Properties

Label 1800.4.f.r
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: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q + 8 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta q^{7} + 28 q^{11} - 13 \beta q^{13} - 31 \beta q^{17} + 68 q^{19} + 104 \beta q^{23} - 58 q^{29} + 160 q^{31} - 135 \beta q^{37} - 282 q^{41} + 38 \beta q^{43} - 140 \beta q^{47} + 87 q^{49} + 105 \beta q^{53} + 196 q^{59} + 742 q^{61} - 418 \beta q^{67} + 504 q^{71} - 531 \beta q^{73} + 224 \beta q^{77} - 768 q^{79} + 526 \beta q^{83} - 726 q^{89} + 416 q^{91} + 703 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 56 q^{11} + 136 q^{19} - 116 q^{29} + 320 q^{31} - 564 q^{41} + 174 q^{49} + 392 q^{59} + 1484 q^{61} + 1008 q^{71} - 1536 q^{79} - 1452 q^{89} + 832 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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 16.0000i 0 0 0
649.2 0 0 0 0 0 16.0000i 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.r 2
3.b odd 2 1 600.4.f.c 2
5.b even 2 1 inner 1800.4.f.r 2
5.c odd 4 1 360.4.a.b 1
5.c odd 4 1 1800.4.a.bb 1
12.b even 2 1 1200.4.f.o 2
15.d odd 2 1 600.4.f.c 2
15.e even 4 1 120.4.a.c 1
15.e even 4 1 600.4.a.q 1
20.e even 4 1 720.4.a.l 1
60.h even 2 1 1200.4.f.o 2
60.l odd 4 1 240.4.a.l 1
60.l odd 4 1 1200.4.a.c 1
120.q odd 4 1 960.4.a.h 1
120.w even 4 1 960.4.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.c 1 15.e even 4 1
240.4.a.l 1 60.l odd 4 1
360.4.a.b 1 5.c odd 4 1
600.4.a.q 1 15.e even 4 1
600.4.f.c 2 3.b odd 2 1
600.4.f.c 2 15.d odd 2 1
720.4.a.l 1 20.e even 4 1
960.4.a.h 1 120.q odd 4 1
960.4.a.u 1 120.w even 4 1
1200.4.a.c 1 60.l odd 4 1
1200.4.f.o 2 12.b even 2 1
1200.4.f.o 2 60.h even 2 1
1800.4.a.bb 1 5.c odd 4 1
1800.4.f.r 2 1.a even 1 1 trivial
1800.4.f.r 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 \(S_{4}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T - 28)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 676 \) Copy content Toggle raw display
$17$ \( T^{2} + 3844 \) Copy content Toggle raw display
$19$ \( (T - 68)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 43264 \) Copy content Toggle raw display
$29$ \( (T + 58)^{2} \) Copy content Toggle raw display
$31$ \( (T - 160)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 72900 \) Copy content Toggle raw display
$41$ \( (T + 282)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 5776 \) Copy content Toggle raw display
$47$ \( T^{2} + 78400 \) Copy content Toggle raw display
$53$ \( T^{2} + 44100 \) Copy content Toggle raw display
$59$ \( (T - 196)^{2} \) Copy content Toggle raw display
$61$ \( (T - 742)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 698896 \) Copy content Toggle raw display
$71$ \( (T - 504)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1127844 \) Copy content Toggle raw display
$79$ \( (T + 768)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1106704 \) Copy content Toggle raw display
$89$ \( (T + 726)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1976836 \) Copy content Toggle raw display
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