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Properties

Label 441.2.f.f
Level $441$
Weight $2$
Character orbit 441.f
Analytic conductor $3.521$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,2,Mod(148,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.148");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.f (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.991381711347.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{8} + \beta_{6} + \beta_{4} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{9} - \beta_{7} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{8} + \beta_{6} + \beta_{4} + \cdots - 1) q^{4}+ \cdots + ( - 3 \beta_{9} + \beta_{7} + \beta_{5} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} + q^{3} - 4 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} + q^{3} - 4 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{8} - 7 q^{9} - 14 q^{10} + 4 q^{11} + 2 q^{12} + 8 q^{13} - 19 q^{15} + 2 q^{16} + 24 q^{17} - 2 q^{18} + 2 q^{19} - 5 q^{20} - q^{22} + 3 q^{23} + 9 q^{24} - q^{25} + 22 q^{26} + 7 q^{27} + 7 q^{29} + 10 q^{30} + 3 q^{31} - 2 q^{32} + 13 q^{33} - 3 q^{34} + 34 q^{36} - 20 q^{38} - 22 q^{39} + 3 q^{40} - 5 q^{41} - 7 q^{43} + 20 q^{44} - 17 q^{45} - 6 q^{46} - 27 q^{47} + 5 q^{48} + 19 q^{50} - 15 q^{51} + 10 q^{52} + 42 q^{53} - 52 q^{54} - 4 q^{55} - 4 q^{57} - 10 q^{58} - 30 q^{59} + 31 q^{60} + 14 q^{61} + 12 q^{62} - 50 q^{64} - 11 q^{65} - 22 q^{66} - 2 q^{67} - 27 q^{68} - 15 q^{69} - 6 q^{71} - 12 q^{72} + 30 q^{73} - 36 q^{74} + 17 q^{75} - 5 q^{76} - 20 q^{78} - 4 q^{79} + 40 q^{80} - 31 q^{81} - 10 q^{82} - 9 q^{83} - 6 q^{85} - 8 q^{86} + 34 q^{87} - 18 q^{88} + 56 q^{89} - 28 q^{90} + 27 q^{92} + 18 q^{93} + 3 q^{94} - 14 q^{95} + 58 q^{96} + 12 q^{97} + 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{9} + 9\nu^{8} - 3\nu^{7} + 95\nu^{6} + 18\nu^{5} + 402\nu^{4} - 87\nu^{3} + 936\nu^{2} + 342\nu + 72 ) / 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{9} + \nu^{8} - 12\nu^{7} - 8\nu^{6} - 68\nu^{5} - 30\nu^{4} - 123\nu^{3} - 204\nu^{2} - 270\nu - 63 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17 \nu^{9} - 24 \nu^{8} + 159 \nu^{7} - 106 \nu^{6} + 786 \nu^{5} - 417 \nu^{4} + 1893 \nu^{3} + \cdots + 639 ) / 567 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} + \cdots - 180 ) / 567 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + \cdots + 504 ) / 567 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{9} - 12\nu^{8} + 69\nu^{7} - 43\nu^{6} + 330\nu^{5} - 219\nu^{4} + 732\nu^{3} - 45\nu^{2} + 477\nu - 306 ) / 189 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 71 \nu^{9} + 123 \nu^{8} - 591 \nu^{7} + 403 \nu^{6} - 2604 \nu^{5} + 1794 \nu^{4} - 5214 \nu^{3} + \cdots - 234 ) / 567 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} + \cdots + 720 ) / 567 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 3\beta_{6} + \beta_{4} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 4\beta_{5} - \beta_{3} - 4\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{8} - 5\beta_{7} - 13\beta_{6} - \beta_{4} - \beta_{3} + 4\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{9} + \beta_{8} - 2\beta_{7} - 7\beta_{6} - 19\beta_{5} - \beta_{4} + \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{9} + 9\beta_{8} + 15\beta_{7} - 10\beta_{5} - 15\beta_{4} + 10\beta_{3} + 9\beta_{2} + 10\beta _1 + 61 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{8} + 11\beta_{7} + 46\beta_{6} + 19\beta_{4} + 43\beta_{3} + 8\beta_{2} + 94\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 73\beta_{9} + 56\beta_{8} + 62\beta_{7} + 298\beta_{6} + 76\beta_{5} + 118\beta_{4} - 118\beta_{2} - 298 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 253 \beta_{9} - 135 \beta_{8} + 48 \beta_{7} + 478 \beta_{5} - 48 \beta_{4} - 253 \beta_{3} - 135 \beta_{2} + \cdots - 295 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1
−1.02682 − 1.77851i
−0.335166 − 0.580525i
0.247934 + 0.429435i
0.920620 + 1.59456i
1.19343 + 2.06709i
−1.02682 + 1.77851i
−0.335166 + 0.580525i
0.247934 − 0.429435i
0.920620 − 1.59456i
1.19343 − 2.06709i
−1.02682 − 1.77851i −0.608729 + 1.62156i −1.10873 + 1.92038i −0.0731228 + 0.126652i 3.50901 − 0.582422i 0 0.446582 −2.25890 − 1.97418i 0.300337
148.2 −0.335166 − 0.580525i 1.27533 − 1.17198i 0.775327 − 1.34291i 0.712469 − 1.23403i −1.10781 − 0.347551i 0 −2.38012 0.252918 − 2.98932i −0.955182
148.3 0.247934 + 0.429435i 1.37706 + 1.05058i 0.877057 − 1.51911i −1.84629 + 3.19787i −0.109735 + 0.851830i 0 1.86155 0.792574 + 2.89341i −1.83103
148.4 0.920620 + 1.59456i −0.195084 − 1.72103i −0.695084 + 1.20392i 0.667377 − 1.15593i 2.56469 − 1.89549i 0 1.12285 −2.92388 + 0.671489i 2.45760
148.5 1.19343 + 2.06709i −1.34857 + 1.08690i −1.84857 + 3.20182i −1.46043 + 2.52954i −3.85615 − 1.49047i 0 −4.05086 0.637290 − 2.93153i −6.97172
295.1 −1.02682 + 1.77851i −0.608729 − 1.62156i −1.10873 − 1.92038i −0.0731228 − 0.126652i 3.50901 + 0.582422i 0 0.446582 −2.25890 + 1.97418i 0.300337
295.2 −0.335166 + 0.580525i 1.27533 + 1.17198i 0.775327 + 1.34291i 0.712469 + 1.23403i −1.10781 + 0.347551i 0 −2.38012 0.252918 + 2.98932i −0.955182
295.3 0.247934 − 0.429435i 1.37706 − 1.05058i 0.877057 + 1.51911i −1.84629 − 3.19787i −0.109735 − 0.851830i 0 1.86155 0.792574 − 2.89341i −1.83103
295.4 0.920620 − 1.59456i −0.195084 + 1.72103i −0.695084 − 1.20392i 0.667377 + 1.15593i 2.56469 + 1.89549i 0 1.12285 −2.92388 − 0.671489i 2.45760
295.5 1.19343 − 2.06709i −1.34857 − 1.08690i −1.84857 − 3.20182i −1.46043 − 2.52954i −3.85615 + 1.49047i 0 −4.05086 0.637290 + 2.93153i −6.97172
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 148.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.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.f.f 10
3.b odd 2 1 1323.2.f.f 10
7.b odd 2 1 441.2.f.e 10
7.c even 3 1 441.2.g.f 10
7.c even 3 1 441.2.h.f 10
7.d odd 6 1 63.2.g.b ✓ 10
7.d odd 6 1 63.2.h.b yes 10
9.c even 3 1 inner 441.2.f.f 10
9.c even 3 1 3969.2.a.ba 5
9.d odd 6 1 1323.2.f.f 10
9.d odd 6 1 3969.2.a.bb 5
21.c even 2 1 1323.2.f.e 10
21.g even 6 1 189.2.g.b 10
21.g even 6 1 189.2.h.b 10
21.h odd 6 1 1323.2.g.f 10
21.h odd 6 1 1323.2.h.f 10
28.f even 6 1 1008.2.q.i 10
28.f even 6 1 1008.2.t.i 10
63.g even 3 1 441.2.h.f 10
63.h even 3 1 441.2.g.f 10
63.i even 6 1 189.2.g.b 10
63.i even 6 1 567.2.e.e 10
63.j odd 6 1 1323.2.g.f 10
63.k odd 6 1 63.2.h.b yes 10
63.k odd 6 1 567.2.e.f 10
63.l odd 6 1 441.2.f.e 10
63.l odd 6 1 3969.2.a.z 5
63.n odd 6 1 1323.2.h.f 10
63.o even 6 1 1323.2.f.e 10
63.o even 6 1 3969.2.a.bc 5
63.s even 6 1 189.2.h.b 10
63.s even 6 1 567.2.e.e 10
63.t odd 6 1 63.2.g.b ✓ 10
63.t odd 6 1 567.2.e.f 10
84.j odd 6 1 3024.2.q.i 10
84.j odd 6 1 3024.2.t.i 10
252.n even 6 1 1008.2.q.i 10
252.r odd 6 1 3024.2.t.i 10
252.bj even 6 1 1008.2.t.i 10
252.bn odd 6 1 3024.2.q.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b ✓ 10 7.d odd 6 1
63.2.g.b ✓ 10 63.t odd 6 1
63.2.h.b yes 10 7.d odd 6 1
63.2.h.b yes 10 63.k odd 6 1
189.2.g.b 10 21.g even 6 1
189.2.g.b 10 63.i even 6 1
189.2.h.b 10 21.g even 6 1
189.2.h.b 10 63.s even 6 1
441.2.f.e 10 7.b odd 2 1
441.2.f.e 10 63.l odd 6 1
441.2.f.f 10 1.a even 1 1 trivial
441.2.f.f 10 9.c even 3 1 inner
441.2.g.f 10 7.c even 3 1
441.2.g.f 10 63.h even 3 1
441.2.h.f 10 7.c even 3 1
441.2.h.f 10 63.g even 3 1
567.2.e.e 10 63.i even 6 1
567.2.e.e 10 63.s even 6 1
567.2.e.f 10 63.k odd 6 1
567.2.e.f 10 63.t odd 6 1
1008.2.q.i 10 28.f even 6 1
1008.2.q.i 10 252.n even 6 1
1008.2.t.i 10 28.f even 6 1
1008.2.t.i 10 252.bj even 6 1
1323.2.f.e 10 21.c even 2 1
1323.2.f.e 10 63.o even 6 1
1323.2.f.f 10 3.b odd 2 1
1323.2.f.f 10 9.d odd 6 1
1323.2.g.f 10 21.h odd 6 1
1323.2.g.f 10 63.j odd 6 1
1323.2.h.f 10 21.h odd 6 1
1323.2.h.f 10 63.n odd 6 1
3024.2.q.i 10 84.j odd 6 1
3024.2.q.i 10 252.bn odd 6 1
3024.2.t.i 10 84.j odd 6 1
3024.2.t.i 10 252.r odd 6 1
3969.2.a.z 5 63.l odd 6 1
3969.2.a.ba 5 9.c even 3 1
3969.2.a.bb 5 9.d odd 6 1
3969.2.a.bc 5 63.o even 6 1

Hecke kernels

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

\( T_{2}^{10} - 2T_{2}^{9} + 9T_{2}^{8} - 8T_{2}^{7} + 40T_{2}^{6} - 36T_{2}^{5} + 90T_{2}^{4} - 3T_{2}^{3} + 36T_{2}^{2} - 9T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{10} + 4T_{5}^{9} + 21T_{5}^{8} + 16T_{5}^{7} + 79T_{5}^{6} - 51T_{5}^{5} + 402T_{5}^{4} - 294T_{5}^{3} + 378T_{5}^{2} + 54T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 2 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} + 4 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 4 T^{9} + \cdots + 225 \) Copy content Toggle raw display
$13$ \( T^{10} - 8 T^{9} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( (T^{5} - 12 T^{4} + 45 T^{3} + \cdots + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - T^{4} - 41 T^{3} + \cdots + 431)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} - 3 T^{9} + \cdots + 2595321 \) Copy content Toggle raw display
$29$ \( T^{10} - 7 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$31$ \( T^{10} - 3 T^{9} + \cdots + 81225 \) Copy content Toggle raw display
$37$ \( (T^{5} - 96 T^{3} + \cdots - 288)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 5 T^{9} + \cdots + 2025 \) Copy content Toggle raw display
$43$ \( T^{10} + 7 T^{9} + \cdots + 687241 \) Copy content Toggle raw display
$47$ \( T^{10} + 27 T^{9} + \cdots + 43758225 \) Copy content Toggle raw display
$53$ \( (T^{5} - 21 T^{4} + \cdots + 423)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 30 T^{9} + \cdots + 31640625 \) Copy content Toggle raw display
$61$ \( T^{10} - 14 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{10} + 2 T^{9} + \cdots + 50708641 \) Copy content Toggle raw display
$71$ \( (T^{5} + 3 T^{4} - 168 T^{3} + \cdots - 81)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 15 T^{4} + \cdots + 879)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 4 T^{9} + \cdots + 37249 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 218123361 \) Copy content Toggle raw display
$89$ \( (T^{5} - 28 T^{4} + \cdots + 2661)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 2307745521 \) Copy content Toggle raw display
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