Properties

Label 441.4.e.o
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: \(\Z[a_1, \ldots, a_{25}]\)
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.

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{2} + ( -8 + 8 \zeta_{6} ) q^{4} + 18 \zeta_{6} q^{5} +O(q^{10})\) \( q + 4 \zeta_{6} q^{2} + ( -8 + 8 \zeta_{6} ) q^{4} + 18 \zeta_{6} q^{5} + ( -72 + 72 \zeta_{6} ) q^{10} + ( -50 + 50 \zeta_{6} ) q^{11} -36 q^{13} + 64 \zeta_{6} q^{16} + ( 126 - 126 \zeta_{6} ) q^{17} + 72 \zeta_{6} q^{19} -144 q^{20} -200 q^{22} + 14 \zeta_{6} q^{23} + ( -199 + 199 \zeta_{6} ) q^{25} -144 \zeta_{6} q^{26} -158 q^{29} + ( 36 - 36 \zeta_{6} ) q^{31} + ( -256 + 256 \zeta_{6} ) q^{32} + 504 q^{34} + 162 \zeta_{6} q^{37} + ( -288 + 288 \zeta_{6} ) q^{38} + 270 q^{41} -324 q^{43} -400 \zeta_{6} q^{44} + ( -56 + 56 \zeta_{6} ) q^{46} -72 \zeta_{6} q^{47} -796 q^{50} + ( 288 - 288 \zeta_{6} ) q^{52} + ( -22 + 22 \zeta_{6} ) q^{53} -900 q^{55} -632 \zeta_{6} q^{58} + ( 468 - 468 \zeta_{6} ) q^{59} -792 \zeta_{6} q^{61} + 144 q^{62} -512 q^{64} -648 \zeta_{6} q^{65} + ( -232 + 232 \zeta_{6} ) q^{67} + 1008 \zeta_{6} q^{68} + 734 q^{71} + ( -180 + 180 \zeta_{6} ) q^{73} + ( -648 + 648 \zeta_{6} ) q^{74} -576 q^{76} -236 \zeta_{6} q^{79} + ( -1152 + 1152 \zeta_{6} ) q^{80} + 1080 \zeta_{6} q^{82} -36 q^{83} + 2268 q^{85} -1296 \zeta_{6} q^{86} + 234 \zeta_{6} q^{89} -112 q^{92} + ( 288 - 288 \zeta_{6} ) q^{94} + ( -1296 + 1296 \zeta_{6} ) q^{95} + 468 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} - 8q^{4} + 18q^{5} + O(q^{10}) \) \( 2q + 4q^{2} - 8q^{4} + 18q^{5} - 72q^{10} - 50q^{11} - 72q^{13} + 64q^{16} + 126q^{17} + 72q^{19} - 288q^{20} - 400q^{22} + 14q^{23} - 199q^{25} - 144q^{26} - 316q^{29} + 36q^{31} - 256q^{32} + 1008q^{34} + 162q^{37} - 288q^{38} + 540q^{41} - 648q^{43} - 400q^{44} - 56q^{46} - 72q^{47} - 1592q^{50} + 288q^{52} - 22q^{53} - 1800q^{55} - 632q^{58} + 468q^{59} - 792q^{61} + 288q^{62} - 1024q^{64} - 648q^{65} - 232q^{67} + 1008q^{68} + 1468q^{71} - 180q^{73} - 648q^{74} - 1152q^{76} - 236q^{79} - 1152q^{80} + 1080q^{82} - 72q^{83} + 4536q^{85} - 1296q^{86} + 234q^{89} - 224q^{92} + 288q^{94} - 1296q^{95} + 936q^{97} + O(q^{100}) \)

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
2.00000 3.46410i 0 −4.00000 6.92820i 9.00000 15.5885i 0 0 0 0 −36.0000 62.3538i
361.1 2.00000 + 3.46410i 0 −4.00000 + 6.92820i 9.00000 + 15.5885i 0 0 0 0 −36.0000 + 62.3538i
\(n\): 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.o 2
3.b odd 2 1 147.4.e.a 2
7.b odd 2 1 441.4.e.l 2
7.c even 3 1 441.4.a.a 1
7.c even 3 1 inner 441.4.e.o 2
7.d odd 6 1 441.4.a.c 1
7.d odd 6 1 441.4.e.l 2
21.c even 2 1 147.4.e.d 2
21.g even 6 1 147.4.a.f 1
21.g even 6 1 147.4.e.d 2
21.h odd 6 1 147.4.a.h yes 1
21.h odd 6 1 147.4.e.a 2
84.j odd 6 1 2352.4.a.t 1
84.n even 6 1 2352.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.f 1 21.g even 6 1
147.4.a.h yes 1 21.h odd 6 1
147.4.e.a 2 3.b odd 2 1
147.4.e.a 2 21.h odd 6 1
147.4.e.d 2 21.c even 2 1
147.4.e.d 2 21.g even 6 1
441.4.a.a 1 7.c even 3 1
441.4.a.c 1 7.d odd 6 1
441.4.e.l 2 7.b odd 2 1
441.4.e.l 2 7.d odd 6 1
441.4.e.o 2 1.a even 1 1 trivial
441.4.e.o 2 7.c even 3 1 inner
2352.4.a.s 1 84.n even 6 1
2352.4.a.t 1 84.j odd 6 1

Hecke kernels

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

\( T_{2}^{2} - 4 T_{2} + 16 \)
\( T_{5}^{2} - 18 T_{5} + 324 \)
\( T_{13} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 4 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 324 - 18 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 2500 + 50 T + T^{2} \)
$13$ \( ( 36 + T )^{2} \)
$17$ \( 15876 - 126 T + T^{2} \)
$19$ \( 5184 - 72 T + T^{2} \)
$23$ \( 196 - 14 T + T^{2} \)
$29$ \( ( 158 + T )^{2} \)
$31$ \( 1296 - 36 T + T^{2} \)
$37$ \( 26244 - 162 T + T^{2} \)
$41$ \( ( -270 + T )^{2} \)
$43$ \( ( 324 + T )^{2} \)
$47$ \( 5184 + 72 T + T^{2} \)
$53$ \( 484 + 22 T + T^{2} \)
$59$ \( 219024 - 468 T + T^{2} \)
$61$ \( 627264 + 792 T + T^{2} \)
$67$ \( 53824 + 232 T + T^{2} \)
$71$ \( ( -734 + T )^{2} \)
$73$ \( 32400 + 180 T + T^{2} \)
$79$ \( 55696 + 236 T + T^{2} \)
$83$ \( ( 36 + T )^{2} \)
$89$ \( 54756 - 234 T + T^{2} \)
$97$ \( ( -468 + T )^{2} \)
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