 49 i q^{49} + ( -7 + 7 i ) q^{53} + 56 i q^{59} + 42 q^{61} + ( 7 + 7 i ) q^{67} -98 q^{71} + ( -49 + 49 i ) q^{73} + ( -70 - 70 i ) q^{77} + 96 i q^{79} + ( -63 + 63 i ) q^{83} + 112 i q^{89} -126 q^{91} + ( -33 - 33 i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{7} + O(q^{10}) \) \( 2q + 14q^{7} - 20q^{11} - 18q^{13} + 2q^{17} - 46q^{23} - 28q^{31} - 66q^{37} + 28q^{41} + 30q^{43} - 78q^{47} - 14q^{53} + 84q^{61} + 14q^{67} - 196q^{71} - 98q^{73} - 140q^{77} - 126q^{83} - 252q^{91} - 66q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
1.00000i
1.00000i
0 0 0 0 0 7.00000 + 7.00000i 0 0 0
793.1 0 0 0 0 0 7.00000 7.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.l.a 2
3.b odd 2 1 100.3.f.a 2
5.b even 2 1 180.3.l.a 2
5.c odd 4 1 180.3.l.a 2
5.c odd 4 1 inner 900.3.l.a 2
12.b even 2 1 400.3.p.d 2
15.d odd 2 1 20.3.f.a 2
15.e even 4 1 20.3.f.a 2
15.e even 4 1 100.3.f.a 2
20.d odd 2 1 720.3.bh.e 2
20.e even 4 1 720.3.bh.e 2
60.h even 2 1 80.3.p.a 2
60.l odd 4 1 80.3.p.a 2
60.l odd 4 1 400.3.p.d 2
105.g even 2 1 980.3.l.a 2
105.k odd 4 1 980.3.l.a 2
120.i odd 2 1 320.3.p.c 2
120.m even 2 1 320.3.p.g 2
120.q odd 4 1 320.3.p.g 2
120.w even 4 1 320.3.p.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.f.a 2 15.d odd 2 1
20.3.f.a 2 15.e even 4 1
80.3.p.a 2 60.h even 2 1
80.3.p.a 2 60.l odd 4 1
100.3.f.a 2 3.b odd 2 1
100.3.f.a 2 15.e even 4 1
180.3.l.a 2 5.b even 2 1
180.3.l.a 2 5.c odd 4 1
320.3.p.c 2 120.i odd 2 1
320.3.p.c 2 120.w even 4 1
320.3.p.g 2 120.m even 2 1
320.3.p.g 2 120.q odd 4 1
400.3.p.d 2 12.b even 2 1
400.3.p.d 2 60.l odd 4 1
720.3.bh.e 2 20.d odd 2 1
720.3.bh.e 2 20.e even 4 1
900.3.l.a 2 1.a even 1 1 trivial
900.3.l.a 2 5.c odd 4 1 inner
980.3.l.a 2 105.g even 2 1
980.3.l.a 2 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{2} - 14 T_{7} + 98 \)
\( T_{11} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 98 - 14 T + T^{2} \)
$11$ \( ( 10 + T )^{2} \)
$13$ \( 162 + 18 T + T^{2} \)
$17$ \( 2 - 2 T + T^{2} \)
$19$ \( 64 + T^{2} \)
$23$ \( 1058 + 46 T + T^{2} \)
$29$ \( 64 + T^{2} \)
$31$ \( ( 14 + T )^{2} \)
$37$ \( 2178 + 66 T + T^{2} \)
$41$ \( ( -14 + T )^{2} \)
$43$ \( 450 - 30 T + T^{2} \)
$47$ \( 3042 + 78 T + T^{2} \)
$53$ \( 98 + 14 T + T^{2} \)
$59$ \( 3136 + T^{2} \)
$61$ \( ( -42 + T )^{2} \)
$67$ \( 98 - 14 T + T^{2} \)
$71$ \( ( 98 + T )^{2} \)
$73$ \( 4802 + 98 T + T^{2} \)
$79$ \( 9216 + T^{2} \)
$83$ \( 7938 + 126 T + T^{2} \)
$89$ \( 12544 + T^{2} \)
$97$ \( 2178 + 66 T + T^{2} \)
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