Properties

Label 1600.2.n.c
Level $1600$
Weight $2$
Character orbit 1600.n
Analytic conductor $12.776$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.n (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - i - 1) q^{3} + (3 i - 3) q^{7} - i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i - 1) q^{3} + (3 i - 3) q^{7} - i q^{9} - 2 i q^{11} + ( - 3 i + 3) q^{13} + ( - i - 1) q^{17} + 4 q^{19} + 6 q^{21} + ( - i - 1) q^{23} + (4 i - 4) q^{27} + 10 i q^{31} + (2 i - 2) q^{33} + ( - i - 1) q^{37} - 6 q^{39} - 10 q^{41} + ( - 5 i - 5) q^{43} + (3 i - 3) q^{47} - 11 i q^{49} + 2 i q^{51} + (5 i - 5) q^{53} + ( - 4 i - 4) q^{57} - 12 q^{59} - 2 q^{61} + (3 i + 3) q^{63} + ( - i + 1) q^{67} + 2 i q^{69} + 2 i q^{71} + (i - 1) q^{73} + (6 i + 6) q^{77} - 8 q^{79} + 5 q^{81} + ( - 5 i - 5) q^{83} + 16 i q^{89} + 18 i q^{91} + ( - 10 i + 10) q^{93} + (3 i + 3) q^{97} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 6 q^{7} + 6 q^{13} - 2 q^{17} + 8 q^{19} + 12 q^{21} - 2 q^{23} - 8 q^{27} - 4 q^{33} - 2 q^{37} - 12 q^{39} - 20 q^{41} - 10 q^{43} - 6 q^{47} - 10 q^{53} - 8 q^{57} - 24 q^{59} - 4 q^{61} + 6 q^{63} + 2 q^{67} - 2 q^{73} + 12 q^{77} - 16 q^{79} + 10 q^{81} - 10 q^{83} + 20 q^{93} + 6 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(i\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1343.1
1.00000i
1.00000i
0 −1.00000 + 1.00000i 0 0 0 −3.00000 3.00000i 0 1.00000i 0
1407.1 0 −1.00000 1.00000i 0 0 0 −3.00000 + 3.00000i 0 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
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.n.c 2
4.b odd 2 1 1600.2.n.m 2
5.b even 2 1 320.2.n.g 2
5.c odd 4 1 320.2.n.b 2
5.c odd 4 1 1600.2.n.m 2
8.b even 2 1 800.2.n.f 2
8.d odd 2 1 800.2.n.e 2
20.d odd 2 1 320.2.n.b 2
20.e even 4 1 320.2.n.g 2
20.e even 4 1 inner 1600.2.n.c 2
40.e odd 2 1 160.2.n.d yes 2
40.f even 2 1 160.2.n.c 2
40.i odd 4 1 160.2.n.d yes 2
40.i odd 4 1 800.2.n.e 2
40.k even 4 1 160.2.n.c 2
40.k even 4 1 800.2.n.f 2
80.i odd 4 1 1280.2.o.l 2
80.j even 4 1 1280.2.o.m 2
80.k odd 4 1 1280.2.o.f 2
80.k odd 4 1 1280.2.o.l 2
80.q even 4 1 1280.2.o.c 2
80.q even 4 1 1280.2.o.m 2
80.s even 4 1 1280.2.o.c 2
80.t odd 4 1 1280.2.o.f 2
120.i odd 2 1 1440.2.x.f 2
120.m even 2 1 1440.2.x.a 2
120.q odd 4 1 1440.2.x.f 2
120.w even 4 1 1440.2.x.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.n.c 2 40.f even 2 1
160.2.n.c 2 40.k even 4 1
160.2.n.d yes 2 40.e odd 2 1
160.2.n.d yes 2 40.i odd 4 1
320.2.n.b 2 5.c odd 4 1
320.2.n.b 2 20.d odd 2 1
320.2.n.g 2 5.b even 2 1
320.2.n.g 2 20.e even 4 1
800.2.n.e 2 8.d odd 2 1
800.2.n.e 2 40.i odd 4 1
800.2.n.f 2 8.b even 2 1
800.2.n.f 2 40.k even 4 1
1280.2.o.c 2 80.q even 4 1
1280.2.o.c 2 80.s even 4 1
1280.2.o.f 2 80.k odd 4 1
1280.2.o.f 2 80.t odd 4 1
1280.2.o.l 2 80.i odd 4 1
1280.2.o.l 2 80.k odd 4 1
1280.2.o.m 2 80.j even 4 1
1280.2.o.m 2 80.q even 4 1
1440.2.x.a 2 120.m even 2 1
1440.2.x.a 2 120.w even 4 1
1440.2.x.f 2 120.i odd 2 1
1440.2.x.f 2 120.q odd 4 1
1600.2.n.c 2 1.a even 1 1 trivial
1600.2.n.c 2 20.e even 4 1 inner
1600.2.n.m 2 4.b odd 2 1
1600.2.n.m 2 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} + 18 \) Copy content Toggle raw display
\( T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 100 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$71$ \( T^{2} + 4 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$89$ \( T^{2} + 256 \) Copy content Toggle raw display
$97$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
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