Properties

Label 1800.4.f.u
Level $1800$
Weight $4$
Character orbit 1800.f
Analytic conductor $106.203$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(106.203438010\)
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 8)
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.

\(f(q)\) \(=\) \( q + 24 i q^{7} +O(q^{10})\) \( q + 24 i q^{7} + 44 q^{11} -22 i q^{13} -50 i q^{17} -44 q^{19} -56 i q^{23} + 198 q^{29} -160 q^{31} -162 i q^{37} + 198 q^{41} -52 i q^{43} -528 i q^{47} -233 q^{49} -242 i q^{53} -668 q^{59} + 550 q^{61} + 188 i q^{67} -728 q^{71} -154 i q^{73} + 1056 i q^{77} + 656 q^{79} + 236 i q^{83} + 714 q^{89} + 528 q^{91} -478 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 88q^{11} - 88q^{19} + 396q^{29} - 320q^{31} + 396q^{41} - 466q^{49} - 1336q^{59} + 1100q^{61} - 1456q^{71} + 1312q^{79} + 1428q^{89} + 1056q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 24.0000i 0 0 0
649.2 0 0 0 0 0 24.0000i 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 1800.4.f.u 2
3.b odd 2 1 200.4.c.e 2
5.b even 2 1 inner 1800.4.f.u 2
5.c odd 4 1 72.4.a.c 1
5.c odd 4 1 1800.4.a.d 1
12.b even 2 1 400.4.c.i 2
15.d odd 2 1 200.4.c.e 2
15.e even 4 1 8.4.a.a 1
15.e even 4 1 200.4.a.g 1
20.e even 4 1 144.4.a.e 1
40.i odd 4 1 576.4.a.k 1
40.k even 4 1 576.4.a.j 1
45.k odd 12 2 648.4.i.e 2
45.l even 12 2 648.4.i.h 2
60.h even 2 1 400.4.c.i 2
60.l odd 4 1 16.4.a.a 1
60.l odd 4 1 400.4.a.g 1
105.k odd 4 1 392.4.a.e 1
105.w odd 12 2 392.4.i.b 2
105.x even 12 2 392.4.i.g 2
120.q odd 4 1 64.4.a.b 1
120.q odd 4 1 1600.4.a.bm 1
120.w even 4 1 64.4.a.d 1
120.w even 4 1 1600.4.a.o 1
165.l odd 4 1 968.4.a.a 1
195.s even 4 1 1352.4.a.a 1
240.z odd 4 1 256.4.b.g 2
240.bb even 4 1 256.4.b.a 2
240.bd odd 4 1 256.4.b.g 2
240.bf even 4 1 256.4.b.a 2
255.o even 4 1 2312.4.a.a 1
420.w even 4 1 784.4.a.e 1
660.q even 4 1 1936.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 15.e even 4 1
16.4.a.a 1 60.l odd 4 1
64.4.a.b 1 120.q odd 4 1
64.4.a.d 1 120.w even 4 1
72.4.a.c 1 5.c odd 4 1
144.4.a.e 1 20.e even 4 1
200.4.a.g 1 15.e even 4 1
200.4.c.e 2 3.b odd 2 1
200.4.c.e 2 15.d odd 2 1
256.4.b.a 2 240.bb even 4 1
256.4.b.a 2 240.bf even 4 1
256.4.b.g 2 240.z odd 4 1
256.4.b.g 2 240.bd odd 4 1
392.4.a.e 1 105.k odd 4 1
392.4.i.b 2 105.w odd 12 2
392.4.i.g 2 105.x even 12 2
400.4.a.g 1 60.l odd 4 1
400.4.c.i 2 12.b even 2 1
400.4.c.i 2 60.h even 2 1
576.4.a.j 1 40.k even 4 1
576.4.a.k 1 40.i odd 4 1
648.4.i.e 2 45.k odd 12 2
648.4.i.h 2 45.l even 12 2
784.4.a.e 1 420.w even 4 1
968.4.a.a 1 165.l odd 4 1
1352.4.a.a 1 195.s even 4 1
1600.4.a.o 1 120.w even 4 1
1600.4.a.bm 1 120.q odd 4 1
1800.4.a.d 1 5.c odd 4 1
1800.4.f.u 2 1.a even 1 1 trivial
1800.4.f.u 2 5.b even 2 1 inner
1936.4.a.l 1 660.q even 4 1
2312.4.a.a 1 255.o even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{2} + 576 \)
\( T_{11} - 44 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ \( 1 - 110 T^{2} + 117649 T^{4} \)
$11$ \( ( 1 - 44 T + 1331 T^{2} )^{2} \)
$13$ \( 1 - 3910 T^{2} + 4826809 T^{4} \)
$17$ \( 1 - 7326 T^{2} + 24137569 T^{4} \)
$19$ \( ( 1 + 44 T + 6859 T^{2} )^{2} \)
$23$ \( 1 - 21198 T^{2} + 148035889 T^{4} \)
$29$ \( ( 1 - 198 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 + 160 T + 29791 T^{2} )^{2} \)
$37$ \( 1 - 75062 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 - 198 T + 68921 T^{2} )^{2} \)
$43$ \( 1 - 156310 T^{2} + 6321363049 T^{4} \)
$47$ \( 1 + 71138 T^{2} + 10779215329 T^{4} \)
$53$ \( 1 - 239190 T^{2} + 22164361129 T^{4} \)
$59$ \( ( 1 + 668 T + 205379 T^{2} )^{2} \)
$61$ \( ( 1 - 550 T + 226981 T^{2} )^{2} \)
$67$ \( 1 - 566182 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 + 728 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 754318 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 - 656 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 1087878 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 - 714 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 1596862 T^{2} + 832972004929 T^{4} \)
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