Properties

Label 441.4.e.h
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 7)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{4} + 16 \zeta_{6} q^{5} -15 q^{8} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{4} + 16 \zeta_{6} q^{5} -15 q^{8} + ( 16 - 16 \zeta_{6} ) q^{10} + ( -8 + 8 \zeta_{6} ) q^{11} + 28 q^{13} -41 \zeta_{6} q^{16} + ( 54 - 54 \zeta_{6} ) q^{17} + 110 \zeta_{6} q^{19} + 112 q^{20} + 8 q^{22} + 48 \zeta_{6} q^{23} + ( -131 + 131 \zeta_{6} ) q^{25} -28 \zeta_{6} q^{26} + 110 q^{29} + ( -12 + 12 \zeta_{6} ) q^{31} + ( -161 + 161 \zeta_{6} ) q^{32} -54 q^{34} + 246 \zeta_{6} q^{37} + ( 110 - 110 \zeta_{6} ) q^{38} -240 \zeta_{6} q^{40} -182 q^{41} + 128 q^{43} + 56 \zeta_{6} q^{44} + ( 48 - 48 \zeta_{6} ) q^{46} + 324 \zeta_{6} q^{47} + 131 q^{50} + ( 196 - 196 \zeta_{6} ) q^{52} + ( -162 + 162 \zeta_{6} ) q^{53} -128 q^{55} -110 \zeta_{6} q^{58} + ( 810 - 810 \zeta_{6} ) q^{59} + 488 \zeta_{6} q^{61} + 12 q^{62} -167 q^{64} + 448 \zeta_{6} q^{65} + ( -244 + 244 \zeta_{6} ) q^{67} -378 \zeta_{6} q^{68} + 768 q^{71} + ( 702 - 702 \zeta_{6} ) q^{73} + ( 246 - 246 \zeta_{6} ) q^{74} + 770 q^{76} -440 \zeta_{6} q^{79} + ( 656 - 656 \zeta_{6} ) q^{80} + 182 \zeta_{6} q^{82} + 1302 q^{83} + 864 q^{85} -128 \zeta_{6} q^{86} + ( 120 - 120 \zeta_{6} ) q^{88} + 730 \zeta_{6} q^{89} + 336 q^{92} + ( 324 - 324 \zeta_{6} ) q^{94} + ( -1760 + 1760 \zeta_{6} ) q^{95} + 294 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 7q^{4} + 16q^{5} - 30q^{8} + O(q^{10}) \) \( 2q - q^{2} + 7q^{4} + 16q^{5} - 30q^{8} + 16q^{10} - 8q^{11} + 56q^{13} - 41q^{16} + 54q^{17} + 110q^{19} + 224q^{20} + 16q^{22} + 48q^{23} - 131q^{25} - 28q^{26} + 220q^{29} - 12q^{31} - 161q^{32} - 108q^{34} + 246q^{37} + 110q^{38} - 240q^{40} - 364q^{41} + 256q^{43} + 56q^{44} + 48q^{46} + 324q^{47} + 262q^{50} + 196q^{52} - 162q^{53} - 256q^{55} - 110q^{58} + 810q^{59} + 488q^{61} + 24q^{62} - 334q^{64} + 448q^{65} - 244q^{67} - 378q^{68} + 1536q^{71} + 702q^{73} + 246q^{74} + 1540q^{76} - 440q^{79} + 656q^{80} + 182q^{82} + 2604q^{83} + 1728q^{85} - 128q^{86} + 120q^{88} + 730q^{89} + 672q^{92} + 324q^{94} - 1760q^{95} + 588q^{97} + O(q^{100}) \)

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 8.00000 13.8564i 0 0 −15.0000 0 8.00000 + 13.8564i
361.1 −0.500000 0.866025i 0 3.50000 6.06218i 8.00000 + 13.8564i 0 0 −15.0000 0 8.00000 13.8564i
\(n\): 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.h 2
3.b odd 2 1 49.4.c.c 2
7.b odd 2 1 441.4.e.e 2
7.c even 3 1 63.4.a.b 1
7.c even 3 1 inner 441.4.e.h 2
7.d odd 6 1 441.4.a.i 1
7.d odd 6 1 441.4.e.e 2
21.c even 2 1 49.4.c.b 2
21.g even 6 1 49.4.a.b 1
21.g even 6 1 49.4.c.b 2
21.h odd 6 1 7.4.a.a 1
21.h odd 6 1 49.4.c.c 2
28.g odd 6 1 1008.4.a.c 1
35.j even 6 1 1575.4.a.e 1
84.j odd 6 1 784.4.a.g 1
84.n even 6 1 112.4.a.f 1
105.o odd 6 1 175.4.a.b 1
105.p even 6 1 1225.4.a.j 1
105.x even 12 2 175.4.b.b 2
168.s odd 6 1 448.4.a.i 1
168.v even 6 1 448.4.a.e 1
231.l even 6 1 847.4.a.b 1
273.w odd 6 1 1183.4.a.b 1
357.q odd 6 1 2023.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 21.h odd 6 1
49.4.a.b 1 21.g even 6 1
49.4.c.b 2 21.c even 2 1
49.4.c.b 2 21.g even 6 1
49.4.c.c 2 3.b odd 2 1
49.4.c.c 2 21.h odd 6 1
63.4.a.b 1 7.c even 3 1
112.4.a.f 1 84.n even 6 1
175.4.a.b 1 105.o odd 6 1
175.4.b.b 2 105.x even 12 2
441.4.a.i 1 7.d odd 6 1
441.4.e.e 2 7.b odd 2 1
441.4.e.e 2 7.d odd 6 1
441.4.e.h 2 1.a even 1 1 trivial
441.4.e.h 2 7.c even 3 1 inner
448.4.a.e 1 168.v even 6 1
448.4.a.i 1 168.s odd 6 1
784.4.a.g 1 84.j odd 6 1
847.4.a.b 1 231.l even 6 1
1008.4.a.c 1 28.g odd 6 1
1183.4.a.b 1 273.w odd 6 1
1225.4.a.j 1 105.p even 6 1
1575.4.a.e 1 35.j even 6 1
2023.4.a.a 1 357.q odd 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \)
\( T_{5}^{2} - 16 T_{5} + 256 \)
\( T_{13} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 256 - 16 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 64 + 8 T + T^{2} \)
$13$ \( ( -28 + T )^{2} \)
$17$ \( 2916 - 54 T + T^{2} \)
$19$ \( 12100 - 110 T + T^{2} \)
$23$ \( 2304 - 48 T + T^{2} \)
$29$ \( ( -110 + T )^{2} \)
$31$ \( 144 + 12 T + T^{2} \)
$37$ \( 60516 - 246 T + T^{2} \)
$41$ \( ( 182 + T )^{2} \)
$43$ \( ( -128 + T )^{2} \)
$47$ \( 104976 - 324 T + T^{2} \)
$53$ \( 26244 + 162 T + T^{2} \)
$59$ \( 656100 - 810 T + T^{2} \)
$61$ \( 238144 - 488 T + T^{2} \)
$67$ \( 59536 + 244 T + T^{2} \)
$71$ \( ( -768 + T )^{2} \)
$73$ \( 492804 - 702 T + T^{2} \)
$79$ \( 193600 + 440 T + T^{2} \)
$83$ \( ( -1302 + T )^{2} \)
$89$ \( 532900 - 730 T + T^{2} \)
$97$ \( ( -294 + T )^{2} \)
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