Properties

Label 441.2.e.g
Level $441$
Weight $2$
Character orbit 441.e
Analytic conductor $3.521$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{5} + ( 3 + \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{5} + ( 3 + \beta_{3} ) q^{8} + ( \beta_{1} + \beta_{3} ) q^{10} + 2 \beta_{2} q^{11} + ( 4 + \beta_{3} ) q^{13} + ( -3 - 3 \beta_{2} ) q^{16} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{17} + 2 \beta_{1} q^{19} + ( 2 - 3 \beta_{3} ) q^{20} + ( 2 + 2 \beta_{3} ) q^{22} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{23} + ( 4 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{25} + ( -6 + 5 \beta_{1} - 6 \beta_{2} ) q^{26} + ( 4 - 2 \beta_{3} ) q^{29} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{32} + ( -8 - 5 \beta_{3} ) q^{34} + ( 4 + 4 \beta_{2} ) q^{37} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{38} + ( 4 + \beta_{1} + 4 \beta_{2} ) q^{40} + ( -2 + 3 \beta_{3} ) q^{41} -4 \beta_{3} q^{43} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{44} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{46} -2 \beta_{1} q^{47} + ( -7 - 3 \beta_{3} ) q^{50} + ( -9 \beta_{1} + 8 \beta_{2} - 9 \beta_{3} ) q^{52} + 2 \beta_{2} q^{53} + ( -4 + 2 \beta_{3} ) q^{55} + 2 \beta_{1} q^{58} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{59} + ( -8 - 3 \beta_{1} - 8 \beta_{2} ) q^{61} + ( -8 - 6 \beta_{3} ) q^{62} + ( -7 - 2 \beta_{3} ) q^{64} + ( 6 + 2 \beta_{1} + 6 \beta_{2} ) q^{65} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{67} + ( 14 - 7 \beta_{1} + 14 \beta_{2} ) q^{68} + ( 2 + 8 \beta_{3} ) q^{71} + ( 7 \beta_{1} + 4 \beta_{2} + 7 \beta_{3} ) q^{73} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{74} + ( 8 + 2 \beta_{3} ) q^{76} + ( -8 - 4 \beta_{1} - 8 \beta_{2} ) q^{79} + ( -3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{80} + ( -4 + \beta_{1} - 4 \beta_{2} ) q^{82} + ( 4 + 8 \beta_{3} ) q^{83} + ( -2 + 4 \beta_{3} ) q^{85} + ( 8 - 4 \beta_{1} + 8 \beta_{2} ) q^{86} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{88} + ( -10 - 3 \beta_{1} - 10 \beta_{2} ) q^{89} -14 q^{92} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{94} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{95} + ( 4 + \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} + 4q^{5} + 12q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} + 4q^{5} + 12q^{8} - 4q^{11} + 16q^{13} - 6q^{16} + 4q^{17} + 8q^{20} + 8q^{22} - 4q^{23} - 2q^{25} - 12q^{26} + 16q^{29} + 8q^{31} + 6q^{32} - 32q^{34} + 8q^{37} - 8q^{38} + 8q^{40} - 8q^{41} - 4q^{44} + 12q^{46} - 28q^{50} - 16q^{52} - 4q^{53} - 16q^{55} - 8q^{59} - 16q^{61} - 32q^{62} - 28q^{64} + 12q^{65} + 28q^{68} + 8q^{71} - 8q^{73} + 8q^{74} + 32q^{76} - 16q^{79} + 12q^{80} - 8q^{82} + 16q^{83} - 8q^{85} + 16q^{86} - 12q^{88} - 20q^{89} - 56q^{92} + 8q^{94} - 8q^{95} + 16q^{97} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−1.20711 + 2.09077i 0 −1.91421 3.31552i 0.292893 0.507306i 0 0 4.41421 0 0.707107 + 1.22474i
226.2 0.207107 0.358719i 0 0.914214 + 1.58346i 1.70711 2.95680i 0 0 1.58579 0 −0.707107 1.22474i
361.1 −1.20711 2.09077i 0 −1.91421 + 3.31552i 0.292893 + 0.507306i 0 0 4.41421 0 0.707107 1.22474i
361.2 0.207107 + 0.358719i 0 0.914214 1.58346i 1.70711 + 2.95680i 0 0 1.58579 0 −0.707107 + 1.22474i
\(n\): 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 441.2.e.g 4
3.b odd 2 1 147.2.e.d 4
7.b odd 2 1 441.2.e.f 4
7.c even 3 1 441.2.a.i 2
7.c even 3 1 inner 441.2.e.g 4
7.d odd 6 1 441.2.a.j 2
7.d odd 6 1 441.2.e.f 4
12.b even 2 1 2352.2.q.bd 4
21.c even 2 1 147.2.e.e 4
21.g even 6 1 147.2.a.d 2
21.g even 6 1 147.2.e.e 4
21.h odd 6 1 147.2.a.e yes 2
21.h odd 6 1 147.2.e.d 4
28.f even 6 1 7056.2.a.cv 2
28.g odd 6 1 7056.2.a.cf 2
84.h odd 2 1 2352.2.q.bb 4
84.j odd 6 1 2352.2.a.be 2
84.j odd 6 1 2352.2.q.bb 4
84.n even 6 1 2352.2.a.bc 2
84.n even 6 1 2352.2.q.bd 4
105.o odd 6 1 3675.2.a.bd 2
105.p even 6 1 3675.2.a.bf 2
168.s odd 6 1 9408.2.a.di 2
168.v even 6 1 9408.2.a.dt 2
168.ba even 6 1 9408.2.a.ef 2
168.be odd 6 1 9408.2.a.dq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 21.g even 6 1
147.2.a.e yes 2 21.h odd 6 1
147.2.e.d 4 3.b odd 2 1
147.2.e.d 4 21.h odd 6 1
147.2.e.e 4 21.c even 2 1
147.2.e.e 4 21.g even 6 1
441.2.a.i 2 7.c even 3 1
441.2.a.j 2 7.d odd 6 1
441.2.e.f 4 7.b odd 2 1
441.2.e.f 4 7.d odd 6 1
441.2.e.g 4 1.a even 1 1 trivial
441.2.e.g 4 7.c even 3 1 inner
2352.2.a.bc 2 84.n even 6 1
2352.2.a.be 2 84.j odd 6 1
2352.2.q.bb 4 84.h odd 2 1
2352.2.q.bb 4 84.j odd 6 1
2352.2.q.bd 4 12.b even 2 1
2352.2.q.bd 4 84.n even 6 1
3675.2.a.bd 2 105.o odd 6 1
3675.2.a.bf 2 105.p even 6 1
7056.2.a.cf 2 28.g odd 6 1
7056.2.a.cv 2 28.f even 6 1
9408.2.a.di 2 168.s odd 6 1
9408.2.a.dq 2 168.be odd 6 1
9408.2.a.dt 2 168.v even 6 1
9408.2.a.ef 2 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{4} + 2 T_{2}^{3} + 5 T_{2}^{2} - 2 T_{2} + 1 \)
\( T_{5}^{4} - 4 T_{5}^{3} + 14 T_{5}^{2} - 8 T_{5} + 4 \)
\( T_{13}^{2} - 8 T_{13} + 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + T^{2} - 2 T^{3} - 3 T^{4} - 4 T^{5} + 4 T^{6} + 16 T^{7} + 16 T^{8} \)
$3$ 1
$5$ \( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 40 T^{5} + 100 T^{6} - 500 T^{7} + 625 T^{8} \)
$7$ 1
$11$ \( ( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 8 T + 40 T^{2} - 104 T^{3} + 169 T^{4} )^{2} \)
$17$ \( 1 - 4 T - 4 T^{2} + 56 T^{3} - 161 T^{4} + 952 T^{5} - 1156 T^{6} - 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \)
$23$ \( 1 + 4 T - 2 T^{2} - 112 T^{3} - 573 T^{4} - 2576 T^{5} - 1058 T^{6} + 48668 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 - 8 T + 66 T^{2} - 232 T^{3} + 841 T^{4} )^{2} \)
$31$ \( 1 - 8 T - 6 T^{2} - 64 T^{3} + 1955 T^{4} - 1984 T^{5} - 5766 T^{6} - 238328 T^{7} + 923521 T^{8} \)
$37$ \( ( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 54 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 + 2 T - 49 T^{2} + 106 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 + 8 T - 62 T^{2} + 64 T^{3} + 8619 T^{4} + 3776 T^{5} - 215822 T^{6} + 1643032 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 16 T + 88 T^{2} + 736 T^{3} + 8887 T^{4} + 44896 T^{5} + 327448 T^{6} + 3631696 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 102 T^{2} + 5915 T^{4} - 457878 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 - 4 T + 18 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 + 8 T - 656 T^{3} - 5905 T^{4} - 47888 T^{5} + 3112136 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 40448 T^{5} + 411906 T^{6} + 7888624 T^{7} + 38950081 T^{8} \)
$83$ \( ( 1 - 8 T + 54 T^{2} - 664 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( 1 + 20 T + 140 T^{2} + 1640 T^{3} + 24079 T^{4} + 145960 T^{5} + 1108940 T^{6} + 14099380 T^{7} + 62742241 T^{8} \)
$97$ \( ( 1 - 8 T + 208 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \)
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