Properties

Label 1800.2.b.f
Level $1800$
Weight $2$
Character orbit 1800.b
Analytic conductor $14.373$
Analytic rank $0$
Dimension $8$
CM discriminant -40
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + 2 q^{4} -\beta_{5} q^{7} + 2 \beta_{2} q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + 2 q^{4} -\beta_{5} q^{7} + 2 \beta_{2} q^{8} -\beta_{3} q^{11} -\beta_{4} q^{13} + ( \beta_{3} + \beta_{6} ) q^{14} + 4 q^{16} + \beta_{7} q^{19} + ( -\beta_{4} + \beta_{5} ) q^{22} + \beta_{1} q^{23} + ( -\beta_{3} + \beta_{6} ) q^{26} -2 \beta_{5} q^{28} + 4 \beta_{2} q^{32} + ( -\beta_{4} - 2 \beta_{5} ) q^{37} -2 \beta_{1} q^{38} + ( 2 \beta_{3} + \beta_{6} ) q^{41} -2 \beta_{3} q^{44} -\beta_{7} q^{46} + 2 \beta_{2} q^{47} + ( -7 - 2 \beta_{7} ) q^{49} -2 \beta_{4} q^{52} + 4 \beta_{2} q^{53} + ( 2 \beta_{3} + 2 \beta_{6} ) q^{56} + ( \beta_{3} - 2 \beta_{6} ) q^{59} + 8 q^{64} + ( \beta_{3} + 3 \beta_{6} ) q^{74} + 2 \beta_{7} q^{76} + ( -\beta_{1} + 8 \beta_{2} ) q^{77} + ( \beta_{4} - 3 \beta_{5} ) q^{82} + ( -2 \beta_{4} + 2 \beta_{5} ) q^{88} + ( -2 \beta_{3} - 3 \beta_{6} ) q^{89} + ( 2 - \beta_{7} ) q^{91} + 2 \beta_{1} q^{92} + 4 q^{94} + ( 4 \beta_{1} - 7 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{4} + O(q^{10}) \) \( 8q + 16q^{4} + 32q^{16} - 56q^{49} + 64q^{64} + 16q^{91} + 32q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4 \nu^{4} + 14 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{7} - 6 \nu^{6} + \nu^{5} + 13 \nu^{3} - 36 \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 4 \nu^{7} - 4 \nu^{6} + \nu^{5} + 29 \nu^{3} - 32 \nu^{2} + 11 \nu \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -4 \nu^{7} - 2 \nu^{6} - \nu^{5} - 29 \nu^{3} - 16 \nu^{2} - 11 \nu \)\()/3\)
\(\beta_{6}\)\(=\)\( -2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu \)
\(\beta_{7}\)\(=\)\((\)\( -8 \nu^{7} + 2 \nu^{5} - 58 \nu^{3} + 22 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + 6 \beta_{2}\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{3}\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - 12 \beta_{2}\)\()/12\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{1} - 14\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-15 \beta_{7} - 22 \beta_{6} + 20 \beta_{5} - 10 \beta_{4} - 66 \beta_{2}\)\()/24\)
\(\nu^{6}\)\(=\)\((\)\(-4 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} - 12 \beta_{3}\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(39 \beta_{7} - 58 \beta_{6} + 52 \beta_{5} - 26 \beta_{4} + 174 \beta_{2}\)\()/24\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
0.437016 0.437016i
−1.14412 1.14412i
−1.14412 + 1.14412i
0.437016 + 0.437016i
1.14412 1.14412i
−0.437016 0.437016i
−0.437016 + 0.437016i
1.14412 + 1.14412i
−1.41421 0 2.00000 0 0 5.16228i −2.82843 0 0
251.2 −1.41421 0 2.00000 0 0 1.16228i −2.82843 0 0
251.3 −1.41421 0 2.00000 0 0 1.16228i −2.82843 0 0
251.4 −1.41421 0 2.00000 0 0 5.16228i −2.82843 0 0
251.5 1.41421 0 2.00000 0 0 5.16228i 2.82843 0 0
251.6 1.41421 0 2.00000 0 0 1.16228i 2.82843 0 0
251.7 1.41421 0 2.00000 0 0 1.16228i 2.82843 0 0
251.8 1.41421 0 2.00000 0 0 5.16228i 2.82843 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.b.f 8
3.b odd 2 1 inner 1800.2.b.f 8
4.b odd 2 1 7200.2.b.f 8
5.b even 2 1 inner 1800.2.b.f 8
5.c odd 4 1 360.2.m.a 4
5.c odd 4 1 360.2.m.b yes 4
8.b even 2 1 7200.2.b.f 8
8.d odd 2 1 inner 1800.2.b.f 8
12.b even 2 1 7200.2.b.f 8
15.d odd 2 1 inner 1800.2.b.f 8
15.e even 4 1 360.2.m.a 4
15.e even 4 1 360.2.m.b yes 4
20.d odd 2 1 7200.2.b.f 8
20.e even 4 1 1440.2.m.a 4
20.e even 4 1 1440.2.m.b 4
24.f even 2 1 inner 1800.2.b.f 8
24.h odd 2 1 7200.2.b.f 8
40.e odd 2 1 CM 1800.2.b.f 8
40.f even 2 1 7200.2.b.f 8
40.i odd 4 1 1440.2.m.a 4
40.i odd 4 1 1440.2.m.b 4
40.k even 4 1 360.2.m.a 4
40.k even 4 1 360.2.m.b yes 4
60.h even 2 1 7200.2.b.f 8
60.l odd 4 1 1440.2.m.a 4
60.l odd 4 1 1440.2.m.b 4
120.i odd 2 1 7200.2.b.f 8
120.m even 2 1 inner 1800.2.b.f 8
120.q odd 4 1 360.2.m.a 4
120.q odd 4 1 360.2.m.b yes 4
120.w even 4 1 1440.2.m.a 4
120.w even 4 1 1440.2.m.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.m.a 4 5.c odd 4 1
360.2.m.a 4 15.e even 4 1
360.2.m.a 4 40.k even 4 1
360.2.m.a 4 120.q odd 4 1
360.2.m.b yes 4 5.c odd 4 1
360.2.m.b yes 4 15.e even 4 1
360.2.m.b yes 4 40.k even 4 1
360.2.m.b yes 4 120.q odd 4 1
1440.2.m.a 4 20.e even 4 1
1440.2.m.a 4 40.i odd 4 1
1440.2.m.a 4 60.l odd 4 1
1440.2.m.a 4 120.w even 4 1
1440.2.m.b 4 20.e even 4 1
1440.2.m.b 4 40.i odd 4 1
1440.2.m.b 4 60.l odd 4 1
1440.2.m.b 4 120.w even 4 1
1800.2.b.f 8 1.a even 1 1 trivial
1800.2.b.f 8 3.b odd 2 1 inner
1800.2.b.f 8 5.b even 2 1 inner
1800.2.b.f 8 8.d odd 2 1 inner
1800.2.b.f 8 15.d odd 2 1 inner
1800.2.b.f 8 24.f even 2 1 inner
1800.2.b.f 8 40.e odd 2 1 CM
1800.2.b.f 8 120.m even 2 1 inner
7200.2.b.f 8 4.b odd 2 1
7200.2.b.f 8 8.b even 2 1
7200.2.b.f 8 12.b even 2 1
7200.2.b.f 8 20.d odd 2 1
7200.2.b.f 8 24.h odd 2 1
7200.2.b.f 8 40.f even 2 1
7200.2.b.f 8 60.h even 2 1
7200.2.b.f 8 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{4} + 28 T_{7}^{2} + 36 \)
\( T_{23}^{2} - 20 \)
\( T_{43} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{4} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 36 + 28 T^{2} + T^{4} )^{2} \)
$11$ \( ( 324 + 44 T^{2} + T^{4} )^{2} \)
$13$ \( ( 36 + 52 T^{2} + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( ( -40 + T^{2} )^{4} \)
$23$ \( ( -20 + T^{2} )^{4} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( ( 2916 + 148 T^{2} + T^{4} )^{2} \)
$41$ \( ( 6084 + 164 T^{2} + T^{4} )^{2} \)
$43$ \( T^{8} \)
$47$ \( ( -8 + T^{2} )^{4} \)
$53$ \( ( -32 + T^{2} )^{4} \)
$59$ \( ( 6084 + 236 T^{2} + T^{4} )^{2} \)
$61$ \( T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 324 + 356 T^{2} + T^{4} )^{2} \)
$97$ \( T^{8} \)
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