Properties

Label 605.1.h.a
Level $605$
Weight $1$
Character orbit 605.h
Analytic conductor $0.302$
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: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 605.h (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.301934332643\)
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 20.10.4936006137688415732421875.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{10}^{3} q^{4} + \zeta_{10} q^{5} + \zeta_{10}^{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{10}^{3} q^{4} + \zeta_{10} q^{5} + \zeta_{10}^{4} q^{9} - \zeta_{10} q^{16} + \zeta_{10}^{4} q^{20} + \zeta_{10}^{2} q^{25} - \zeta_{10}^{4} q^{31} - \zeta_{10}^{2} q^{36} - q^{45} + \zeta_{10} q^{49} - \zeta_{10}^{3} q^{59} - \zeta_{10}^{4} q^{64} - \zeta_{10} q^{71} - \zeta_{10}^{2} q^{80} - \zeta_{10}^{3} q^{81} - q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{4} + q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{4} + q^{5} - q^{9} - q^{16} - q^{20} - q^{25} + 2 q^{31} + q^{36} - 4 q^{45} + q^{49} - 2 q^{59} + q^{64} - 2 q^{71} + q^{80} - q^{81} - 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.309017 0.951057i
0.809017 0.587785i
0 0 0.809017 0.587785i −0.309017 + 0.951057i 0 0 0 0.309017 + 0.951057i 0
239.1 0 0 −0.309017 + 0.951057i 0.809017 + 0.587785i 0 0 0 −0.809017 + 0.587785i 0
354.1 0 0 0.809017 + 0.587785i −0.309017 0.951057i 0 0 0 0.309017 0.951057i 0
524.1 0 0 −0.309017 0.951057i 0.809017 0.587785i 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 605.1.h.a 4
5.b even 2 1 RM 605.1.h.a 4
5.c odd 4 2 3025.1.x.a 4
11.b odd 2 1 CM 605.1.h.a 4
11.c even 5 1 55.1.d.a 1
11.c even 5 3 inner 605.1.h.a 4
11.d odd 10 1 55.1.d.a 1
11.d odd 10 3 inner 605.1.h.a 4
33.f even 10 1 495.1.h.a 1
33.h odd 10 1 495.1.h.a 1
44.g even 10 1 880.1.i.a 1
44.h odd 10 1 880.1.i.a 1
55.d odd 2 1 CM 605.1.h.a 4
55.e even 4 2 3025.1.x.a 4
55.h odd 10 1 55.1.d.a 1
55.h odd 10 3 inner 605.1.h.a 4
55.j even 10 1 55.1.d.a 1
55.j even 10 3 inner 605.1.h.a 4
55.k odd 20 2 275.1.c.a 1
55.k odd 20 6 3025.1.x.a 4
55.l even 20 2 275.1.c.a 1
55.l even 20 6 3025.1.x.a 4
77.j odd 10 1 2695.1.g.c 1
77.l even 10 1 2695.1.g.c 1
77.m even 15 2 2695.1.q.c 2
77.n even 30 2 2695.1.q.b 2
77.o odd 30 2 2695.1.q.c 2
77.p odd 30 2 2695.1.q.b 2
88.k even 10 1 3520.1.i.a 1
88.l odd 10 1 3520.1.i.a 1
88.o even 10 1 3520.1.i.b 1
88.p odd 10 1 3520.1.i.b 1
165.o odd 10 1 495.1.h.a 1
165.r even 10 1 495.1.h.a 1
165.u odd 20 2 2475.1.b.a 1
165.v even 20 2 2475.1.b.a 1
220.n odd 10 1 880.1.i.a 1
220.o even 10 1 880.1.i.a 1
385.v even 10 1 2695.1.g.c 1
385.y odd 10 1 2695.1.g.c 1
385.bm even 30 2 2695.1.q.c 2
385.bn odd 30 2 2695.1.q.b 2
385.bp odd 30 2 2695.1.q.c 2
385.br even 30 2 2695.1.q.b 2
440.ba odd 10 1 3520.1.i.b 1
440.bd even 10 1 3520.1.i.b 1
440.bh odd 10 1 3520.1.i.a 1
440.bm even 10 1 3520.1.i.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.1.d.a 1 11.c even 5 1
55.1.d.a 1 11.d odd 10 1
55.1.d.a 1 55.h odd 10 1
55.1.d.a 1 55.j even 10 1
275.1.c.a 1 55.k odd 20 2
275.1.c.a 1 55.l even 20 2
495.1.h.a 1 33.f even 10 1
495.1.h.a 1 33.h odd 10 1
495.1.h.a 1 165.o odd 10 1
495.1.h.a 1 165.r even 10 1
605.1.h.a 4 1.a even 1 1 trivial
605.1.h.a 4 5.b even 2 1 RM
605.1.h.a 4 11.b odd 2 1 CM
605.1.h.a 4 11.c even 5 3 inner
605.1.h.a 4 11.d odd 10 3 inner
605.1.h.a 4 55.d odd 2 1 CM
605.1.h.a 4 55.h odd 10 3 inner
605.1.h.a 4 55.j even 10 3 inner
880.1.i.a 1 44.g even 10 1
880.1.i.a 1 44.h odd 10 1
880.1.i.a 1 220.n odd 10 1
880.1.i.a 1 220.o even 10 1
2475.1.b.a 1 165.u odd 20 2
2475.1.b.a 1 165.v even 20 2
2695.1.g.c 1 77.j odd 10 1
2695.1.g.c 1 77.l even 10 1
2695.1.g.c 1 385.v even 10 1
2695.1.g.c 1 385.y odd 10 1
2695.1.q.b 2 77.n even 30 2
2695.1.q.b 2 77.p odd 30 2
2695.1.q.b 2 385.bn odd 30 2
2695.1.q.b 2 385.br even 30 2
2695.1.q.c 2 77.m even 15 2
2695.1.q.c 2 77.o odd 30 2
2695.1.q.c 2 385.bm even 30 2
2695.1.q.c 2 385.bp odd 30 2
3025.1.x.a 4 5.c odd 4 2
3025.1.x.a 4 55.e even 4 2
3025.1.x.a 4 55.k odd 20 6
3025.1.x.a 4 55.l even 20 6
3520.1.i.a 1 88.k even 10 1
3520.1.i.a 1 88.l odd 10 1
3520.1.i.a 1 440.bh odd 10 1
3520.1.i.a 1 440.bm even 10 1
3520.1.i.b 1 88.o even 10 1
3520.1.i.b 1 88.p odd 10 1
3520.1.i.b 1 440.ba odd 10 1
3520.1.i.b 1 440.bd even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(605, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} + T^{2} - T + 1 \) 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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