Properties

Label 775.1.w.a
Level $775$
Weight $1$
Character orbit 775.w
Analytic conductor $0.387$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -31
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 775.w (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.386775384791\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.375390625.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( - \zeta_{10} + 1) q^{2} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{4} + \zeta_{10}^{2} q^{5} + (\zeta_{10}^{4} - \zeta_{10}) q^{7} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{8} + \zeta_{10}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10} + 1) q^{2} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{4} + \zeta_{10}^{2} q^{5} + (\zeta_{10}^{4} - \zeta_{10}) q^{7} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{8} + \zeta_{10}^{2} q^{9} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{10} + (\zeta_{10}^{4} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{14} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{16} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{18} + ( - \zeta_{10}^{3} + 1) q^{19} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{20} + \zeta_{10}^{4} q^{25} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{28} + \zeta_{10}^{4} q^{31} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{32} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{35} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{36} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{38} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{40} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{41} + \zeta_{10}^{4} q^{45} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{47} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{49} + (\zeta_{10}^{4} + 1) q^{50} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{56} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{59} + (\zeta_{10}^{4} + 1) q^{62} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{63} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{64} + ( - \zeta_{10}^{3} + 1) q^{67} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{70} - \zeta_{10} q^{71} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{72} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{76} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{80} + \zeta_{10}^{4} q^{81} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{82} + (\zeta_{10}^{4} + 1) q^{90} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + 1) q^{94} + (\zeta_{10}^{2} + 1) q^{95} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{97} + (\zeta_{10}^{4} - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} + q^{8} - q^{9} - 2 q^{10} + q^{14} - 2 q^{18} + 3 q^{19} - 3 q^{20} - q^{25} - q^{28} - q^{31} + 4 q^{32} - 2 q^{35} - 3 q^{36} + q^{38} + q^{40} - 2 q^{41} - q^{45} - 2 q^{47} + 2 q^{49} + 3 q^{50} - 3 q^{56} - 2 q^{59} + 3 q^{62} - 2 q^{63} + 3 q^{64} + 3 q^{67} - 4 q^{70} - 2 q^{71} + q^{72} + 4 q^{76} - q^{81} - 4 q^{82} + 3 q^{90} + q^{94} + 3 q^{95} - 2 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-1\) \(-\zeta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−0.309017 + 0.951057i
−0.309017 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
1.30902 0.951057i 0 0.500000 1.53884i −0.809017 0.587785i 0 0.618034 −0.309017 0.951057i −0.809017 0.587785i −1.61803
216.1 1.30902 + 0.951057i 0 0.500000 + 1.53884i −0.809017 + 0.587785i 0 0.618034 −0.309017 + 0.951057i −0.809017 + 0.587785i −1.61803
371.1 0.190983 0.587785i 0 0.500000 + 0.363271i 0.309017 + 0.951057i 0 −1.61803 0.809017 0.587785i 0.309017 + 0.951057i 0.618034
681.1 0.190983 + 0.587785i 0 0.500000 0.363271i 0.309017 0.951057i 0 −1.61803 0.809017 + 0.587785i 0.309017 0.951057i 0.618034
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
25.d even 5 1 inner
775.w odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.1.w.a 4
5.b even 2 1 3875.1.w.a 4
5.c odd 4 2 3875.1.z.c 8
25.d even 5 1 inner 775.1.w.a 4
25.e even 10 1 3875.1.w.a 4
25.f odd 20 2 3875.1.z.c 8
31.b odd 2 1 CM 775.1.w.a 4
155.c odd 2 1 3875.1.w.a 4
155.f even 4 2 3875.1.z.c 8
775.w odd 10 1 inner 775.1.w.a 4
775.z odd 10 1 3875.1.w.a 4
775.ca even 20 2 3875.1.z.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.1.w.a 4 1.a even 1 1 trivial
775.1.w.a 4 25.d even 5 1 inner
775.1.w.a 4 31.b odd 2 1 CM
775.1.w.a 4 775.w odd 10 1 inner
3875.1.w.a 4 5.b even 2 1
3875.1.w.a 4 25.e even 10 1
3875.1.w.a 4 155.c odd 2 1
3875.1.w.a 4 775.z odd 10 1
3875.1.z.c 8 5.c odd 4 2
3875.1.z.c 8 25.f odd 20 2
3875.1.z.c 8 155.f even 4 2
3875.1.z.c 8 775.ca even 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 3T_{2}^{3} + 4T_{2}^{2} - 2T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
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