Properties

Label 405.4.e.j
Level $405$
Weight $4$
Character orbit 405.e
Analytic conductor $23.896$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} + 4 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + 24 q^{8} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} + 4 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + 24 q^{8} -10 q^{10} + ( -10 + 10 \zeta_{6} ) q^{11} + 80 \zeta_{6} q^{13} + ( 16 - 16 \zeta_{6} ) q^{16} + 7 q^{17} -113 q^{19} + ( 20 - 20 \zeta_{6} ) q^{20} + 20 \zeta_{6} q^{22} + 81 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} + 160 q^{26} + ( 220 - 220 \zeta_{6} ) q^{29} + 189 \zeta_{6} q^{31} + 160 \zeta_{6} q^{32} + ( 14 - 14 \zeta_{6} ) q^{34} + 170 q^{37} + ( -226 + 226 \zeta_{6} ) q^{38} -120 \zeta_{6} q^{40} + 130 \zeta_{6} q^{41} + ( -10 + 10 \zeta_{6} ) q^{43} -40 q^{44} + 162 q^{46} + ( -160 + 160 \zeta_{6} ) q^{47} + 343 \zeta_{6} q^{49} + 50 \zeta_{6} q^{50} + ( -320 + 320 \zeta_{6} ) q^{52} + 631 q^{53} + 50 q^{55} -440 \zeta_{6} q^{58} + 560 \zeta_{6} q^{59} + ( -229 + 229 \zeta_{6} ) q^{61} + 378 q^{62} + 448 q^{64} + ( 400 - 400 \zeta_{6} ) q^{65} -750 \zeta_{6} q^{67} + 28 \zeta_{6} q^{68} + 890 q^{71} -890 q^{73} + ( 340 - 340 \zeta_{6} ) q^{74} -452 \zeta_{6} q^{76} + ( 27 - 27 \zeta_{6} ) q^{79} -80 q^{80} + 260 q^{82} + ( -429 + 429 \zeta_{6} ) q^{83} -35 \zeta_{6} q^{85} + 20 \zeta_{6} q^{86} + ( -240 + 240 \zeta_{6} ) q^{88} -750 q^{89} + ( -324 + 324 \zeta_{6} ) q^{92} + 320 \zeta_{6} q^{94} + 565 \zeta_{6} q^{95} + ( 1480 - 1480 \zeta_{6} ) q^{97} + 686 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{4} - 5 q^{5} + 48 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 4 q^{4} - 5 q^{5} + 48 q^{8} - 20 q^{10} - 10 q^{11} + 80 q^{13} + 16 q^{16} + 14 q^{17} - 226 q^{19} + 20 q^{20} + 20 q^{22} + 81 q^{23} - 25 q^{25} + 320 q^{26} + 220 q^{29} + 189 q^{31} + 160 q^{32} + 14 q^{34} + 340 q^{37} - 226 q^{38} - 120 q^{40} + 130 q^{41} - 10 q^{43} - 80 q^{44} + 324 q^{46} - 160 q^{47} + 343 q^{49} + 50 q^{50} - 320 q^{52} + 1262 q^{53} + 100 q^{55} - 440 q^{58} + 560 q^{59} - 229 q^{61} + 756 q^{62} + 896 q^{64} + 400 q^{65} - 750 q^{67} + 28 q^{68} + 1780 q^{71} - 1780 q^{73} + 340 q^{74} - 452 q^{76} + 27 q^{79} - 160 q^{80} + 520 q^{82} - 429 q^{83} - 35 q^{85} + 20 q^{86} - 240 q^{88} - 1500 q^{89} - 324 q^{92} + 320 q^{94} + 565 q^{95} + 1480 q^{97} + 1372 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 1.73205i 0 2.00000 + 3.46410i −2.50000 4.33013i 0 0 24.0000 0 −10.0000
271.1 1.00000 + 1.73205i 0 2.00000 3.46410i −2.50000 + 4.33013i 0 0 24.0000 0 −10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.4.e.j 2
3.b odd 2 1 405.4.e.e 2
9.c even 3 1 135.4.a.a 1
9.c even 3 1 inner 405.4.e.j 2
9.d odd 6 1 135.4.a.d yes 1
9.d odd 6 1 405.4.e.e 2
36.f odd 6 1 2160.4.a.n 1
36.h even 6 1 2160.4.a.d 1
45.h odd 6 1 675.4.a.b 1
45.j even 6 1 675.4.a.i 1
45.k odd 12 2 675.4.b.d 2
45.l even 12 2 675.4.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.a 1 9.c even 3 1
135.4.a.d yes 1 9.d odd 6 1
405.4.e.e 2 3.b odd 2 1
405.4.e.e 2 9.d odd 6 1
405.4.e.j 2 1.a even 1 1 trivial
405.4.e.j 2 9.c even 3 1 inner
675.4.a.b 1 45.h odd 6 1
675.4.a.i 1 45.j even 6 1
675.4.b.c 2 45.l even 12 2
675.4.b.d 2 45.k odd 12 2
2160.4.a.d 1 36.h even 6 1
2160.4.a.n 1 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{2} - 2 T_{2} + 4 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 25 + 5 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 100 + 10 T + T^{2} \)
$13$ \( 6400 - 80 T + T^{2} \)
$17$ \( ( -7 + T )^{2} \)
$19$ \( ( 113 + T )^{2} \)
$23$ \( 6561 - 81 T + T^{2} \)
$29$ \( 48400 - 220 T + T^{2} \)
$31$ \( 35721 - 189 T + T^{2} \)
$37$ \( ( -170 + T )^{2} \)
$41$ \( 16900 - 130 T + T^{2} \)
$43$ \( 100 + 10 T + T^{2} \)
$47$ \( 25600 + 160 T + T^{2} \)
$53$ \( ( -631 + T )^{2} \)
$59$ \( 313600 - 560 T + T^{2} \)
$61$ \( 52441 + 229 T + T^{2} \)
$67$ \( 562500 + 750 T + T^{2} \)
$71$ \( ( -890 + T )^{2} \)
$73$ \( ( 890 + T )^{2} \)
$79$ \( 729 - 27 T + T^{2} \)
$83$ \( 184041 + 429 T + T^{2} \)
$89$ \( ( 750 + T )^{2} \)
$97$ \( 2190400 - 1480 T + T^{2} \)
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