Properties

Label 209.1.h.a
Level $209$
Weight $1$
Character orbit 209.h
Analytic conductor $0.104$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 209.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.104304587640\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.43681.1
Artin image: $\SL(2,3):C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{2} q^{3} -\zeta_{12}^{2} q^{5} + \zeta_{12} q^{6} -\zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{2} q^{3} -\zeta_{12}^{2} q^{5} + \zeta_{12} q^{6} -\zeta_{12}^{3} q^{8} + \zeta_{12} q^{10} -\zeta_{12}^{3} q^{11} -\zeta_{12} q^{13} + \zeta_{12}^{4} q^{15} + \zeta_{12}^{2} q^{16} -\zeta_{12}^{5} q^{17} + \zeta_{12}^{3} q^{19} + \zeta_{12}^{2} q^{22} + \zeta_{12}^{4} q^{23} + \zeta_{12}^{5} q^{24} + q^{26} - q^{27} + \zeta_{12} q^{29} -\zeta_{12}^{3} q^{30} + \zeta_{12}^{5} q^{33} + \zeta_{12}^{4} q^{34} -\zeta_{12}^{2} q^{38} + \zeta_{12}^{3} q^{39} + \zeta_{12}^{5} q^{40} + \zeta_{12}^{5} q^{41} -\zeta_{12}^{5} q^{43} -\zeta_{12}^{3} q^{46} + \zeta_{12}^{4} q^{47} -\zeta_{12}^{4} q^{48} + q^{49} -\zeta_{12} q^{51} -\zeta_{12}^{4} q^{53} -\zeta_{12}^{5} q^{54} + \zeta_{12}^{5} q^{55} -\zeta_{12}^{5} q^{57} - q^{58} -\zeta_{12}^{2} q^{59} + \zeta_{12} q^{61} - q^{64} + \zeta_{12}^{3} q^{65} -\zeta_{12}^{4} q^{66} -\zeta_{12}^{4} q^{67} + q^{69} + \zeta_{12}^{2} q^{71} -\zeta_{12}^{5} q^{73} -\zeta_{12}^{2} q^{78} -\zeta_{12}^{5} q^{79} -\zeta_{12}^{4} q^{80} + \zeta_{12}^{2} q^{81} -\zeta_{12}^{4} q^{82} -\zeta_{12} q^{85} + \zeta_{12}^{4} q^{86} -\zeta_{12}^{3} q^{87} - q^{88} -\zeta_{12}^{4} q^{89} -\zeta_{12}^{3} q^{94} -\zeta_{12}^{5} q^{95} + \zeta_{12}^{2} q^{97} + \zeta_{12}^{5} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} + O(q^{10}) \) \( 4 q - 2 q^{3} - 2 q^{5} - 2 q^{15} + 2 q^{16} + 2 q^{22} - 2 q^{23} + 4 q^{26} - 4 q^{27} - 2 q^{34} - 2 q^{38} - 2 q^{47} + 2 q^{48} + 4 q^{49} + 2 q^{53} - 4 q^{58} - 2 q^{59} - 4 q^{64} + 2 q^{66} + 2 q^{67} + 4 q^{69} + 2 q^{71} - 2 q^{78} + 2 q^{80} + 2 q^{81} + 2 q^{82} - 2 q^{86} - 4 q^{88} + 2 q^{89} + 2 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/209\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(134\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
87.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 0.500000i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.866025 0.500000i 0 1.00000i 0 0.866025 0.500000i
87.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.866025 + 0.500000i 0 1.00000i 0 −0.866025 + 0.500000i
197.1 −0.866025 + 0.500000i −0.500000 0.866025i 0 −0.500000 0.866025i 0.866025 + 0.500000i 0 1.00000i 0 0.866025 + 0.500000i
197.2 0.866025 0.500000i −0.500000 0.866025i 0 −0.500000 0.866025i −0.866025 0.500000i 0 1.00000i 0 −0.866025 0.500000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
19.c even 3 1 inner
209.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.1.h.a 4
3.b odd 2 1 1881.1.bc.a 4
4.b odd 2 1 3344.1.bb.a 4
11.b odd 2 1 inner 209.1.h.a 4
11.c even 5 4 2299.1.w.a 16
11.d odd 10 4 2299.1.w.a 16
19.b odd 2 1 3971.1.h.d 4
19.c even 3 1 inner 209.1.h.a 4
19.c even 3 1 3971.1.c.e 2
19.d odd 6 1 3971.1.c.b 2
19.d odd 6 1 3971.1.h.d 4
19.e even 9 6 3971.1.q.e 12
19.f odd 18 6 3971.1.q.d 12
33.d even 2 1 1881.1.bc.a 4
44.c even 2 1 3344.1.bb.a 4
57.h odd 6 1 1881.1.bc.a 4
76.g odd 6 1 3344.1.bb.a 4
209.d even 2 1 3971.1.h.d 4
209.g even 6 1 3971.1.c.b 2
209.g even 6 1 3971.1.h.d 4
209.h odd 6 1 inner 209.1.h.a 4
209.h odd 6 1 3971.1.c.e 2
209.n even 15 4 2299.1.w.a 16
209.p even 18 6 3971.1.q.d 12
209.q odd 18 6 3971.1.q.e 12
209.s odd 30 4 2299.1.w.a 16
627.l even 6 1 1881.1.bc.a 4
836.q even 6 1 3344.1.bb.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.1.h.a 4 1.a even 1 1 trivial
209.1.h.a 4 11.b odd 2 1 inner
209.1.h.a 4 19.c even 3 1 inner
209.1.h.a 4 209.h odd 6 1 inner
1881.1.bc.a 4 3.b odd 2 1
1881.1.bc.a 4 33.d even 2 1
1881.1.bc.a 4 57.h odd 6 1
1881.1.bc.a 4 627.l even 6 1
2299.1.w.a 16 11.c even 5 4
2299.1.w.a 16 11.d odd 10 4
2299.1.w.a 16 209.n even 15 4
2299.1.w.a 16 209.s odd 30 4
3344.1.bb.a 4 4.b odd 2 1
3344.1.bb.a 4 44.c even 2 1
3344.1.bb.a 4 76.g odd 6 1
3344.1.bb.a 4 836.q even 6 1
3971.1.c.b 2 19.d odd 6 1
3971.1.c.b 2 209.g even 6 1
3971.1.c.e 2 19.c even 3 1
3971.1.c.e 2 209.h odd 6 1
3971.1.h.d 4 19.b odd 2 1
3971.1.h.d 4 19.d odd 6 1
3971.1.h.d 4 209.d even 2 1
3971.1.h.d 4 209.g even 6 1
3971.1.q.d 12 19.f odd 18 6
3971.1.q.d 12 209.p even 18 6
3971.1.q.e 12 19.e even 9 6
3971.1.q.e 12 209.q odd 18 6

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(209, [\chi])\).

Hecke characteristic polynomials

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