Properties

Label 612.1.l.a
Level $612$
Weight $1$
Character orbit 612.l
Analytic conductor $0.305$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
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.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.305427787731\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.19652.1
Artin image: $C_4{\rm wrC}_2$
Artin field: Galois closure of 8.0.101875968.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + ( 1 - i ) q^{5} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} + ( 1 - i ) q^{5} -i q^{8} + ( 1 + i ) q^{10} + q^{16} + q^{17} + ( -1 + i ) q^{20} -i q^{25} + ( -1 + i ) q^{29} + i q^{32} + i q^{34} + ( -1 + i ) q^{37} + ( -1 - i ) q^{40} + ( -1 - i ) q^{41} + i q^{49} + q^{50} + ( -1 - i ) q^{58} + ( -1 - i ) q^{61} - q^{64} - q^{68} + ( 1 - i ) q^{73} + ( -1 - i ) q^{74} + ( 1 - i ) q^{80} + ( 1 - i ) q^{82} + ( 1 - i ) q^{85} + ( -1 + i ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{5} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{5} + 2q^{10} + 2q^{16} + 2q^{17} - 2q^{20} - 2q^{29} - 2q^{37} - 2q^{40} - 2q^{41} + 2q^{50} - 2q^{58} - 2q^{61} - 2q^{64} - 2q^{68} + 2q^{73} - 2q^{74} + 2q^{80} + 2q^{82} + 2q^{85} - 2q^{97} - 2q^{98} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
1.00000i
1.00000i
1.00000i 0 −1.00000 1.00000 1.00000i 0 0 1.00000i 0 1.00000 + 1.00000i
523.1 1.00000i 0 −1.00000 1.00000 + 1.00000i 0 0 1.00000i 0 1.00000 1.00000i
\(n\): 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.c even 4 1 inner
68.f odd 4 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

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