Properties

Label 405.4.e.g
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} + 7 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( 24 - 24 \zeta_{6} ) q^{7} -15 q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} + 7 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( 24 - 24 \zeta_{6} ) q^{7} -15 q^{8} + 5 q^{10} + ( -52 + 52 \zeta_{6} ) q^{11} -22 \zeta_{6} q^{13} + 24 \zeta_{6} q^{14} + ( -41 + 41 \zeta_{6} ) q^{16} -14 q^{17} -20 q^{19} + ( 35 - 35 \zeta_{6} ) q^{20} -52 \zeta_{6} q^{22} + 168 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} + 22 q^{26} + 168 q^{28} + ( -230 + 230 \zeta_{6} ) q^{29} + 288 \zeta_{6} q^{31} -161 \zeta_{6} q^{32} + ( 14 - 14 \zeta_{6} ) q^{34} -120 q^{35} -34 q^{37} + ( 20 - 20 \zeta_{6} ) q^{38} + 75 \zeta_{6} q^{40} -122 \zeta_{6} q^{41} + ( 188 - 188 \zeta_{6} ) q^{43} -364 q^{44} -168 q^{46} + ( -256 + 256 \zeta_{6} ) q^{47} -233 \zeta_{6} q^{49} -25 \zeta_{6} q^{50} + ( 154 - 154 \zeta_{6} ) q^{52} -338 q^{53} + 260 q^{55} + ( -360 + 360 \zeta_{6} ) q^{56} -230 \zeta_{6} q^{58} -100 \zeta_{6} q^{59} + ( -742 + 742 \zeta_{6} ) q^{61} -288 q^{62} -167 q^{64} + ( -110 + 110 \zeta_{6} ) q^{65} + 84 \zeta_{6} q^{67} -98 \zeta_{6} q^{68} + ( 120 - 120 \zeta_{6} ) q^{70} -328 q^{71} -38 q^{73} + ( 34 - 34 \zeta_{6} ) q^{74} -140 \zeta_{6} q^{76} + 1248 \zeta_{6} q^{77} + ( 240 - 240 \zeta_{6} ) q^{79} + 205 q^{80} + 122 q^{82} + ( -1212 + 1212 \zeta_{6} ) q^{83} + 70 \zeta_{6} q^{85} + 188 \zeta_{6} q^{86} + ( 780 - 780 \zeta_{6} ) q^{88} + 330 q^{89} -528 q^{91} + ( -1176 + 1176 \zeta_{6} ) q^{92} -256 \zeta_{6} q^{94} + 100 \zeta_{6} q^{95} + ( -866 + 866 \zeta_{6} ) q^{97} + 233 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 7q^{4} - 5q^{5} + 24q^{7} - 30q^{8} + O(q^{10}) \) \( 2q - q^{2} + 7q^{4} - 5q^{5} + 24q^{7} - 30q^{8} + 10q^{10} - 52q^{11} - 22q^{13} + 24q^{14} - 41q^{16} - 28q^{17} - 40q^{19} + 35q^{20} - 52q^{22} + 168q^{23} - 25q^{25} + 44q^{26} + 336q^{28} - 230q^{29} + 288q^{31} - 161q^{32} + 14q^{34} - 240q^{35} - 68q^{37} + 20q^{38} + 75q^{40} - 122q^{41} + 188q^{43} - 728q^{44} - 336q^{46} - 256q^{47} - 233q^{49} - 25q^{50} + 154q^{52} - 676q^{53} + 520q^{55} - 360q^{56} - 230q^{58} - 100q^{59} - 742q^{61} - 576q^{62} - 334q^{64} - 110q^{65} + 84q^{67} - 98q^{68} + 120q^{70} - 656q^{71} - 76q^{73} + 34q^{74} - 140q^{76} + 1248q^{77} + 240q^{79} + 410q^{80} + 244q^{82} - 1212q^{83} + 70q^{85} + 188q^{86} + 780q^{88} + 660q^{89} - 1056q^{91} - 1176q^{92} - 256q^{94} + 100q^{95} - 866q^{97} + 466q^{98} + O(q^{100}) \)

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
−0.500000 + 0.866025i 0 3.50000 + 6.06218i −2.50000 4.33013i 0 12.0000 20.7846i −15.0000 0 5.00000
271.1 −0.500000 0.866025i 0 3.50000 6.06218i −2.50000 + 4.33013i 0 12.0000 + 20.7846i −15.0000 0 5.00000
\(n\): 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.g 2
3.b odd 2 1 405.4.e.i 2
9.c even 3 1 15.4.a.a 1
9.c even 3 1 inner 405.4.e.g 2
9.d odd 6 1 45.4.a.c 1
9.d odd 6 1 405.4.e.i 2
36.f odd 6 1 240.4.a.e 1
36.h even 6 1 720.4.a.n 1
45.h odd 6 1 225.4.a.f 1
45.j even 6 1 75.4.a.b 1
45.k odd 12 2 75.4.b.b 2
45.l even 12 2 225.4.b.e 2
63.l odd 6 1 735.4.a.e 1
63.o even 6 1 2205.4.a.l 1
72.n even 6 1 960.4.a.b 1
72.p odd 6 1 960.4.a.ba 1
99.h odd 6 1 1815.4.a.e 1
180.p odd 6 1 1200.4.a.t 1
180.x even 12 2 1200.4.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 9.c even 3 1
45.4.a.c 1 9.d odd 6 1
75.4.a.b 1 45.j even 6 1
75.4.b.b 2 45.k odd 12 2
225.4.a.f 1 45.h odd 6 1
225.4.b.e 2 45.l even 12 2
240.4.a.e 1 36.f odd 6 1
405.4.e.g 2 1.a even 1 1 trivial
405.4.e.g 2 9.c even 3 1 inner
405.4.e.i 2 3.b odd 2 1
405.4.e.i 2 9.d odd 6 1
720.4.a.n 1 36.h even 6 1
735.4.a.e 1 63.l odd 6 1
960.4.a.b 1 72.n even 6 1
960.4.a.ba 1 72.p odd 6 1
1200.4.a.t 1 180.p odd 6 1
1200.4.f.b 2 180.x even 12 2
1815.4.a.e 1 99.h odd 6 1
2205.4.a.l 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 \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{2} + T_{2} + 1 \)
\( T_{7}^{2} - 24 T_{7} + 576 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 25 + 5 T + T^{2} \)
$7$ \( 576 - 24 T + T^{2} \)
$11$ \( 2704 + 52 T + T^{2} \)
$13$ \( 484 + 22 T + T^{2} \)
$17$ \( ( 14 + T )^{2} \)
$19$ \( ( 20 + T )^{2} \)
$23$ \( 28224 - 168 T + T^{2} \)
$29$ \( 52900 + 230 T + T^{2} \)
$31$ \( 82944 - 288 T + T^{2} \)
$37$ \( ( 34 + T )^{2} \)
$41$ \( 14884 + 122 T + T^{2} \)
$43$ \( 35344 - 188 T + T^{2} \)
$47$ \( 65536 + 256 T + T^{2} \)
$53$ \( ( 338 + T )^{2} \)
$59$ \( 10000 + 100 T + T^{2} \)
$61$ \( 550564 + 742 T + T^{2} \)
$67$ \( 7056 - 84 T + T^{2} \)
$71$ \( ( 328 + T )^{2} \)
$73$ \( ( 38 + T )^{2} \)
$79$ \( 57600 - 240 T + T^{2} \)
$83$ \( 1468944 + 1212 T + T^{2} \)
$89$ \( ( -330 + T )^{2} \)
$97$ \( 749956 + 866 T + T^{2} \)
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