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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: \(x^{2} - x + 1\)
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.

\(f(q)\) \(=\) \( q + ( -4 + 4 \zeta_{6} ) q^{2} -8 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( -6 + 6 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -4 + 4 \zeta_{6} ) q^{2} -8 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( -6 + 6 \zeta_{6} ) q^{7} + 20 q^{10} + ( 32 - 32 \zeta_{6} ) q^{11} + 38 \zeta_{6} q^{13} -24 \zeta_{6} q^{14} + ( 64 - 64 \zeta_{6} ) q^{16} -26 q^{17} + 100 q^{19} + ( -40 + 40 \zeta_{6} ) q^{20} + 128 \zeta_{6} q^{22} -78 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} -152 q^{26} + 48 q^{28} + ( -50 + 50 \zeta_{6} ) q^{29} + 108 \zeta_{6} q^{31} + 256 \zeta_{6} q^{32} + ( 104 - 104 \zeta_{6} ) q^{34} + 30 q^{35} + 266 q^{37} + ( -400 + 400 \zeta_{6} ) q^{38} + 22 \zeta_{6} q^{41} + ( -442 + 442 \zeta_{6} ) q^{43} -256 q^{44} + 312 q^{46} + ( -514 + 514 \zeta_{6} ) q^{47} + 307 \zeta_{6} q^{49} -100 \zeta_{6} q^{50} + ( 304 - 304 \zeta_{6} ) q^{52} -2 q^{53} -160 q^{55} -200 \zeta_{6} q^{58} + 500 \zeta_{6} q^{59} + ( 518 - 518 \zeta_{6} ) q^{61} -432 q^{62} -512 q^{64} + ( 190 - 190 \zeta_{6} ) q^{65} -126 \zeta_{6} q^{67} + 208 \zeta_{6} q^{68} + ( -120 + 120 \zeta_{6} ) q^{70} -412 q^{71} -878 q^{73} + ( -1064 + 1064 \zeta_{6} ) q^{74} -800 \zeta_{6} q^{76} + 192 \zeta_{6} q^{77} + ( -600 + 600 \zeta_{6} ) q^{79} -320 q^{80} -88 q^{82} + ( 282 - 282 \zeta_{6} ) q^{83} + 130 \zeta_{6} q^{85} -1768 \zeta_{6} q^{86} + 150 q^{89} -228 q^{91} + ( -624 + 624 \zeta_{6} ) q^{92} -2056 \zeta_{6} q^{94} -500 \zeta_{6} q^{95} + ( -386 + 386 \zeta_{6} ) q^{97} -1228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} - 8q^{4} - 5q^{5} - 6q^{7} + O(q^{10}) \) \( 2q - 4q^{2} - 8q^{4} - 5q^{5} - 6q^{7} + 40q^{10} + 32q^{11} + 38q^{13} - 24q^{14} + 64q^{16} - 52q^{17} + 200q^{19} - 40q^{20} + 128q^{22} - 78q^{23} - 25q^{25} - 304q^{26} + 96q^{28} - 50q^{29} + 108q^{31} + 256q^{32} + 104q^{34} + 60q^{35} + 532q^{37} - 400q^{38} + 22q^{41} - 442q^{43} - 512q^{44} + 624q^{46} - 514q^{47} + 307q^{49} - 100q^{50} + 304q^{52} - 4q^{53} - 320q^{55} - 200q^{58} + 500q^{59} + 518q^{61} - 864q^{62} - 1024q^{64} + 190q^{65} - 126q^{67} + 208q^{68} - 120q^{70} - 824q^{71} - 1756q^{73} - 1064q^{74} - 800q^{76} + 192q^{77} - 600q^{79} - 640q^{80} - 176q^{82} + 282q^{83} + 130q^{85} - 1768q^{86} + 300q^{89} - 456q^{91} - 624q^{92} - 2056q^{94} - 500q^{95} - 386q^{97} - 2456q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{2} + 4 T_{2} + 16 \)
\( T_{7}^{2} + 6 T_{7} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 8 T^{2} + 32 T^{3} + 64 T^{4} \)
$3$ 1
$5$ \( 1 + 5 T + 25 T^{2} \)
$7$ \( 1 + 6 T - 307 T^{2} + 2058 T^{3} + 117649 T^{4} \)
$11$ \( 1 - 32 T - 307 T^{2} - 42592 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 38 T - 753 T^{2} - 83486 T^{3} + 4826809 T^{4} \)
$17$ \( ( 1 + 26 T + 4913 T^{2} )^{2} \)
$19$ \( ( 1 - 100 T + 6859 T^{2} )^{2} \)
$23$ \( 1 + 78 T - 6083 T^{2} + 949026 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 50 T - 21889 T^{2} + 1219450 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 108 T - 18127 T^{2} - 3217428 T^{3} + 887503681 T^{4} \)
$37$ \( ( 1 - 266 T + 50653 T^{2} )^{2} \)
$41$ \( 1 - 22 T - 68437 T^{2} - 1516262 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 442 T + 115857 T^{2} + 35142094 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 + 514 T + 160373 T^{2} + 53365022 T^{3} + 10779215329 T^{4} \)
$53$ \( ( 1 + 2 T + 148877 T^{2} )^{2} \)
$59$ \( 1 - 500 T + 44621 T^{2} - 102689500 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 518 T + 41343 T^{2} - 117576158 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 126 T - 284887 T^{2} + 37896138 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 + 412 T + 357911 T^{2} )^{2} \)
$73$ \( ( 1 + 878 T + 389017 T^{2} )^{2} \)
$79$ \( 1 + 600 T - 133039 T^{2} + 295823400 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 282 T - 492263 T^{2} - 161243934 T^{3} + 326940373369 T^{4} \)
$89$ \( ( 1 - 150 T + 704969 T^{2} )^{2} \)
$97$ \( 1 + 386 T - 763677 T^{2} + 352291778 T^{3} + 832972004929 T^{4} \)
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