Properties

Label 405.4.e.b
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
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.

\(f(q)\) \(=\) \( q + ( -5 + 5 \zeta_{6} ) q^{2} -17 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( 30 - 30 \zeta_{6} ) q^{7} + 45 q^{8} +O(q^{10})\) \( q + ( -5 + 5 \zeta_{6} ) q^{2} -17 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( 30 - 30 \zeta_{6} ) q^{7} + 45 q^{8} -25 q^{10} + ( -50 + 50 \zeta_{6} ) q^{11} + 20 \zeta_{6} q^{13} + 150 \zeta_{6} q^{14} + ( -89 + 89 \zeta_{6} ) q^{16} -10 q^{17} -44 q^{19} + ( 85 - 85 \zeta_{6} ) q^{20} -250 \zeta_{6} q^{22} -120 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} -100 q^{26} -510 q^{28} + ( 50 - 50 \zeta_{6} ) q^{29} -108 \zeta_{6} q^{31} -85 \zeta_{6} q^{32} + ( 50 - 50 \zeta_{6} ) q^{34} + 150 q^{35} -40 q^{37} + ( 220 - 220 \zeta_{6} ) q^{38} + 225 \zeta_{6} q^{40} -400 \zeta_{6} q^{41} + ( -280 + 280 \zeta_{6} ) q^{43} + 850 q^{44} + 600 q^{46} + ( 280 - 280 \zeta_{6} ) q^{47} -557 \zeta_{6} q^{49} -125 \zeta_{6} q^{50} + ( 340 - 340 \zeta_{6} ) q^{52} -610 q^{53} -250 q^{55} + ( 1350 - 1350 \zeta_{6} ) q^{56} + 250 \zeta_{6} q^{58} -50 \zeta_{6} q^{59} + ( 518 - 518 \zeta_{6} ) q^{61} + 540 q^{62} -287 q^{64} + ( -100 + 100 \zeta_{6} ) q^{65} + 180 \zeta_{6} q^{67} + 170 \zeta_{6} q^{68} + ( -750 + 750 \zeta_{6} ) q^{70} + 700 q^{71} -410 q^{73} + ( 200 - 200 \zeta_{6} ) q^{74} + 748 \zeta_{6} q^{76} + 1500 \zeta_{6} q^{77} + ( 516 - 516 \zeta_{6} ) q^{79} -445 q^{80} + 2000 q^{82} + ( -660 + 660 \zeta_{6} ) q^{83} -50 \zeta_{6} q^{85} -1400 \zeta_{6} q^{86} + ( -2250 + 2250 \zeta_{6} ) q^{88} -1500 q^{89} + 600 q^{91} + ( -2040 + 2040 \zeta_{6} ) q^{92} + 1400 \zeta_{6} q^{94} -220 \zeta_{6} q^{95} + ( 1630 - 1630 \zeta_{6} ) q^{97} + 2785 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{2} - 17 q^{4} + 5 q^{5} + 30 q^{7} + 90 q^{8} + O(q^{10}) \) \( 2 q - 5 q^{2} - 17 q^{4} + 5 q^{5} + 30 q^{7} + 90 q^{8} - 50 q^{10} - 50 q^{11} + 20 q^{13} + 150 q^{14} - 89 q^{16} - 20 q^{17} - 88 q^{19} + 85 q^{20} - 250 q^{22} - 120 q^{23} - 25 q^{25} - 200 q^{26} - 1020 q^{28} + 50 q^{29} - 108 q^{31} - 85 q^{32} + 50 q^{34} + 300 q^{35} - 80 q^{37} + 220 q^{38} + 225 q^{40} - 400 q^{41} - 280 q^{43} + 1700 q^{44} + 1200 q^{46} + 280 q^{47} - 557 q^{49} - 125 q^{50} + 340 q^{52} - 1220 q^{53} - 500 q^{55} + 1350 q^{56} + 250 q^{58} - 50 q^{59} + 518 q^{61} + 1080 q^{62} - 574 q^{64} - 100 q^{65} + 180 q^{67} + 170 q^{68} - 750 q^{70} + 1400 q^{71} - 820 q^{73} + 200 q^{74} + 748 q^{76} + 1500 q^{77} + 516 q^{79} - 890 q^{80} + 4000 q^{82} - 660 q^{83} - 50 q^{85} - 1400 q^{86} - 2250 q^{88} - 3000 q^{89} + 1200 q^{91} - 2040 q^{92} + 1400 q^{94} - 220 q^{95} + 1630 q^{97} + 5570 q^{98} + O(q^{100}) \)

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
−2.50000 + 4.33013i 0 −8.50000 14.7224i 2.50000 + 4.33013i 0 15.0000 25.9808i 45.0000 0 −25.0000
271.1 −2.50000 4.33013i 0 −8.50000 + 14.7224i 2.50000 4.33013i 0 15.0000 + 25.9808i 45.0000 0 −25.0000
\(n\): 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.b 2
3.b odd 2 1 405.4.e.n 2
9.c even 3 1 45.4.a.e yes 1
9.c even 3 1 inner 405.4.e.b 2
9.d odd 6 1 45.4.a.a 1
9.d odd 6 1 405.4.e.n 2
36.f odd 6 1 720.4.a.o 1
36.h even 6 1 720.4.a.bc 1
45.h odd 6 1 225.4.a.h 1
45.j even 6 1 225.4.a.a 1
45.k odd 12 2 225.4.b.b 2
45.l even 12 2 225.4.b.a 2
63.l odd 6 1 2205.4.a.t 1
63.o even 6 1 2205.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 9.d odd 6 1
45.4.a.e yes 1 9.c even 3 1
225.4.a.a 1 45.j even 6 1
225.4.a.h 1 45.h odd 6 1
225.4.b.a 2 45.l even 12 2
225.4.b.b 2 45.k odd 12 2
405.4.e.b 2 1.a even 1 1 trivial
405.4.e.b 2 9.c even 3 1 inner
405.4.e.n 2 3.b odd 2 1
405.4.e.n 2 9.d odd 6 1
720.4.a.o 1 36.f odd 6 1
720.4.a.bc 1 36.h even 6 1
2205.4.a.a 1 63.o even 6 1
2205.4.a.t 1 63.l odd 6 1

Hecke kernels

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

\( T_{2}^{2} + 5 T_{2} + 25 \)
\( T_{7}^{2} - 30 T_{7} + 900 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 + 5 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 25 - 5 T + T^{2} \)
$7$ \( 900 - 30 T + T^{2} \)
$11$ \( 2500 + 50 T + T^{2} \)
$13$ \( 400 - 20 T + T^{2} \)
$17$ \( ( 10 + T )^{2} \)
$19$ \( ( 44 + T )^{2} \)
$23$ \( 14400 + 120 T + T^{2} \)
$29$ \( 2500 - 50 T + T^{2} \)
$31$ \( 11664 + 108 T + T^{2} \)
$37$ \( ( 40 + T )^{2} \)
$41$ \( 160000 + 400 T + T^{2} \)
$43$ \( 78400 + 280 T + T^{2} \)
$47$ \( 78400 - 280 T + T^{2} \)
$53$ \( ( 610 + T )^{2} \)
$59$ \( 2500 + 50 T + T^{2} \)
$61$ \( 268324 - 518 T + T^{2} \)
$67$ \( 32400 - 180 T + T^{2} \)
$71$ \( ( -700 + T )^{2} \)
$73$ \( ( 410 + T )^{2} \)
$79$ \( 266256 - 516 T + T^{2} \)
$83$ \( 435600 + 660 T + T^{2} \)
$89$ \( ( 1500 + T )^{2} \)
$97$ \( 2656900 - 1630 T + T^{2} \)
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