Properties

Label 3025.1.x.a
Level $3025$
Weight $1$
Character orbit 3025.x
Analytic conductor $1.510$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -11, -55, 5
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3025.x (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50967166321\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\)
Artin image: $C_5\times D_4$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q - \zeta_{10}^{3} q^{4} - \zeta_{10}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{10}^{3} q^{4} - \zeta_{10}^{4} q^{9} - \zeta_{10} q^{16} - \zeta_{10}^{4} q^{31} - \zeta_{10}^{2} q^{36} - \zeta_{10} q^{49} + \zeta_{10}^{3} q^{59} + \zeta_{10}^{4} q^{64} - \zeta_{10} q^{71} - \zeta_{10}^{3} q^{81} + q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{4} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{4} + q^{9} - q^{16} + 2 q^{31} + q^{36} - q^{49} + 2 q^{59} - q^{64} - 2 q^{71} - q^{81} + 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3025\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(2301\)
\(\chi(n)\) \(1\) \(\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1201.1
−0.309017 0.951057i
0.809017 0.587785i
−0.309017 + 0.951057i
0.809017 + 0.587785i
0 0 −0.809017 0.587785i 0 0 0 0 −0.309017 + 0.951057i 0
1976.1 0 0 0.309017 + 0.951057i 0 0 0 0 0.809017 + 0.587785i 0
2151.1 0 0 −0.809017 + 0.587785i 0 0 0 0 −0.309017 0.951057i 0
2901.1 0 0 0.309017 0.951057i 0 0 0 0 0.809017 0.587785i 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.b even 2 1 RM by \(\Q(\sqrt{5}) \)
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
11.c even 5 3 inner
11.d odd 10 3 inner
55.h odd 10 3 inner
55.j even 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3025.1.x.a 4
5.b even 2 1 RM 3025.1.x.a 4
5.c odd 4 2 605.1.h.a 4
11.b odd 2 1 CM 3025.1.x.a 4
11.c even 5 1 275.1.c.a 1
11.c even 5 3 inner 3025.1.x.a 4
11.d odd 10 1 275.1.c.a 1
11.d odd 10 3 inner 3025.1.x.a 4
33.f even 10 1 2475.1.b.a 1
33.h odd 10 1 2475.1.b.a 1
55.d odd 2 1 CM 3025.1.x.a 4
55.e even 4 2 605.1.h.a 4
55.h odd 10 1 275.1.c.a 1
55.h odd 10 3 inner 3025.1.x.a 4
55.j even 10 1 275.1.c.a 1
55.j even 10 3 inner 3025.1.x.a 4
55.k odd 20 2 55.1.d.a 1
55.k odd 20 6 605.1.h.a 4
55.l even 20 2 55.1.d.a 1
55.l even 20 6 605.1.h.a 4
165.o odd 10 1 2475.1.b.a 1
165.r even 10 1 2475.1.b.a 1
165.u odd 20 2 495.1.h.a 1
165.v even 20 2 495.1.h.a 1
220.v even 20 2 880.1.i.a 1
220.w odd 20 2 880.1.i.a 1
385.bi odd 20 2 2695.1.g.c 1
385.bk even 20 2 2695.1.g.c 1
385.bs odd 60 4 2695.1.q.b 2
385.bt odd 60 4 2695.1.q.c 2
385.bu even 60 4 2695.1.q.b 2
385.bv even 60 4 2695.1.q.c 2
440.bp odd 20 2 3520.1.i.b 1
440.br odd 20 2 3520.1.i.a 1
440.bs even 20 2 3520.1.i.a 1
440.bu even 20 2 3520.1.i.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.1.d.a 1 55.k odd 20 2
55.1.d.a 1 55.l even 20 2
275.1.c.a 1 11.c even 5 1
275.1.c.a 1 11.d odd 10 1
275.1.c.a 1 55.h odd 10 1
275.1.c.a 1 55.j even 10 1
495.1.h.a 1 165.u odd 20 2
495.1.h.a 1 165.v even 20 2
605.1.h.a 4 5.c odd 4 2
605.1.h.a 4 55.e even 4 2
605.1.h.a 4 55.k odd 20 6
605.1.h.a 4 55.l even 20 6
880.1.i.a 1 220.v even 20 2
880.1.i.a 1 220.w odd 20 2
2475.1.b.a 1 33.f even 10 1
2475.1.b.a 1 33.h odd 10 1
2475.1.b.a 1 165.o odd 10 1
2475.1.b.a 1 165.r even 10 1
2695.1.g.c 1 385.bi odd 20 2
2695.1.g.c 1 385.bk even 20 2
2695.1.q.b 2 385.bs odd 60 4
2695.1.q.b 2 385.bu even 60 4
2695.1.q.c 2 385.bt odd 60 4
2695.1.q.c 2 385.bv even 60 4
3025.1.x.a 4 1.a even 1 1 trivial
3025.1.x.a 4 5.b even 2 1 RM
3025.1.x.a 4 11.b odd 2 1 CM
3025.1.x.a 4 11.c even 5 3 inner
3025.1.x.a 4 11.d odd 10 3 inner
3025.1.x.a 4 55.d odd 2 1 CM
3025.1.x.a 4 55.h odd 10 3 inner
3025.1.x.a 4 55.j even 10 3 inner
3520.1.i.a 1 440.br odd 20 2
3520.1.i.a 1 440.bs even 20 2
3520.1.i.b 1 440.bp odd 20 2
3520.1.i.b 1 440.bu even 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3025, [\chi])\).

Hecke characteristic polynomials

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