Properties

Label 900.4.d.c
Level $900$
Weight $4$
Character orbit 900.d
Analytic conductor $53.102$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(53.1017190052\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 12)
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.

\(f(q)\) \(=\) \( q + 8 i q^{7} +O(q^{10})\) \( q + 8 i q^{7} -36 q^{11} + 10 i q^{13} -18 i q^{17} + 100 q^{19} + 72 i q^{23} -234 q^{29} -16 q^{31} -226 i q^{37} -90 q^{41} -452 i q^{43} -432 i q^{47} + 279 q^{49} + 414 i q^{53} -684 q^{59} + 422 q^{61} + 332 i q^{67} + 360 q^{71} -26 i q^{73} -288 i q^{77} -512 q^{79} -1188 i q^{83} -630 q^{89} -80 q^{91} -1054 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 72q^{11} + 200q^{19} - 468q^{29} - 32q^{31} - 180q^{41} + 558q^{49} - 1368q^{59} + 844q^{61} + 720q^{71} - 1024q^{79} - 1260q^{89} - 160q^{91} + O(q^{100}) \)

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 8.00000i 0 0 0
649.2 0 0 0 0 0 8.00000i 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 900.4.d.c 2
3.b odd 2 1 300.4.d.e 2
5.b even 2 1 inner 900.4.d.c 2
5.c odd 4 1 36.4.a.a 1
5.c odd 4 1 900.4.a.g 1
12.b even 2 1 1200.4.f.d 2
15.d odd 2 1 300.4.d.e 2
15.e even 4 1 12.4.a.a 1
15.e even 4 1 300.4.a.b 1
20.e even 4 1 144.4.a.g 1
35.f even 4 1 1764.4.a.b 1
35.k even 12 2 1764.4.k.o 2
35.l odd 12 2 1764.4.k.b 2
40.i odd 4 1 576.4.a.b 1
40.k even 4 1 576.4.a.a 1
45.k odd 12 2 324.4.e.a 2
45.l even 12 2 324.4.e.h 2
60.h even 2 1 1200.4.f.d 2
60.l odd 4 1 48.4.a.a 1
60.l odd 4 1 1200.4.a.be 1
105.k odd 4 1 588.4.a.c 1
105.w odd 12 2 588.4.i.e 2
105.x even 12 2 588.4.i.d 2
120.q odd 4 1 192.4.a.l 1
120.w even 4 1 192.4.a.f 1
165.l odd 4 1 1452.4.a.d 1
195.j odd 4 1 2028.4.b.c 2
195.s even 4 1 2028.4.a.c 1
195.u odd 4 1 2028.4.b.c 2
240.z odd 4 1 768.4.d.j 2
240.bb even 4 1 768.4.d.g 2
240.bd odd 4 1 768.4.d.j 2
240.bf even 4 1 768.4.d.g 2
420.w even 4 1 2352.4.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 15.e even 4 1
36.4.a.a 1 5.c odd 4 1
48.4.a.a 1 60.l odd 4 1
144.4.a.g 1 20.e even 4 1
192.4.a.f 1 120.w even 4 1
192.4.a.l 1 120.q odd 4 1
300.4.a.b 1 15.e even 4 1
300.4.d.e 2 3.b odd 2 1
300.4.d.e 2 15.d odd 2 1
324.4.e.a 2 45.k odd 12 2
324.4.e.h 2 45.l even 12 2
576.4.a.a 1 40.k even 4 1
576.4.a.b 1 40.i odd 4 1
588.4.a.c 1 105.k odd 4 1
588.4.i.d 2 105.x even 12 2
588.4.i.e 2 105.w odd 12 2
768.4.d.g 2 240.bb even 4 1
768.4.d.g 2 240.bf even 4 1
768.4.d.j 2 240.z odd 4 1
768.4.d.j 2 240.bd odd 4 1
900.4.a.g 1 5.c odd 4 1
900.4.d.c 2 1.a even 1 1 trivial
900.4.d.c 2 5.b even 2 1 inner
1200.4.a.be 1 60.l odd 4 1
1200.4.f.d 2 12.b even 2 1
1200.4.f.d 2 60.h even 2 1
1452.4.a.d 1 165.l odd 4 1
1764.4.a.b 1 35.f even 4 1
1764.4.k.b 2 35.l odd 12 2
1764.4.k.o 2 35.k even 12 2
2028.4.a.c 1 195.s even 4 1
2028.4.b.c 2 195.j odd 4 1
2028.4.b.c 2 195.u odd 4 1
2352.4.a.bk 1 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{2} + 64 \)
\( T_{11} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 64 + T^{2} \)
$11$ \( ( 36 + T )^{2} \)
$13$ \( 100 + T^{2} \)
$17$ \( 324 + T^{2} \)
$19$ \( ( -100 + T )^{2} \)
$23$ \( 5184 + T^{2} \)
$29$ \( ( 234 + T )^{2} \)
$31$ \( ( 16 + T )^{2} \)
$37$ \( 51076 + T^{2} \)
$41$ \( ( 90 + T )^{2} \)
$43$ \( 204304 + T^{2} \)
$47$ \( 186624 + T^{2} \)
$53$ \( 171396 + T^{2} \)
$59$ \( ( 684 + T )^{2} \)
$61$ \( ( -422 + T )^{2} \)
$67$ \( 110224 + T^{2} \)
$71$ \( ( -360 + T )^{2} \)
$73$ \( 676 + T^{2} \)
$79$ \( ( 512 + T )^{2} \)
$83$ \( 1411344 + T^{2} \)
$89$ \( ( 630 + T )^{2} \)
$97$ \( 1110916 + T^{2} \)
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