Properties

Label 405.4.e.c
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
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.

\(f(q)\) \(=\) \( q + (4 \zeta_{6} - 4) q^{2} - 8 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + (6 \zeta_{6} - 6) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + (4 \zeta_{6} - 4) q^{2} - 8 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + (6 \zeta_{6} - 6) q^{7} + 20 q^{10} + ( - 32 \zeta_{6} + 32) q^{11} + 38 \zeta_{6} q^{13} - 24 \zeta_{6} q^{14} + ( - 64 \zeta_{6} + 64) q^{16} - 26 q^{17} + 100 q^{19} + (40 \zeta_{6} - 40) q^{20} + 128 \zeta_{6} q^{22} - 78 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 152 q^{26} + 48 q^{28} + (50 \zeta_{6} - 50) q^{29} + 108 \zeta_{6} q^{31} + 256 \zeta_{6} q^{32} + ( - 104 \zeta_{6} + 104) q^{34} + 30 q^{35} + 266 q^{37} + (400 \zeta_{6} - 400) q^{38} + 22 \zeta_{6} q^{41} + (442 \zeta_{6} - 442) q^{43} - 256 q^{44} + 312 q^{46} + (514 \zeta_{6} - 514) q^{47} + 307 \zeta_{6} q^{49} - 100 \zeta_{6} q^{50} + ( - 304 \zeta_{6} + 304) q^{52} - 2 q^{53} - 160 q^{55} - 200 \zeta_{6} q^{58} + 500 \zeta_{6} q^{59} + ( - 518 \zeta_{6} + 518) q^{61} - 432 q^{62} - 512 q^{64} + ( - 190 \zeta_{6} + 190) q^{65} - 126 \zeta_{6} q^{67} + 208 \zeta_{6} q^{68} + (120 \zeta_{6} - 120) q^{70} - 412 q^{71} - 878 q^{73} + (1064 \zeta_{6} - 1064) q^{74} - 800 \zeta_{6} q^{76} + 192 \zeta_{6} q^{77} + (600 \zeta_{6} - 600) q^{79} - 320 q^{80} - 88 q^{82} + ( - 282 \zeta_{6} + 282) q^{83} + 130 \zeta_{6} q^{85} - 1768 \zeta_{6} q^{86} + 150 q^{89} - 228 q^{91} + (624 \zeta_{6} - 624) q^{92} - 2056 \zeta_{6} q^{94} - 500 \zeta_{6} q^{95} + (386 \zeta_{6} - 386) q^{97} - 1228 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 8 q^{4} - 5 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 8 q^{4} - 5 q^{5} - 6 q^{7} + 40 q^{10} + 32 q^{11} + 38 q^{13} - 24 q^{14} + 64 q^{16} - 52 q^{17} + 200 q^{19} - 40 q^{20} + 128 q^{22} - 78 q^{23} - 25 q^{25} - 304 q^{26} + 96 q^{28} - 50 q^{29} + 108 q^{31} + 256 q^{32} + 104 q^{34} + 60 q^{35} + 532 q^{37} - 400 q^{38} + 22 q^{41} - 442 q^{43} - 512 q^{44} + 624 q^{46} - 514 q^{47} + 307 q^{49} - 100 q^{50} + 304 q^{52} - 4 q^{53} - 320 q^{55} - 200 q^{58} + 500 q^{59} + 518 q^{61} - 864 q^{62} - 1024 q^{64} + 190 q^{65} - 126 q^{67} + 208 q^{68} - 120 q^{70} - 824 q^{71} - 1756 q^{73} - 1064 q^{74} - 800 q^{76} + 192 q^{77} - 600 q^{79} - 640 q^{80} - 176 q^{82} + 282 q^{83} + 130 q^{85} - 1768 q^{86} + 300 q^{89} - 456 q^{91} - 624 q^{92} - 2056 q^{94} - 500 q^{95} - 386 q^{97} - 2456 q^{98}+O(q^{100}) \) 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
−2.00000 + 3.46410i 0 −4.00000 6.92820i −2.50000 4.33013i 0 −3.00000 + 5.19615i 0 0 20.0000
271.1 −2.00000 3.46410i 0 −4.00000 + 6.92820i −2.50000 + 4.33013i 0 −3.00000 5.19615i 0 0 20.0000
\(n\): 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.c 2
3.b odd 2 1 405.4.e.l 2
9.c even 3 1 45.4.a.d 1
9.c even 3 1 inner 405.4.e.c 2
9.d odd 6 1 5.4.a.a 1
9.d odd 6 1 405.4.e.l 2
36.f odd 6 1 720.4.a.u 1
36.h even 6 1 80.4.a.d 1
45.h odd 6 1 25.4.a.c 1
45.j even 6 1 225.4.a.b 1
45.k odd 12 2 225.4.b.c 2
45.l even 12 2 25.4.b.a 2
63.i even 6 1 245.4.e.g 2
63.j odd 6 1 245.4.e.f 2
63.l odd 6 1 2205.4.a.q 1
63.n odd 6 1 245.4.e.f 2
63.o even 6 1 245.4.a.a 1
63.s even 6 1 245.4.e.g 2
72.j odd 6 1 320.4.a.g 1
72.l even 6 1 320.4.a.h 1
99.g even 6 1 605.4.a.d 1
117.n odd 6 1 845.4.a.b 1
144.u even 12 2 1280.4.d.l 2
144.w odd 12 2 1280.4.d.e 2
153.i odd 6 1 1445.4.a.a 1
171.l even 6 1 1805.4.a.h 1
180.n even 6 1 400.4.a.m 1
180.v odd 12 2 400.4.c.k 2
315.z even 6 1 1225.4.a.k 1
360.bd even 6 1 1600.4.a.s 1
360.bh odd 6 1 1600.4.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 9.d odd 6 1
25.4.a.c 1 45.h odd 6 1
25.4.b.a 2 45.l even 12 2
45.4.a.d 1 9.c even 3 1
80.4.a.d 1 36.h even 6 1
225.4.a.b 1 45.j even 6 1
225.4.b.c 2 45.k odd 12 2
245.4.a.a 1 63.o even 6 1
245.4.e.f 2 63.j odd 6 1
245.4.e.f 2 63.n odd 6 1
245.4.e.g 2 63.i even 6 1
245.4.e.g 2 63.s even 6 1
320.4.a.g 1 72.j odd 6 1
320.4.a.h 1 72.l even 6 1
400.4.a.m 1 180.n even 6 1
400.4.c.k 2 180.v odd 12 2
405.4.e.c 2 1.a even 1 1 trivial
405.4.e.c 2 9.c even 3 1 inner
405.4.e.l 2 3.b odd 2 1
405.4.e.l 2 9.d odd 6 1
605.4.a.d 1 99.g even 6 1
720.4.a.u 1 36.f odd 6 1
845.4.a.b 1 117.n odd 6 1
1225.4.a.k 1 315.z even 6 1
1280.4.d.e 2 144.w odd 12 2
1280.4.d.l 2 144.u even 12 2
1445.4.a.a 1 153.i odd 6 1
1600.4.a.s 1 360.bd even 6 1
1600.4.a.bi 1 360.bh odd 6 1
1805.4.a.h 1 171.l even 6 1
2205.4.a.q 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 \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{2} + 4T_{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$11$ \( T^{2} - 32T + 1024 \) Copy content Toggle raw display
$13$ \( T^{2} - 38T + 1444 \) Copy content Toggle raw display
$17$ \( (T + 26)^{2} \) Copy content Toggle raw display
$19$ \( (T - 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 78T + 6084 \) 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
$37$ \( (T - 266)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 22T + 484 \) Copy content Toggle raw display
$43$ \( T^{2} + 442T + 195364 \) Copy content Toggle raw display
$47$ \( T^{2} + 514T + 264196 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 500T + 250000 \) Copy content Toggle raw display
$61$ \( T^{2} - 518T + 268324 \) Copy content Toggle raw display
$67$ \( T^{2} + 126T + 15876 \) Copy content Toggle raw display
$71$ \( (T + 412)^{2} \) Copy content Toggle raw display
$73$ \( (T + 878)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 600T + 360000 \) Copy content Toggle raw display
$83$ \( T^{2} - 282T + 79524 \) Copy content Toggle raw display
$89$ \( (T - 150)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 386T + 148996 \) Copy content Toggle raw display
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