Properties

Label 612.1.e.a
Level $612$
Weight $1$
Character orbit 612.e
Self dual yes
Analytic conductor $0.305$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -4, -68, 17
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 612 = 2^{2} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 612.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.305427787731\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{17})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.2448.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{8} - 2 q^{13} + q^{16} - q^{17} + q^{25} - 2 q^{26} + q^{32} - q^{34} - q^{49} + q^{50} - 2 q^{52} - 2 q^{53} + q^{64} - q^{68} + 2 q^{89} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/612\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\) \(307\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
0
1.00000 0 1.00000 0 0 0 1.00000 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
17.b even 2 1 RM by \(\Q(\sqrt{17}) \)
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 612.1.e.a 1
3.b odd 2 1 68.1.d.a 1
4.b odd 2 1 CM 612.1.e.a 1
12.b even 2 1 68.1.d.a 1
15.d odd 2 1 1700.1.h.d 1
15.e even 4 2 1700.1.d.b 2
17.b even 2 1 RM 612.1.e.a 1
21.c even 2 1 3332.1.g.a 1
21.g even 6 2 3332.1.o.d 2
21.h odd 6 2 3332.1.o.c 2
24.f even 2 1 1088.1.g.a 1
24.h odd 2 1 1088.1.g.a 1
51.c odd 2 1 68.1.d.a 1
51.f odd 4 2 1156.1.c.a 1
51.g odd 8 4 1156.1.f.a 2
51.i even 16 8 1156.1.g.a 4
60.h even 2 1 1700.1.h.d 1
60.l odd 4 2 1700.1.d.b 2
68.d odd 2 1 CM 612.1.e.a 1
84.h odd 2 1 3332.1.g.a 1
84.j odd 6 2 3332.1.o.d 2
84.n even 6 2 3332.1.o.c 2
204.h even 2 1 68.1.d.a 1
204.l even 4 2 1156.1.c.a 1
204.p even 8 4 1156.1.f.a 2
204.t odd 16 8 1156.1.g.a 4
255.h odd 2 1 1700.1.h.d 1
255.o even 4 2 1700.1.d.b 2
357.c even 2 1 3332.1.g.a 1
357.q odd 6 2 3332.1.o.c 2
357.s even 6 2 3332.1.o.d 2
408.b odd 2 1 1088.1.g.a 1
408.h even 2 1 1088.1.g.a 1
1020.b even 2 1 1700.1.h.d 1
1020.x odd 4 2 1700.1.d.b 2
1428.b odd 2 1 3332.1.g.a 1
1428.be even 6 2 3332.1.o.c 2
1428.bl odd 6 2 3332.1.o.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a 1 3.b odd 2 1
68.1.d.a 1 12.b even 2 1
68.1.d.a 1 51.c odd 2 1
68.1.d.a 1 204.h even 2 1
612.1.e.a 1 1.a even 1 1 trivial
612.1.e.a 1 4.b odd 2 1 CM
612.1.e.a 1 17.b even 2 1 RM
612.1.e.a 1 68.d odd 2 1 CM
1088.1.g.a 1 24.f even 2 1
1088.1.g.a 1 24.h odd 2 1
1088.1.g.a 1 408.b odd 2 1
1088.1.g.a 1 408.h even 2 1
1156.1.c.a 1 51.f odd 4 2
1156.1.c.a 1 204.l even 4 2
1156.1.f.a 2 51.g odd 8 4
1156.1.f.a 2 204.p even 8 4
1156.1.g.a 4 51.i even 16 8
1156.1.g.a 4 204.t odd 16 8
1700.1.d.b 2 15.e even 4 2
1700.1.d.b 2 60.l odd 4 2
1700.1.d.b 2 255.o even 4 2
1700.1.d.b 2 1020.x odd 4 2
1700.1.h.d 1 15.d odd 2 1
1700.1.h.d 1 60.h even 2 1
1700.1.h.d 1 255.h odd 2 1
1700.1.h.d 1 1020.b even 2 1
3332.1.g.a 1 21.c even 2 1
3332.1.g.a 1 84.h odd 2 1
3332.1.g.a 1 357.c even 2 1
3332.1.g.a 1 1428.b odd 2 1
3332.1.o.c 2 21.h odd 6 2
3332.1.o.c 2 84.n even 6 2
3332.1.o.c 2 357.q odd 6 2
3332.1.o.c 2 1428.be even 6 2
3332.1.o.d 2 21.g even 6 2
3332.1.o.d 2 84.j odd 6 2
3332.1.o.d 2 357.s even 6 2
3332.1.o.d 2 1428.bl odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(612, [\chi])\).

Hecke characteristic polynomials

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