Properties

Label 900.2.d.c
Level 900
Weight 2
Character orbit 900.d
Analytic conductor 7.187
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q + 2 i q^{7} +O(q^{10})\) \( q + 2 i q^{7} -2 i q^{13} + 6 i q^{17} + 4 q^{19} + 6 i q^{23} + 6 q^{29} -4 q^{31} + 2 i q^{37} -6 q^{41} + 10 i q^{43} + 6 i q^{47} + 3 q^{49} -6 i q^{53} + 12 q^{59} + 2 q^{61} + 2 i q^{67} + 12 q^{71} -2 i q^{73} -8 q^{79} + 6 i q^{83} -6 q^{89} + 4 q^{91} + 2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 8q^{19} + 12q^{29} - 8q^{31} - 12q^{41} + 6q^{49} + 24q^{59} + 4q^{61} + 24q^{71} - 16q^{79} - 12q^{89} + 8q^{91} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
0 0 0 0 0 2.00000i 0 0 0
649.2 0 0 0 0 0 2.00000i 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.d.c 2
3.b odd 2 1 100.2.c.a 2
4.b odd 2 1 3600.2.f.j 2
5.b even 2 1 inner 900.2.d.c 2
5.c odd 4 1 180.2.a.a 1
5.c odd 4 1 900.2.a.b 1
12.b even 2 1 400.2.c.b 2
15.d odd 2 1 100.2.c.a 2
15.e even 4 1 20.2.a.a 1
15.e even 4 1 100.2.a.a 1
20.d odd 2 1 3600.2.f.j 2
20.e even 4 1 720.2.a.h 1
20.e even 4 1 3600.2.a.be 1
21.c even 2 1 4900.2.e.f 2
24.f even 2 1 1600.2.c.e 2
24.h odd 2 1 1600.2.c.d 2
35.f even 4 1 8820.2.a.g 1
40.i odd 4 1 2880.2.a.m 1
40.k even 4 1 2880.2.a.f 1
45.k odd 12 2 1620.2.i.b 2
45.l even 12 2 1620.2.i.h 2
60.h even 2 1 400.2.c.b 2
60.l odd 4 1 80.2.a.b 1
60.l odd 4 1 400.2.a.c 1
105.g even 2 1 4900.2.e.f 2
105.k odd 4 1 980.2.a.h 1
105.k odd 4 1 4900.2.a.e 1
105.w odd 12 2 980.2.i.c 2
105.x even 12 2 980.2.i.i 2
120.i odd 2 1 1600.2.c.d 2
120.m even 2 1 1600.2.c.e 2
120.q odd 4 1 320.2.a.a 1
120.q odd 4 1 1600.2.a.w 1
120.w even 4 1 320.2.a.f 1
120.w even 4 1 1600.2.a.c 1
165.l odd 4 1 2420.2.a.a 1
195.j odd 4 1 3380.2.f.b 2
195.s even 4 1 3380.2.a.c 1
195.u odd 4 1 3380.2.f.b 2
240.z odd 4 1 1280.2.d.g 2
240.bb even 4 1 1280.2.d.c 2
240.bd odd 4 1 1280.2.d.g 2
240.bf even 4 1 1280.2.d.c 2
255.k even 4 1 5780.2.c.a 2
255.o even 4 1 5780.2.a.f 1
255.r even 4 1 5780.2.c.a 2
285.j odd 4 1 7220.2.a.f 1
420.w even 4 1 3920.2.a.h 1
660.q even 4 1 9680.2.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 15.e even 4 1
80.2.a.b 1 60.l odd 4 1
100.2.a.a 1 15.e even 4 1
100.2.c.a 2 3.b odd 2 1
100.2.c.a 2 15.d odd 2 1
180.2.a.a 1 5.c odd 4 1
320.2.a.a 1 120.q odd 4 1
320.2.a.f 1 120.w even 4 1
400.2.a.c 1 60.l odd 4 1
400.2.c.b 2 12.b even 2 1
400.2.c.b 2 60.h even 2 1
720.2.a.h 1 20.e even 4 1
900.2.a.b 1 5.c odd 4 1
900.2.d.c 2 1.a even 1 1 trivial
900.2.d.c 2 5.b even 2 1 inner
980.2.a.h 1 105.k odd 4 1
980.2.i.c 2 105.w odd 12 2
980.2.i.i 2 105.x even 12 2
1280.2.d.c 2 240.bb even 4 1
1280.2.d.c 2 240.bf even 4 1
1280.2.d.g 2 240.z odd 4 1
1280.2.d.g 2 240.bd odd 4 1
1600.2.a.c 1 120.w even 4 1
1600.2.a.w 1 120.q odd 4 1
1600.2.c.d 2 24.h odd 2 1
1600.2.c.d 2 120.i odd 2 1
1600.2.c.e 2 24.f even 2 1
1600.2.c.e 2 120.m even 2 1
1620.2.i.b 2 45.k odd 12 2
1620.2.i.h 2 45.l even 12 2
2420.2.a.a 1 165.l odd 4 1
2880.2.a.f 1 40.k even 4 1
2880.2.a.m 1 40.i odd 4 1
3380.2.a.c 1 195.s even 4 1
3380.2.f.b 2 195.j odd 4 1
3380.2.f.b 2 195.u odd 4 1
3600.2.a.be 1 20.e even 4 1
3600.2.f.j 2 4.b odd 2 1
3600.2.f.j 2 20.d odd 2 1
3920.2.a.h 1 420.w even 4 1
4900.2.a.e 1 105.k odd 4 1
4900.2.e.f 2 21.c even 2 1
4900.2.e.f 2 105.g even 2 1
5780.2.a.f 1 255.o even 4 1
5780.2.c.a 2 255.k even 4 1
5780.2.c.a 2 255.r even 4 1
7220.2.a.f 1 285.j odd 4 1
8820.2.a.g 1 35.f even 4 1
9680.2.a.ba 1 660.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 10 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 14 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 58 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 - 12 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 130 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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