Properties

Label 405.4.e.k
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 15)
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.

\(f(q)\) \(=\) \( q + ( 3 - 3 \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( -20 + 20 \zeta_{6} ) q^{7} + 21 q^{8} +O(q^{10})\) \( q + ( 3 - 3 \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( -20 + 20 \zeta_{6} ) q^{7} + 21 q^{8} -15 q^{10} + ( -24 + 24 \zeta_{6} ) q^{11} -74 \zeta_{6} q^{13} + 60 \zeta_{6} q^{14} + ( 71 - 71 \zeta_{6} ) q^{16} -54 q^{17} -124 q^{19} + ( -5 + 5 \zeta_{6} ) q^{20} + 72 \zeta_{6} q^{22} -120 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} -222 q^{26} + 20 q^{28} + ( -78 + 78 \zeta_{6} ) q^{29} -200 \zeta_{6} q^{31} -45 \zeta_{6} q^{32} + ( -162 + 162 \zeta_{6} ) q^{34} + 100 q^{35} -70 q^{37} + ( -372 + 372 \zeta_{6} ) q^{38} -105 \zeta_{6} q^{40} + 330 \zeta_{6} q^{41} + ( -92 + 92 \zeta_{6} ) q^{43} + 24 q^{44} -360 q^{46} + ( -24 + 24 \zeta_{6} ) q^{47} -57 \zeta_{6} q^{49} + 75 \zeta_{6} q^{50} + ( -74 + 74 \zeta_{6} ) q^{52} -450 q^{53} + 120 q^{55} + ( -420 + 420 \zeta_{6} ) q^{56} + 234 \zeta_{6} q^{58} + 24 \zeta_{6} q^{59} + ( 322 - 322 \zeta_{6} ) q^{61} -600 q^{62} + 433 q^{64} + ( -370 + 370 \zeta_{6} ) q^{65} + 196 \zeta_{6} q^{67} + 54 \zeta_{6} q^{68} + ( 300 - 300 \zeta_{6} ) q^{70} + 288 q^{71} -430 q^{73} + ( -210 + 210 \zeta_{6} ) q^{74} + 124 \zeta_{6} q^{76} -480 \zeta_{6} q^{77} + ( 520 - 520 \zeta_{6} ) q^{79} -355 q^{80} + 990 q^{82} + ( 156 - 156 \zeta_{6} ) q^{83} + 270 \zeta_{6} q^{85} + 276 \zeta_{6} q^{86} + ( -504 + 504 \zeta_{6} ) q^{88} -1026 q^{89} + 1480 q^{91} + ( -120 + 120 \zeta_{6} ) q^{92} + 72 \zeta_{6} q^{94} + 620 \zeta_{6} q^{95} + ( 286 - 286 \zeta_{6} ) q^{97} -171 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - q^{4} - 5 q^{5} - 20 q^{7} + 42 q^{8} + O(q^{10}) \) \( 2 q + 3 q^{2} - q^{4} - 5 q^{5} - 20 q^{7} + 42 q^{8} - 30 q^{10} - 24 q^{11} - 74 q^{13} + 60 q^{14} + 71 q^{16} - 108 q^{17} - 248 q^{19} - 5 q^{20} + 72 q^{22} - 120 q^{23} - 25 q^{25} - 444 q^{26} + 40 q^{28} - 78 q^{29} - 200 q^{31} - 45 q^{32} - 162 q^{34} + 200 q^{35} - 140 q^{37} - 372 q^{38} - 105 q^{40} + 330 q^{41} - 92 q^{43} + 48 q^{44} - 720 q^{46} - 24 q^{47} - 57 q^{49} + 75 q^{50} - 74 q^{52} - 900 q^{53} + 240 q^{55} - 420 q^{56} + 234 q^{58} + 24 q^{59} + 322 q^{61} - 1200 q^{62} + 866 q^{64} - 370 q^{65} + 196 q^{67} + 54 q^{68} + 300 q^{70} + 576 q^{71} - 860 q^{73} - 210 q^{74} + 124 q^{76} - 480 q^{77} + 520 q^{79} - 710 q^{80} + 1980 q^{82} + 156 q^{83} + 270 q^{85} + 276 q^{86} - 504 q^{88} - 2052 q^{89} + 2960 q^{91} - 120 q^{92} + 72 q^{94} + 620 q^{95} + 286 q^{97} - 342 q^{98} + O(q^{100}) \)

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
1.50000 2.59808i 0 −0.500000 0.866025i −2.50000 4.33013i 0 −10.0000 + 17.3205i 21.0000 0 −15.0000
271.1 1.50000 + 2.59808i 0 −0.500000 + 0.866025i −2.50000 + 4.33013i 0 −10.0000 17.3205i 21.0000 0 −15.0000
\(n\): 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.k 2
3.b odd 2 1 405.4.e.d 2
9.c even 3 1 45.4.a.b 1
9.c even 3 1 inner 405.4.e.k 2
9.d odd 6 1 15.4.a.b 1
9.d odd 6 1 405.4.e.d 2
36.f odd 6 1 720.4.a.r 1
36.h even 6 1 240.4.a.f 1
45.h odd 6 1 75.4.a.a 1
45.j even 6 1 225.4.a.g 1
45.k odd 12 2 225.4.b.d 2
45.l even 12 2 75.4.b.a 2
63.l odd 6 1 2205.4.a.c 1
63.o even 6 1 735.4.a.i 1
72.j odd 6 1 960.4.a.bi 1
72.l even 6 1 960.4.a.l 1
99.g even 6 1 1815.4.a.a 1
180.n even 6 1 1200.4.a.o 1
180.v odd 12 2 1200.4.f.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 9.d odd 6 1
45.4.a.b 1 9.c even 3 1
75.4.a.a 1 45.h odd 6 1
75.4.b.a 2 45.l even 12 2
225.4.a.g 1 45.j even 6 1
225.4.b.d 2 45.k odd 12 2
240.4.a.f 1 36.h even 6 1
405.4.e.d 2 3.b odd 2 1
405.4.e.d 2 9.d odd 6 1
405.4.e.k 2 1.a even 1 1 trivial
405.4.e.k 2 9.c even 3 1 inner
720.4.a.r 1 36.f odd 6 1
735.4.a.i 1 63.o even 6 1
960.4.a.l 1 72.l even 6 1
960.4.a.bi 1 72.j odd 6 1
1200.4.a.o 1 180.n even 6 1
1200.4.f.m 2 180.v odd 12 2
1815.4.a.a 1 99.g even 6 1
2205.4.a.c 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 \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{2} - 3 T_{2} + 9 \)
\( T_{7}^{2} + 20 T_{7} + 400 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 - 3 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 25 + 5 T + T^{2} \)
$7$ \( 400 + 20 T + T^{2} \)
$11$ \( 576 + 24 T + T^{2} \)
$13$ \( 5476 + 74 T + T^{2} \)
$17$ \( ( 54 + T )^{2} \)
$19$ \( ( 124 + T )^{2} \)
$23$ \( 14400 + 120 T + T^{2} \)
$29$ \( 6084 + 78 T + T^{2} \)
$31$ \( 40000 + 200 T + T^{2} \)
$37$ \( ( 70 + T )^{2} \)
$41$ \( 108900 - 330 T + T^{2} \)
$43$ \( 8464 + 92 T + T^{2} \)
$47$ \( 576 + 24 T + T^{2} \)
$53$ \( ( 450 + T )^{2} \)
$59$ \( 576 - 24 T + T^{2} \)
$61$ \( 103684 - 322 T + T^{2} \)
$67$ \( 38416 - 196 T + T^{2} \)
$71$ \( ( -288 + T )^{2} \)
$73$ \( ( 430 + T )^{2} \)
$79$ \( 270400 - 520 T + T^{2} \)
$83$ \( 24336 - 156 T + T^{2} \)
$89$ \( ( 1026 + T )^{2} \)
$97$ \( 81796 - 286 T + T^{2} \)
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