Properties

Label 1800.3.v.n
Level $1800$
Weight $3$
Character orbit 1800.v
Analytic conductor $49.046$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1800.v (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(49.0464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 4 - \beta_{3} ) q^{11} + ( -12 - 12 \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -6 + 6 \beta_{1} - 7 \beta_{2} + 7 \beta_{3} ) q^{17} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{19} + ( -14 - 14 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{23} + ( 22 \beta_{1} + 9 \beta_{2} ) q^{29} + ( -20 - 10 \beta_{3} ) q^{31} + ( -28 + 28 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{37} + ( 14 + 14 \beta_{3} ) q^{41} + ( 2 + 2 \beta_{1} + 14 \beta_{2} + 14 \beta_{3} ) q^{43} + ( -4 + 4 \beta_{1} - 19 \beta_{2} + 19 \beta_{3} ) q^{47} + ( 19 \beta_{1} + 12 \beta_{2} ) q^{49} + ( -30 - 30 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} ) q^{53} + ( 4 \beta_{1} - 21 \beta_{2} ) q^{59} + ( -6 - 22 \beta_{3} ) q^{61} + ( 2 - 2 \beta_{1} + 34 \beta_{2} - 34 \beta_{3} ) q^{67} + ( -68 - 8 \beta_{3} ) q^{71} + ( -27 - 27 \beta_{1} - 20 \beta_{2} - 20 \beta_{3} ) q^{73} + ( 18 - 18 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{77} + ( 112 \beta_{1} - 6 \beta_{2} ) q^{79} + ( 68 + 68 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} ) q^{83} + ( -62 \beta_{1} + 4 \beta_{2} ) q^{89} + ( -84 + 30 \beta_{3} ) q^{91} + ( 37 - 37 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{7} + O(q^{10}) \) \( 4 q + 12 q^{7} + 16 q^{11} - 48 q^{13} - 24 q^{17} - 56 q^{23} - 80 q^{31} - 112 q^{37} + 56 q^{41} + 8 q^{43} - 16 q^{47} - 120 q^{53} - 24 q^{61} + 8 q^{67} - 272 q^{71} - 108 q^{73} + 72 q^{77} + 272 q^{83} - 336 q^{91} + 148 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 3 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 3 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 3 \beta_{2}\)\()/2\)

Character values

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

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-\beta_{1}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
793.1
1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 1.22474i
−1.22474 + 1.22474i
0 0 0 0 0 0.550510 0.550510i 0 0 0
793.2 0 0 0 0 0 5.44949 5.44949i 0 0 0
1657.1 0 0 0 0 0 0.550510 + 0.550510i 0 0 0
1657.2 0 0 0 0 0 5.44949 + 5.44949i 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.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.3.v.n 4
3.b odd 2 1 600.3.u.e 4
5.b even 2 1 360.3.v.b 4
5.c odd 4 1 360.3.v.b 4
5.c odd 4 1 inner 1800.3.v.n 4
12.b even 2 1 1200.3.bg.e 4
15.d odd 2 1 120.3.u.a 4
15.e even 4 1 120.3.u.a 4
15.e even 4 1 600.3.u.e 4
20.d odd 2 1 720.3.bh.g 4
20.e even 4 1 720.3.bh.g 4
60.h even 2 1 240.3.bg.c 4
60.l odd 4 1 240.3.bg.c 4
60.l odd 4 1 1200.3.bg.e 4
120.i odd 2 1 960.3.bg.c 4
120.m even 2 1 960.3.bg.d 4
120.q odd 4 1 960.3.bg.d 4
120.w even 4 1 960.3.bg.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.u.a 4 15.d odd 2 1
120.3.u.a 4 15.e even 4 1
240.3.bg.c 4 60.h even 2 1
240.3.bg.c 4 60.l odd 4 1
360.3.v.b 4 5.b even 2 1
360.3.v.b 4 5.c odd 4 1
600.3.u.e 4 3.b odd 2 1
600.3.u.e 4 15.e even 4 1
720.3.bh.g 4 20.d odd 2 1
720.3.bh.g 4 20.e even 4 1
960.3.bg.c 4 120.i odd 2 1
960.3.bg.c 4 120.w even 4 1
960.3.bg.d 4 120.m even 2 1
960.3.bg.d 4 120.q odd 4 1
1200.3.bg.e 4 12.b even 2 1
1200.3.bg.e 4 60.l odd 4 1
1800.3.v.n 4 1.a even 1 1 trivial
1800.3.v.n 4 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{4} - 12 T_{7}^{3} + 72 T_{7}^{2} - 72 T_{7} + 36 \)
\( T_{11}^{2} - 8 T_{11} + 10 \)
\( T_{17}^{4} + 24 T_{17}^{3} + 288 T_{17}^{2} - 12384 T_{17} + 266256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 36 - 72 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$11$ \( ( 10 - 8 T + T^{2} )^{2} \)
$13$ \( 76176 + 13248 T + 1152 T^{2} + 48 T^{3} + T^{4} \)
$17$ \( 266256 - 12384 T + 288 T^{2} + 24 T^{3} + T^{4} \)
$19$ \( 44944 + 440 T^{2} + T^{4} \)
$23$ \( 80656 + 15904 T + 1568 T^{2} + 56 T^{3} + T^{4} \)
$29$ \( 4 + 1940 T^{2} + T^{4} \)
$31$ \( ( -200 + 40 T + T^{2} )^{2} \)
$37$ \( 2131600 + 163520 T + 6272 T^{2} + 112 T^{3} + T^{4} \)
$41$ \( ( -980 - 28 T + T^{2} )^{2} \)
$43$ \( 5494336 + 18752 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$47$ \( 18490000 - 68800 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$53$ \( 360000 + 72000 T + 7200 T^{2} + 120 T^{3} + T^{4} \)
$59$ \( 6916900 + 5324 T^{2} + T^{4} \)
$61$ \( ( -2868 + 12 T + T^{2} )^{2} \)
$67$ \( 192210496 + 110912 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( ( 4240 + 136 T + T^{2} )^{2} \)
$73$ \( 11168964 - 360936 T + 5832 T^{2} + 108 T^{3} + T^{4} \)
$79$ \( 151979584 + 25520 T^{2} + T^{4} \)
$83$ \( 60777616 - 2120512 T + 36992 T^{2} - 272 T^{3} + T^{4} \)
$89$ \( 14047504 + 7880 T^{2} + T^{4} \)
$97$ \( ( 2738 - 74 T + T^{2} )^{2} \)
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