Properties

Label 441.4.e.g
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0198423125\)
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 147)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{4} + 12 \zeta_{6} q^{5} -15 q^{8} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{4} + 12 \zeta_{6} q^{5} -15 q^{8} + ( 12 - 12 \zeta_{6} ) q^{10} + ( 20 - 20 \zeta_{6} ) q^{11} -84 q^{13} -41 \zeta_{6} q^{16} + ( -96 + 96 \zeta_{6} ) q^{17} -12 \zeta_{6} q^{19} + 84 q^{20} -20 q^{22} -176 \zeta_{6} q^{23} + ( -19 + 19 \zeta_{6} ) q^{25} + 84 \zeta_{6} q^{26} -58 q^{29} + ( 264 - 264 \zeta_{6} ) q^{31} + ( -161 + 161 \zeta_{6} ) q^{32} + 96 q^{34} -258 \zeta_{6} q^{37} + ( -12 + 12 \zeta_{6} ) q^{38} -180 \zeta_{6} q^{40} + 156 q^{43} -140 \zeta_{6} q^{44} + ( -176 + 176 \zeta_{6} ) q^{46} -408 \zeta_{6} q^{47} + 19 q^{50} + ( -588 + 588 \zeta_{6} ) q^{52} + ( -722 + 722 \zeta_{6} ) q^{53} + 240 q^{55} + 58 \zeta_{6} q^{58} + ( 492 - 492 \zeta_{6} ) q^{59} + 492 \zeta_{6} q^{61} -264 q^{62} -167 q^{64} -1008 \zeta_{6} q^{65} + ( -412 + 412 \zeta_{6} ) q^{67} + 672 \zeta_{6} q^{68} -296 q^{71} + ( -240 + 240 \zeta_{6} ) q^{73} + ( -258 + 258 \zeta_{6} ) q^{74} -84 q^{76} -776 \zeta_{6} q^{79} + ( 492 - 492 \zeta_{6} ) q^{80} -924 q^{83} -1152 q^{85} -156 \zeta_{6} q^{86} + ( -300 + 300 \zeta_{6} ) q^{88} -744 \zeta_{6} q^{89} -1232 q^{92} + ( -408 + 408 \zeta_{6} ) q^{94} + ( 144 - 144 \zeta_{6} ) q^{95} -168 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 7q^{4} + 12q^{5} - 30q^{8} + O(q^{10}) \) \( 2q - q^{2} + 7q^{4} + 12q^{5} - 30q^{8} + 12q^{10} + 20q^{11} - 168q^{13} - 41q^{16} - 96q^{17} - 12q^{19} + 168q^{20} - 40q^{22} - 176q^{23} - 19q^{25} + 84q^{26} - 116q^{29} + 264q^{31} - 161q^{32} + 192q^{34} - 258q^{37} - 12q^{38} - 180q^{40} + 312q^{43} - 140q^{44} - 176q^{46} - 408q^{47} + 38q^{50} - 588q^{52} - 722q^{53} + 480q^{55} + 58q^{58} + 492q^{59} + 492q^{61} - 528q^{62} - 334q^{64} - 1008q^{65} - 412q^{67} + 672q^{68} - 592q^{71} - 240q^{73} - 258q^{74} - 168q^{76} - 776q^{79} + 492q^{80} - 1848q^{83} - 2304q^{85} - 156q^{86} - 300q^{88} - 744q^{89} - 2464q^{92} - 408q^{94} + 144q^{95} - 336q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0 3.50000 + 6.06218i 6.00000 10.3923i 0 0 −15.0000 0 6.00000 + 10.3923i
361.1 −0.500000 0.866025i 0 3.50000 6.06218i 6.00000 + 10.3923i 0 0 −15.0000 0 6.00000 10.3923i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.g 2
3.b odd 2 1 147.4.e.f 2
7.b odd 2 1 441.4.e.f 2
7.c even 3 1 441.4.a.g 1
7.c even 3 1 inner 441.4.e.g 2
7.d odd 6 1 441.4.a.h 1
7.d odd 6 1 441.4.e.f 2
21.c even 2 1 147.4.e.e 2
21.g even 6 1 147.4.a.e yes 1
21.g even 6 1 147.4.e.e 2
21.h odd 6 1 147.4.a.d 1
21.h odd 6 1 147.4.e.f 2
84.j odd 6 1 2352.4.a.b 1
84.n even 6 1 2352.4.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.d 1 21.h odd 6 1
147.4.a.e yes 1 21.g even 6 1
147.4.e.e 2 21.c even 2 1
147.4.e.e 2 21.g even 6 1
147.4.e.f 2 3.b odd 2 1
147.4.e.f 2 21.h odd 6 1
441.4.a.g 1 7.c even 3 1
441.4.a.h 1 7.d odd 6 1
441.4.e.f 2 7.b odd 2 1
441.4.e.f 2 7.d odd 6 1
441.4.e.g 2 1.a even 1 1 trivial
441.4.e.g 2 7.c even 3 1 inner
2352.4.a.b 1 84.j odd 6 1
2352.4.a.bi 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{2} + T_{2} + 1 \)
\( T_{5}^{2} - 12 T_{5} + 144 \)
\( T_{13} + 84 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 144 - 12 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 400 - 20 T + T^{2} \)
$13$ \( ( 84 + T )^{2} \)
$17$ \( 9216 + 96 T + T^{2} \)
$19$ \( 144 + 12 T + T^{2} \)
$23$ \( 30976 + 176 T + T^{2} \)
$29$ \( ( 58 + T )^{2} \)
$31$ \( 69696 - 264 T + T^{2} \)
$37$ \( 66564 + 258 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -156 + T )^{2} \)
$47$ \( 166464 + 408 T + T^{2} \)
$53$ \( 521284 + 722 T + T^{2} \)
$59$ \( 242064 - 492 T + T^{2} \)
$61$ \( 242064 - 492 T + T^{2} \)
$67$ \( 169744 + 412 T + T^{2} \)
$71$ \( ( 296 + T )^{2} \)
$73$ \( 57600 + 240 T + T^{2} \)
$79$ \( 602176 + 776 T + T^{2} \)
$83$ \( ( 924 + T )^{2} \)
$89$ \( 553536 + 744 T + T^{2} \)
$97$ \( ( 168 + T )^{2} \)
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