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Properties

Label 441.2.h.g
Level $441$
Weight $2$
Character orbit 441.h
Analytic conductor $3.521$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,2,Mod(214,441)]
 
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([4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.214");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.h (of order \(3\), degree \(2\), not 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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{10} q^{3} + (\beta_{4} + \beta_{2} + 1) q^{4} + (\beta_{8} - \beta_1) q^{5} + (\beta_{11} - \beta_{10} + \cdots + 2 \beta_1) q^{6}+ \cdots + (\beta_{5} - \beta_{4} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{10} q^{3} + (\beta_{4} + \beta_{2} + 1) q^{4} + (\beta_{8} - \beta_1) q^{5} + (\beta_{11} - \beta_{10} + \cdots + 2 \beta_1) q^{6}+ \cdots + (2 \beta_{7} - \beta_{5} + 3 \beta_{4} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 12 q^{4} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} + 12 q^{4} + 24 q^{8} - 8 q^{11} + 18 q^{15} + 12 q^{16} + 24 q^{18} - 6 q^{22} - 4 q^{23} - 12 q^{25} - 22 q^{29} + 6 q^{30} + 32 q^{32} - 30 q^{36} + 6 q^{37} - 48 q^{39} - 6 q^{43} + 14 q^{44} - 12 q^{46} - 56 q^{50} + 36 q^{51} - 28 q^{53} - 6 q^{57} - 18 q^{58} + 18 q^{60} - 48 q^{64} - 12 q^{65} + 76 q^{71} - 30 q^{72} - 36 q^{74} + 36 q^{78} - 12 q^{79} + 24 q^{81} + 30 q^{85} + 36 q^{86} + 6 q^{88} - 62 q^{92} - 84 q^{93} + 120 q^{95} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\nu^{11} + 556\nu^{9} - 3553\nu^{7} + 18231\nu^{5} - 32493\nu^{3} + 26811\nu ) / 12897 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -49\nu^{10} + 259\nu^{8} - 1369\nu^{6} + 861\nu^{4} - 252\nu^{2} - 7266 ) / 4299 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 148\nu^{10} - 987\nu^{8} + 5217\nu^{6} - 10175\nu^{4} + 17343\nu^{2} - 5076 ) / 4299 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 161\nu^{10} - 851\nu^{8} + 3884\nu^{6} - 2829\nu^{4} + 828\nu^{2} + 6678 ) / 4299 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -296\nu^{10} + 1974\nu^{8} - 10434\nu^{6} + 20350\nu^{4} - 30387\nu^{2} + 1554 ) / 4299 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -322\nu^{11} + 1702\nu^{9} - 7768\nu^{7} + 1359\nu^{5} + 15540\nu^{3} - 90738\nu ) / 12897 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -120\nu^{10} + 839\nu^{8} - 4230\nu^{6} + 8250\nu^{4} - 10034\nu^{2} + 630 ) / 1433 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 455\nu^{11} - 2405\nu^{9} + 12098\nu^{7} - 12294\nu^{5} + 19536\nu^{3} + 37377\nu ) / 12897 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 461\nu^{11} - 3665\nu^{9} + 18758\nu^{7} - 44949\nu^{5} + 54573\nu^{3} - 33588\nu ) / 12897 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -1804\nu^{11} + 11992\nu^{9} - 62158\nu^{7} + 118293\nu^{5} - 178167\nu^{3} + 13770\nu ) / 12897 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2051\nu^{11} - 15140\nu^{9} + 81254\nu^{7} - 186504\nu^{5} + 298581\nu^{3} - 112869\nu ) / 12897 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{9} + 3\beta_{8} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} - 3\beta_{10} - 2\beta_{9} + \beta_{6} - 6\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + 5\beta_{5} - \beta_{4} + 7\beta_{3} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{11} - 24\beta_{10} - 28\beta_{9} - 54\beta_{8} - 7\beta_{6} + 12\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{4} - 23\beta_{2} - 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 37\beta_{11} - 30\beta_{10} - 104\beta_{9} - 222\beta_{8} - 53\beta_{6} + 171\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 37\beta_{7} - 104\beta_{5} - 118\beta_{3} - 118 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 245\beta_{11} + 333\beta_{10} + 44\beta_{9} + 111\beta_{8} - 67\beta_{6} + 534\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 178\beta_{7} - 467\beta_{5} + 178\beta_{4} - 511\beta_{3} + 467\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 289\beta_{11} + 1956\beta_{10} + 2245\beta_{9} + 4869\beta_{8} + 823\beta_{6} - 978\beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(\beta_{3}\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1
−0.474636 + 0.274031i
0.474636 − 0.274031i
−1.29589 + 0.748185i
1.29589 − 0.748185i
1.82904 − 1.05600i
−1.82904 + 1.05600i
−0.474636 − 0.274031i
0.474636 + 0.274031i
−1.29589 − 0.748185i
1.29589 + 0.748185i
1.82904 + 1.05600i
−1.82904 − 1.05600i
−1.69963 −0.175815 − 1.72310i 0.888736 −0.474636 + 0.822093i 0.298820 + 2.92864i 0 1.88874 −2.93818 + 0.605896i 0.806704 − 1.39725i
214.2 −1.69963 0.175815 + 1.72310i 0.888736 0.474636 − 0.822093i −0.298820 − 2.92864i 0 1.88874 −2.93818 + 0.605896i −0.806704 + 1.39725i
214.3 0.239123 −1.70316 + 0.315036i −1.94282 −1.29589 + 2.24456i −0.407265 + 0.0753324i 0 −0.942820 2.80150 − 1.07311i −0.309879 + 0.536725i
214.4 0.239123 1.70316 − 0.315036i −1.94282 1.29589 − 2.24456i 0.407265 − 0.0753324i 0 −0.942820 2.80150 − 1.07311i 0.309879 − 0.536725i
214.5 2.46050 −1.25233 + 1.19652i 4.05408 1.82904 − 3.16799i −3.08137 + 2.94405i 0 5.05408 0.136673 − 2.99689i 4.50036 − 7.79485i
214.6 2.46050 1.25233 − 1.19652i 4.05408 −1.82904 + 3.16799i 3.08137 − 2.94405i 0 5.05408 0.136673 − 2.99689i −4.50036 + 7.79485i
373.1 −1.69963 −0.175815 + 1.72310i 0.888736 −0.474636 − 0.822093i 0.298820 − 2.92864i 0 1.88874 −2.93818 − 0.605896i 0.806704 + 1.39725i
373.2 −1.69963 0.175815 − 1.72310i 0.888736 0.474636 + 0.822093i −0.298820 + 2.92864i 0 1.88874 −2.93818 − 0.605896i −0.806704 − 1.39725i
373.3 0.239123 −1.70316 − 0.315036i −1.94282 −1.29589 − 2.24456i −0.407265 − 0.0753324i 0 −0.942820 2.80150 + 1.07311i −0.309879 − 0.536725i
373.4 0.239123 1.70316 + 0.315036i −1.94282 1.29589 + 2.24456i 0.407265 + 0.0753324i 0 −0.942820 2.80150 + 1.07311i 0.309879 + 0.536725i
373.5 2.46050 −1.25233 − 1.19652i 4.05408 1.82904 + 3.16799i −3.08137 − 2.94405i 0 5.05408 0.136673 + 2.99689i 4.50036 + 7.79485i
373.6 2.46050 1.25233 + 1.19652i 4.05408 −1.82904 − 3.16799i 3.08137 + 2.94405i 0 5.05408 0.136673 + 2.99689i −4.50036 − 7.79485i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 214.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.h even 3 1 inner
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.h.g 12
3.b odd 2 1 1323.2.h.g 12
7.b odd 2 1 inner 441.2.h.g 12
7.c even 3 1 441.2.f.g ✓ 12
7.c even 3 1 441.2.g.g 12
7.d odd 6 1 441.2.f.g ✓ 12
7.d odd 6 1 441.2.g.g 12
9.c even 3 1 441.2.g.g 12
9.d odd 6 1 1323.2.g.g 12
21.c even 2 1 1323.2.h.g 12
21.g even 6 1 1323.2.f.g 12
21.g even 6 1 1323.2.g.g 12
21.h odd 6 1 1323.2.f.g 12
21.h odd 6 1 1323.2.g.g 12
63.g even 3 1 441.2.f.g ✓ 12
63.h even 3 1 inner 441.2.h.g 12
63.h even 3 1 3969.2.a.be 6
63.i even 6 1 1323.2.h.g 12
63.i even 6 1 3969.2.a.bd 6
63.j odd 6 1 1323.2.h.g 12
63.j odd 6 1 3969.2.a.bd 6
63.k odd 6 1 441.2.f.g ✓ 12
63.l odd 6 1 441.2.g.g 12
63.n odd 6 1 1323.2.f.g 12
63.o even 6 1 1323.2.g.g 12
63.s even 6 1 1323.2.f.g 12
63.t odd 6 1 inner 441.2.h.g 12
63.t odd 6 1 3969.2.a.be 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.g ✓ 12 7.c even 3 1
441.2.f.g ✓ 12 7.d odd 6 1
441.2.f.g ✓ 12 63.g even 3 1
441.2.f.g ✓ 12 63.k odd 6 1
441.2.g.g 12 7.c even 3 1
441.2.g.g 12 7.d odd 6 1
441.2.g.g 12 9.c even 3 1
441.2.g.g 12 63.l odd 6 1
441.2.h.g 12 1.a even 1 1 trivial
441.2.h.g 12 7.b odd 2 1 inner
441.2.h.g 12 63.h even 3 1 inner
441.2.h.g 12 63.t odd 6 1 inner
1323.2.f.g 12 21.g even 6 1
1323.2.f.g 12 21.h odd 6 1
1323.2.f.g 12 63.n odd 6 1
1323.2.f.g 12 63.s even 6 1
1323.2.g.g 12 9.d odd 6 1
1323.2.g.g 12 21.g even 6 1
1323.2.g.g 12 21.h odd 6 1
1323.2.g.g 12 63.o even 6 1
1323.2.h.g 12 3.b odd 2 1
1323.2.h.g 12 21.c even 2 1
1323.2.h.g 12 63.i even 6 1
1323.2.h.g 12 63.j odd 6 1
3969.2.a.bd 6 63.i even 6 1
3969.2.a.bd 6 63.j odd 6 1
3969.2.a.be 6 63.h even 3 1
3969.2.a.be 6 63.t odd 6 1

Hecke kernels

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

\( T_{2}^{3} - T_{2}^{2} - 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{12} + 21T_{5}^{10} + 333T_{5}^{8} + 2106T_{5}^{6} + 9963T_{5}^{4} + 8748T_{5}^{2} + 6561 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} - T^{2} - 4 T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{12} - 6 T^{8} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 21 T^{10} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + 4 T^{5} + 17 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 39 T^{10} + \cdots + 6561 \) Copy content Toggle raw display
$17$ \( T^{12} + 84 T^{10} + \cdots + 15752961 \) Copy content Toggle raw display
$19$ \( T^{12} + 75 T^{10} + \cdots + 15752961 \) Copy content Toggle raw display
$23$ \( (T^{6} + 2 T^{5} + \cdots + 3481)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 11 T^{5} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 129 T^{4} + \cdots - 77841)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 3 T^{5} + \cdots + 729)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 162 T^{10} + \cdots + 43046721 \) Copy content Toggle raw display
$43$ \( (T^{6} + 3 T^{5} + \cdots + 729)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 183 T^{4} + \cdots - 194481)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 14 T^{5} + \cdots + 69169)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 183 T^{4} + \cdots - 149769)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 264 T^{4} + \cdots - 558009)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 111 T + 353)^{4} \) Copy content Toggle raw display
$71$ \( (T^{3} - 19 T^{2} + \cdots - 227)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + 75 T^{10} + \cdots + 15752961 \) Copy content Toggle raw display
$79$ \( (T^{3} + 3 T^{2} + \cdots - 107)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 51769445841 \) Copy content Toggle raw display
$89$ \( T^{12} + 246 T^{10} + \cdots + 15752961 \) Copy content Toggle raw display
$97$ \( T^{12} + 111 T^{10} + \cdots + 96059601 \) Copy content Toggle raw display
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