Properties

Label 403.1.p.a
Level 403
Weight 1
Character orbit 403.p
Analytic conductor 0.201
Analytic rank 0
Dimension 4
Projective image \(A_{4}\)
CM/RM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 403.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.201123200091\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.162409.1
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.

\(f(q)\) \(=\) \( q -\zeta_{12}^{2} q^{2} + \zeta_{12}^{5} q^{3} + \zeta_{12} q^{6} -\zeta_{12}^{4} q^{7} - q^{8} +O(q^{10})\) \( q -\zeta_{12}^{2} q^{2} + \zeta_{12}^{5} q^{3} + \zeta_{12} q^{6} -\zeta_{12}^{4} q^{7} - q^{8} -\zeta_{12}^{5} q^{11} -\zeta_{12}^{3} q^{13} - q^{14} + \zeta_{12}^{2} q^{16} -\zeta_{12} q^{17} -\zeta_{12}^{4} q^{19} + \zeta_{12}^{3} q^{21} -\zeta_{12} q^{22} -\zeta_{12}^{5} q^{23} -\zeta_{12}^{5} q^{24} - q^{25} + \zeta_{12}^{5} q^{26} -\zeta_{12}^{3} q^{27} + \zeta_{12}^{5} q^{29} + \zeta_{12}^{3} q^{31} + \zeta_{12}^{4} q^{33} + \zeta_{12}^{3} q^{34} -\zeta_{12}^{5} q^{37} - q^{38} + \zeta_{12}^{2} q^{39} + \zeta_{12}^{2} q^{41} -\zeta_{12}^{5} q^{42} + \zeta_{12} q^{43} -\zeta_{12} q^{46} -\zeta_{12} q^{48} + \zeta_{12}^{2} q^{50} + q^{51} + 2 \zeta_{12}^{3} q^{53} + \zeta_{12}^{5} q^{54} + \zeta_{12}^{4} q^{56} + \zeta_{12}^{3} q^{57} + \zeta_{12} q^{58} -\zeta_{12}^{4} q^{59} -\zeta_{12} q^{61} -\zeta_{12}^{5} q^{62} + q^{64} + q^{66} + \zeta_{12}^{2} q^{67} + \zeta_{12}^{4} q^{69} + \zeta_{12}^{4} q^{71} -\zeta_{12} q^{74} -\zeta_{12}^{5} q^{75} -\zeta_{12}^{3} q^{77} -\zeta_{12}^{4} q^{78} + \zeta_{12}^{2} q^{81} -\zeta_{12}^{4} q^{82} -\zeta_{12}^{3} q^{86} -\zeta_{12}^{4} q^{87} + \zeta_{12}^{5} q^{88} -\zeta_{12}^{5} q^{89} -\zeta_{12} q^{91} -\zeta_{12}^{2} q^{93} + \zeta_{12}^{4} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 2q^{7} - 4q^{8} + O(q^{10}) \) \( 4q - 2q^{2} + 2q^{7} - 4q^{8} - 4q^{14} + 2q^{16} + 2q^{19} - 4q^{25} - 2q^{33} - 4q^{38} + 2q^{39} + 2q^{41} + 2q^{50} + 4q^{51} - 2q^{56} + 2q^{59} + 4q^{64} + 4q^{66} + 2q^{67} - 2q^{69} - 2q^{71} + 2q^{78} + 2q^{81} + 2q^{82} + 2q^{87} - 2q^{93} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\) \(249\) \(313\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.500000 + 0.866025i −0.866025 0.500000i 0 0 0.866025 0.500000i 0.500000 + 0.866025i −1.00000 0 0
61.2 −0.500000 + 0.866025i 0.866025 + 0.500000i 0 0 −0.866025 + 0.500000i 0.500000 + 0.866025i −1.00000 0 0
185.1 −0.500000 0.866025i −0.866025 + 0.500000i 0 0 0.866025 + 0.500000i 0.500000 0.866025i −1.00000 0 0
185.2 −0.500000 0.866025i 0.866025 0.500000i 0 0 −0.866025 0.500000i 0.500000 0.866025i −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
13.c even 3 1 inner
31.b odd 2 1 inner
403.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.1.p.a 4
3.b odd 2 1 3627.1.cl.a 4
13.c even 3 1 inner 403.1.p.a 4
31.b odd 2 1 inner 403.1.p.a 4
39.i odd 6 1 3627.1.cl.a 4
93.c even 2 1 3627.1.cl.a 4
403.p odd 6 1 inner 403.1.p.a 4
1209.be even 6 1 3627.1.cl.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.1.p.a 4 1.a even 1 1 trivial
403.1.p.a 4 13.c even 3 1 inner
403.1.p.a 4 31.b odd 2 1 inner
403.1.p.a 4 403.p odd 6 1 inner
3627.1.cl.a 4 3.b odd 2 1
3627.1.cl.a 4 39.i odd 6 1
3627.1.cl.a 4 93.c even 2 1
3627.1.cl.a 4 1209.be even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(403, [\chi])\).

Hecke characteristic polynomials

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