Properties

Label 405.3.i.b
Level $405$
Weight $3$
Character orbit 405.i
Analytic conductor $11.035$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 6 - 6 \beta_{2} ) q^{7} -3 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 6 - 6 \beta_{2} ) q^{7} -3 \beta_{3} q^{8} + 5 q^{10} -2 \beta_{1} q^{11} -16 \beta_{2} q^{13} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{14} + ( 19 - 19 \beta_{2} ) q^{16} + 2 \beta_{3} q^{17} -2 q^{19} + \beta_{1} q^{20} -10 \beta_{2} q^{22} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{23} + ( 5 - 5 \beta_{2} ) q^{25} -16 \beta_{3} q^{26} + 6 q^{28} + 14 \beta_{1} q^{29} + 18 \beta_{2} q^{31} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{32} + ( -10 + 10 \beta_{2} ) q^{34} -6 \beta_{3} q^{35} -16 q^{37} -2 \beta_{1} q^{38} -15 \beta_{2} q^{40} + ( 28 \beta_{1} - 28 \beta_{3} ) q^{41} + ( -16 + 16 \beta_{2} ) q^{43} -2 \beta_{3} q^{44} + 30 q^{46} + 22 \beta_{1} q^{47} + 13 \beta_{2} q^{49} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{50} + ( 16 - 16 \beta_{2} ) q^{52} + 2 \beta_{3} q^{53} -10 q^{55} -18 \beta_{1} q^{56} + 70 \beta_{2} q^{58} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{59} + ( -82 + 82 \beta_{2} ) q^{61} + 18 \beta_{3} q^{62} -41 q^{64} -16 \beta_{1} q^{65} -24 \beta_{2} q^{67} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{68} + ( 30 - 30 \beta_{2} ) q^{70} -56 \beta_{3} q^{71} -74 q^{73} -16 \beta_{1} q^{74} -2 \beta_{2} q^{76} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{77} + ( -138 + 138 \beta_{2} ) q^{79} -19 \beta_{3} q^{80} + 140 q^{82} + 42 \beta_{1} q^{83} + 10 \beta_{2} q^{85} + ( -16 \beta_{1} + 16 \beta_{3} ) q^{86} + ( -30 + 30 \beta_{2} ) q^{88} + 48 \beta_{3} q^{89} -96 q^{91} + 6 \beta_{1} q^{92} + 110 \beta_{2} q^{94} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{95} + ( 166 - 166 \beta_{2} ) q^{97} + 13 \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 12q^{7} + O(q^{10}) \) \( 4q + 2q^{4} + 12q^{7} + 20q^{10} - 32q^{13} + 38q^{16} - 8q^{19} - 20q^{22} + 10q^{25} + 24q^{28} + 36q^{31} - 20q^{34} - 64q^{37} - 30q^{40} - 32q^{43} + 120q^{46} + 26q^{49} + 32q^{52} - 40q^{55} + 140q^{58} - 164q^{61} - 164q^{64} - 48q^{67} + 60q^{70} - 296q^{73} - 4q^{76} - 276q^{79} + 560q^{82} + 20q^{85} - 60q^{88} - 384q^{91} + 220q^{94} + 332q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.93649 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 1.11803i
−1.93649 1.11803i 0 0.500000 + 0.866025i −1.93649 + 1.11803i 0 3.00000 5.19615i 6.70820i 0 5.00000
26.2 1.93649 + 1.11803i 0 0.500000 + 0.866025i 1.93649 1.11803i 0 3.00000 5.19615i 6.70820i 0 5.00000
296.1 −1.93649 + 1.11803i 0 0.500000 0.866025i −1.93649 1.11803i 0 3.00000 + 5.19615i 6.70820i 0 5.00000
296.2 1.93649 1.11803i 0 0.500000 0.866025i 1.93649 + 1.11803i 0 3.00000 + 5.19615i 6.70820i 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
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.i.b 4
3.b odd 2 1 inner 405.3.i.b 4
9.c even 3 1 15.3.c.a 2
9.c even 3 1 inner 405.3.i.b 4
9.d odd 6 1 15.3.c.a 2
9.d odd 6 1 inner 405.3.i.b 4
36.f odd 6 1 240.3.l.b 2
36.h even 6 1 240.3.l.b 2
45.h odd 6 1 75.3.c.e 2
45.j even 6 1 75.3.c.e 2
45.k odd 12 2 75.3.d.b 4
45.l even 12 2 75.3.d.b 4
72.j odd 6 1 960.3.l.c 2
72.l even 6 1 960.3.l.b 2
72.n even 6 1 960.3.l.c 2
72.p odd 6 1 960.3.l.b 2
180.n even 6 1 1200.3.l.g 2
180.p odd 6 1 1200.3.l.g 2
180.v odd 12 2 1200.3.c.f 4
180.x even 12 2 1200.3.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.c.a 2 9.c even 3 1
15.3.c.a 2 9.d odd 6 1
75.3.c.e 2 45.h odd 6 1
75.3.c.e 2 45.j even 6 1
75.3.d.b 4 45.k odd 12 2
75.3.d.b 4 45.l even 12 2
240.3.l.b 2 36.f odd 6 1
240.3.l.b 2 36.h even 6 1
405.3.i.b 4 1.a even 1 1 trivial
405.3.i.b 4 3.b odd 2 1 inner
405.3.i.b 4 9.c even 3 1 inner
405.3.i.b 4 9.d odd 6 1 inner
960.3.l.b 2 72.l even 6 1
960.3.l.b 2 72.p odd 6 1
960.3.l.c 2 72.j odd 6 1
960.3.l.c 2 72.n even 6 1
1200.3.c.f 4 180.v odd 12 2
1200.3.c.f 4 180.x even 12 2
1200.3.l.g 2 180.n even 6 1
1200.3.l.g 2 180.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{4} - 5 T_{2}^{2} + 25 \)
\( T_{7}^{2} - 6 T_{7} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 - 5 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 - 5 T^{2} + T^{4} \)
$7$ \( ( 36 - 6 T + T^{2} )^{2} \)
$11$ \( 400 - 20 T^{2} + T^{4} \)
$13$ \( ( 256 + 16 T + T^{2} )^{2} \)
$17$ \( ( 20 + T^{2} )^{2} \)
$19$ \( ( 2 + T )^{4} \)
$23$ \( 32400 - 180 T^{2} + T^{4} \)
$29$ \( 960400 - 980 T^{2} + T^{4} \)
$31$ \( ( 324 - 18 T + T^{2} )^{2} \)
$37$ \( ( 16 + T )^{4} \)
$41$ \( 15366400 - 3920 T^{2} + T^{4} \)
$43$ \( ( 256 + 16 T + T^{2} )^{2} \)
$47$ \( 5856400 - 2420 T^{2} + T^{4} \)
$53$ \( ( 20 + T^{2} )^{2} \)
$59$ \( 400 - 20 T^{2} + T^{4} \)
$61$ \( ( 6724 + 82 T + T^{2} )^{2} \)
$67$ \( ( 576 + 24 T + T^{2} )^{2} \)
$71$ \( ( 15680 + T^{2} )^{2} \)
$73$ \( ( 74 + T )^{4} \)
$79$ \( ( 19044 + 138 T + T^{2} )^{2} \)
$83$ \( 77792400 - 8820 T^{2} + T^{4} \)
$89$ \( ( 11520 + T^{2} )^{2} \)
$97$ \( ( 27556 - 166 T + T^{2} )^{2} \)
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