Properties

Label 405.4.e.d
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 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.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} - 3) q^{2} - \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (20 \zeta_{6} - 20) q^{7} - 21 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} - 3) q^{2} - \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (20 \zeta_{6} - 20) q^{7} - 21 q^{8} - 15 q^{10} + ( - 24 \zeta_{6} + 24) q^{11} - 74 \zeta_{6} q^{13} - 60 \zeta_{6} q^{14} + ( - 71 \zeta_{6} + 71) q^{16} + 54 q^{17} - 124 q^{19} + ( - 5 \zeta_{6} + 5) q^{20} + 72 \zeta_{6} q^{22} + 120 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 222 q^{26} + 20 q^{28} + ( - 78 \zeta_{6} + 78) q^{29} - 200 \zeta_{6} q^{31} + 45 \zeta_{6} q^{32} + (162 \zeta_{6} - 162) q^{34} - 100 q^{35} - 70 q^{37} + ( - 372 \zeta_{6} + 372) q^{38} - 105 \zeta_{6} q^{40} - 330 \zeta_{6} q^{41} + (92 \zeta_{6} - 92) q^{43} - 24 q^{44} - 360 q^{46} + ( - 24 \zeta_{6} + 24) q^{47} - 57 \zeta_{6} q^{49} - 75 \zeta_{6} q^{50} + (74 \zeta_{6} - 74) q^{52} + 450 q^{53} + 120 q^{55} + ( - 420 \zeta_{6} + 420) q^{56} + 234 \zeta_{6} q^{58} - 24 \zeta_{6} q^{59} + ( - 322 \zeta_{6} + 322) q^{61} + 600 q^{62} + 433 q^{64} + ( - 370 \zeta_{6} + 370) q^{65} + 196 \zeta_{6} q^{67} - 54 \zeta_{6} q^{68} + ( - 300 \zeta_{6} + 300) q^{70} - 288 q^{71} - 430 q^{73} + ( - 210 \zeta_{6} + 210) q^{74} + 124 \zeta_{6} q^{76} + 480 \zeta_{6} q^{77} + ( - 520 \zeta_{6} + 520) q^{79} + 355 q^{80} + 990 q^{82} + (156 \zeta_{6} - 156) q^{83} + 270 \zeta_{6} q^{85} - 276 \zeta_{6} q^{86} + (504 \zeta_{6} - 504) q^{88} + 1026 q^{89} + 1480 q^{91} + ( - 120 \zeta_{6} + 120) q^{92} + 72 \zeta_{6} q^{94} - 620 \zeta_{6} q^{95} + ( - 286 \zeta_{6} + 286) q^{97} + 171 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - q^{4} + 5 q^{5} - 20 q^{7} - 42 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - q^{4} + 5 q^{5} - 20 q^{7} - 42 q^{8} - 30 q^{10} + 24 q^{11} - 74 q^{13} - 60 q^{14} + 71 q^{16} + 108 q^{17} - 248 q^{19} + 5 q^{20} + 72 q^{22} + 120 q^{23} - 25 q^{25} + 444 q^{26} + 40 q^{28} + 78 q^{29} - 200 q^{31} + 45 q^{32} - 162 q^{34} - 200 q^{35} - 140 q^{37} + 372 q^{38} - 105 q^{40} - 330 q^{41} - 92 q^{43} - 48 q^{44} - 720 q^{46} + 24 q^{47} - 57 q^{49} - 75 q^{50} - 74 q^{52} + 900 q^{53} + 240 q^{55} + 420 q^{56} + 234 q^{58} - 24 q^{59} + 322 q^{61} + 1200 q^{62} + 866 q^{64} + 370 q^{65} + 196 q^{67} - 54 q^{68} + 300 q^{70} - 576 q^{71} - 860 q^{73} + 210 q^{74} + 124 q^{76} + 480 q^{77} + 520 q^{79} + 710 q^{80} + 1980 q^{82} - 156 q^{83} + 270 q^{85} - 276 q^{86} - 504 q^{88} + 2052 q^{89} + 2960 q^{91} + 120 q^{92} + 72 q^{94} - 620 q^{95} + 286 q^{97} + 342 q^{98}+O(q^{100}) \) 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
−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.d 2
3.b odd 2 1 405.4.e.k 2
9.c even 3 1 15.4.a.b 1
9.c even 3 1 inner 405.4.e.d 2
9.d odd 6 1 45.4.a.b 1
9.d odd 6 1 405.4.e.k 2
36.f odd 6 1 240.4.a.f 1
36.h even 6 1 720.4.a.r 1
45.h odd 6 1 225.4.a.g 1
45.j even 6 1 75.4.a.a 1
45.k odd 12 2 75.4.b.a 2
45.l even 12 2 225.4.b.d 2
63.l odd 6 1 735.4.a.i 1
63.o even 6 1 2205.4.a.c 1
72.n even 6 1 960.4.a.bi 1
72.p odd 6 1 960.4.a.l 1
99.h odd 6 1 1815.4.a.a 1
180.p odd 6 1 1200.4.a.o 1
180.x even 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.c even 3 1
45.4.a.b 1 9.d odd 6 1
75.4.a.a 1 45.j even 6 1
75.4.b.a 2 45.k odd 12 2
225.4.a.g 1 45.h odd 6 1
225.4.b.d 2 45.l even 12 2
240.4.a.f 1 36.f odd 6 1
405.4.e.d 2 1.a even 1 1 trivial
405.4.e.d 2 9.c even 3 1 inner
405.4.e.k 2 3.b odd 2 1
405.4.e.k 2 9.d odd 6 1
720.4.a.r 1 36.h even 6 1
735.4.a.i 1 63.l odd 6 1
960.4.a.l 1 72.p odd 6 1
960.4.a.bi 1 72.n even 6 1
1200.4.a.o 1 180.p odd 6 1
1200.4.f.m 2 180.x even 12 2
1815.4.a.a 1 99.h odd 6 1
2205.4.a.c 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 \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{2} + 3T_{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} + 20T_{7} + 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

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