Properties

Label 900.6.d.a
Level $900$
Weight $6$
Character orbit 900.d
Analytic conductor $144.345$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 900.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(144.345437832\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4)
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.

\(f(q)\) \(=\) \( q + 88 i q^{7} +O(q^{10})\) \( q + 88 i q^{7} -540 q^{11} -418 i q^{13} + 594 i q^{17} -836 q^{19} + 4104 i q^{23} -594 q^{29} + 4256 q^{31} + 298 i q^{37} -17226 q^{41} -12100 i q^{43} -1296 i q^{47} + 9063 q^{49} -19494 i q^{53} -7668 q^{59} -34738 q^{61} -21812 i q^{67} + 46872 q^{71} + 67562 i q^{73} -47520 i q^{77} + 76912 q^{79} -67716 i q^{83} + 29754 q^{89} + 36784 q^{91} + 122398 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + O(q^{10}) \) \( 2 q - 1080 q^{11} - 1672 q^{19} - 1188 q^{29} + 8512 q^{31} - 34452 q^{41} + 18126 q^{49} - 15336 q^{59} - 69476 q^{61} + 93744 q^{71} + 153824 q^{79} + 59508 q^{89} + 73568 q^{91} + O(q^{100}) \)

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 88.0000i 0 0 0
649.2 0 0 0 0 0 88.0000i 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 900.6.d.a 2
3.b odd 2 1 100.6.c.b 2
5.b even 2 1 inner 900.6.d.a 2
5.c odd 4 1 36.6.a.a 1
5.c odd 4 1 900.6.a.h 1
12.b even 2 1 400.6.c.f 2
15.d odd 2 1 100.6.c.b 2
15.e even 4 1 4.6.a.a 1
15.e even 4 1 100.6.a.b 1
20.e even 4 1 144.6.a.c 1
40.i odd 4 1 576.6.a.bc 1
40.k even 4 1 576.6.a.bd 1
45.k odd 12 2 324.6.e.d 2
45.l even 12 2 324.6.e.a 2
60.h even 2 1 400.6.c.f 2
60.l odd 4 1 16.6.a.b 1
60.l odd 4 1 400.6.a.d 1
105.k odd 4 1 196.6.a.e 1
105.w odd 12 2 196.6.e.d 2
105.x even 12 2 196.6.e.g 2
120.q odd 4 1 64.6.a.b 1
120.w even 4 1 64.6.a.f 1
165.l odd 4 1 484.6.a.a 1
195.j odd 4 1 676.6.d.a 2
195.s even 4 1 676.6.a.a 1
195.u odd 4 1 676.6.d.a 2
240.z odd 4 1 256.6.b.c 2
240.bb even 4 1 256.6.b.g 2
240.bd odd 4 1 256.6.b.c 2
240.bf even 4 1 256.6.b.g 2
420.w even 4 1 784.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 15.e even 4 1
16.6.a.b 1 60.l odd 4 1
36.6.a.a 1 5.c odd 4 1
64.6.a.b 1 120.q odd 4 1
64.6.a.f 1 120.w even 4 1
100.6.a.b 1 15.e even 4 1
100.6.c.b 2 3.b odd 2 1
100.6.c.b 2 15.d odd 2 1
144.6.a.c 1 20.e even 4 1
196.6.a.e 1 105.k odd 4 1
196.6.e.d 2 105.w odd 12 2
196.6.e.g 2 105.x even 12 2
256.6.b.c 2 240.z odd 4 1
256.6.b.c 2 240.bd odd 4 1
256.6.b.g 2 240.bb even 4 1
256.6.b.g 2 240.bf even 4 1
324.6.e.a 2 45.l even 12 2
324.6.e.d 2 45.k odd 12 2
400.6.a.d 1 60.l odd 4 1
400.6.c.f 2 12.b even 2 1
400.6.c.f 2 60.h even 2 1
484.6.a.a 1 165.l odd 4 1
576.6.a.bc 1 40.i odd 4 1
576.6.a.bd 1 40.k even 4 1
676.6.a.a 1 195.s even 4 1
676.6.d.a 2 195.j odd 4 1
676.6.d.a 2 195.u odd 4 1
784.6.a.d 1 420.w even 4 1
900.6.a.h 1 5.c odd 4 1
900.6.d.a 2 1.a even 1 1 trivial
900.6.d.a 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{2} + 7744 \)
\( T_{11} + 540 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7744 + T^{2} \)
$11$ \( ( 540 + T )^{2} \)
$13$ \( 174724 + T^{2} \)
$17$ \( 352836 + T^{2} \)
$19$ \( ( 836 + T )^{2} \)
$23$ \( 16842816 + T^{2} \)
$29$ \( ( 594 + T )^{2} \)
$31$ \( ( -4256 + T )^{2} \)
$37$ \( 88804 + T^{2} \)
$41$ \( ( 17226 + T )^{2} \)
$43$ \( 146410000 + T^{2} \)
$47$ \( 1679616 + T^{2} \)
$53$ \( 380016036 + T^{2} \)
$59$ \( ( 7668 + T )^{2} \)
$61$ \( ( 34738 + T )^{2} \)
$67$ \( 475763344 + T^{2} \)
$71$ \( ( -46872 + T )^{2} \)
$73$ \( 4564623844 + T^{2} \)
$79$ \( ( -76912 + T )^{2} \)
$83$ \( 4585456656 + T^{2} \)
$89$ \( ( -29754 + T )^{2} \)
$97$ \( 14981270404 + T^{2} \)
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