Properties

Label 1200.2.v.c
Level $1200$
Weight $2$
Character orbit 1200.v
Analytic conductor $9.582$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.v (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{8}^{2} + \zeta_{8} - 1) q^{3} + (3 \zeta_{8}^{2} + 3) q^{7} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} - 2 \zeta_{8}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{8}^{2} + \zeta_{8} - 1) q^{3} + (3 \zeta_{8}^{2} + 3) q^{7} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} - 2 \zeta_{8}) q^{9} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{11} + 6 \zeta_{8}^{3} q^{17} + 4 \zeta_{8}^{2} q^{19} + (3 \zeta_{8}^{3} + 3 \zeta_{8} - 6) q^{21} + 4 \zeta_{8} q^{23} + ( - 5 \zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{27} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{29} + 2 q^{31} + ( - \zeta_{8}^{2} + 2 \zeta_{8} + 1) q^{33} + (2 \zeta_{8}^{2} + 2) q^{37} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{41} + (2 \zeta_{8}^{2} - 2) q^{43} + 8 \zeta_{8}^{3} q^{47} + 11 \zeta_{8}^{2} q^{49} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8} - 6) q^{51} - 12 \zeta_{8} q^{53} + (4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 4) q^{57} + (\zeta_{8}^{3} - \zeta_{8}) q^{59} - 6 q^{61} + ( - 3 \zeta_{8}^{2} - 12 \zeta_{8} + 3) q^{63} + (4 \zeta_{8}^{2} + 4) q^{67} + (4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 4 \zeta_{8}) q^{69} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{71} + (3 \zeta_{8}^{2} - 3) q^{73} - 6 \zeta_{8}^{3} q^{77} + 10 \zeta_{8}^{2} q^{79} + (4 \zeta_{8}^{3} + 4 \zeta_{8} + 7) q^{81} - 4 \zeta_{8} q^{83} + (2 \zeta_{8}^{3} + \zeta_{8}^{2} + 1) q^{87} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{89} + (2 \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{93} + (13 \zeta_{8}^{2} + 13) q^{97} + (\zeta_{8}^{3} + 4 \zeta_{8}^{2} - \zeta_{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 12 q^{7} - 24 q^{21} - 4 q^{27} + 8 q^{31} + 4 q^{33} + 8 q^{37} - 8 q^{43} - 24 q^{51} - 16 q^{57} - 24 q^{61} + 12 q^{63} + 16 q^{67} - 12 q^{73} + 28 q^{81} + 4 q^{87} - 8 q^{93} + 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(\zeta_{8}^{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 −1.70711 + 0.292893i 0 0 0 3.00000 + 3.00000i 0 2.82843 1.00000i 0
257.2 0 −0.292893 + 1.70711i 0 0 0 3.00000 + 3.00000i 0 −2.82843 1.00000i 0
593.1 0 −1.70711 0.292893i 0 0 0 3.00000 3.00000i 0 2.82843 + 1.00000i 0
593.2 0 −0.292893 1.70711i 0 0 0 3.00000 3.00000i 0 −2.82843 + 1.00000i 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.v.c 4
3.b odd 2 1 inner 1200.2.v.c 4
4.b odd 2 1 600.2.r.d 4
5.b even 2 1 240.2.v.d 4
5.c odd 4 1 240.2.v.d 4
5.c odd 4 1 inner 1200.2.v.c 4
12.b even 2 1 600.2.r.d 4
15.d odd 2 1 240.2.v.d 4
15.e even 4 1 240.2.v.d 4
15.e even 4 1 inner 1200.2.v.c 4
20.d odd 2 1 120.2.r.a 4
20.e even 4 1 120.2.r.a 4
20.e even 4 1 600.2.r.d 4
40.e odd 2 1 960.2.v.l 4
40.f even 2 1 960.2.v.b 4
40.i odd 4 1 960.2.v.b 4
40.k even 4 1 960.2.v.l 4
60.h even 2 1 120.2.r.a 4
60.l odd 4 1 120.2.r.a 4
60.l odd 4 1 600.2.r.d 4
120.i odd 2 1 960.2.v.b 4
120.m even 2 1 960.2.v.l 4
120.q odd 4 1 960.2.v.l 4
120.w even 4 1 960.2.v.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.r.a 4 20.d odd 2 1
120.2.r.a 4 20.e even 4 1
120.2.r.a 4 60.h even 2 1
120.2.r.a 4 60.l odd 4 1
240.2.v.d 4 5.b even 2 1
240.2.v.d 4 5.c odd 4 1
240.2.v.d 4 15.d odd 2 1
240.2.v.d 4 15.e even 4 1
600.2.r.d 4 4.b odd 2 1
600.2.r.d 4 12.b even 2 1
600.2.r.d 4 20.e even 4 1
600.2.r.d 4 60.l odd 4 1
960.2.v.b 4 40.f even 2 1
960.2.v.b 4 40.i odd 4 1
960.2.v.b 4 120.i odd 2 1
960.2.v.b 4 120.w even 4 1
960.2.v.l 4 40.e odd 2 1
960.2.v.l 4 40.k even 4 1
960.2.v.l 4 120.m even 2 1
960.2.v.l 4 120.q odd 4 1
1200.2.v.c 4 1.a even 1 1 trivial
1200.2.v.c 4 3.b odd 2 1 inner
1200.2.v.c 4 5.c odd 4 1 inner
1200.2.v.c 4 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} - 6T_{7} + 18 \) Copy content Toggle raw display
\( T_{11}^{2} + 2 \) Copy content Toggle raw display
\( T_{17}^{4} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + 8 T^{2} + 12 T + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4096 \) Copy content Toggle raw display
$53$ \( T^{4} + 20736 \) Copy content Toggle raw display
$59$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$61$ \( (T + 6)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T + 32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 256 \) Copy content Toggle raw display
$89$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 26 T + 338)^{2} \) Copy content Toggle raw display
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