Properties

Label 405.4.e.n
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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.n 2
3.b odd 2 1 405.4.e.b 2
9.c even 3 1 45.4.a.a 1
9.c even 3 1 inner 405.4.e.n 2
9.d odd 6 1 45.4.a.e yes 1
9.d odd 6 1 405.4.e.b 2
36.f odd 6 1 720.4.a.bc 1
36.h even 6 1 720.4.a.o 1
45.h odd 6 1 225.4.a.a 1
45.j even 6 1 225.4.a.h 1
45.k odd 12 2 225.4.b.a 2
45.l even 12 2 225.4.b.b 2
63.l odd 6 1 2205.4.a.a 1
63.o even 6 1 2205.4.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 9.c even 3 1
45.4.a.e yes 1 9.d odd 6 1
225.4.a.a 1 45.h odd 6 1
225.4.a.h 1 45.j even 6 1
225.4.b.a 2 45.k odd 12 2
225.4.b.b 2 45.l even 12 2
405.4.e.b 2 3.b odd 2 1
405.4.e.b 2 9.d odd 6 1
405.4.e.n 2 1.a even 1 1 trivial
405.4.e.n 2 9.c even 3 1 inner
720.4.a.o 1 36.h even 6 1
720.4.a.bc 1 36.f odd 6 1
2205.4.a.a 1 63.l odd 6 1
2205.4.a.t 1 63.o even 6 1

Hecke kernels

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

\( T_{2}^{2} - 5T_{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{2} - 30T_{7} + 900 \) Copy content Toggle raw display

Hecke characteristic polynomials

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