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Properties

Label 441.3.r.e
Level $441$
Weight $3$
Character orbit 441.r
Analytic conductor $12.016$
Analytic rank $0$
Dimension $16$
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] = [441,3,Mod(50,441)]
 
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([5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.50");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.r (of order \(6\), 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: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 30x^{14} + 699x^{12} + 5328x^{10} + 29790x^{8} + 65691x^{6} + 106920x^{4} + 28431x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{9} q^{3} + ( - \beta_{8} + 3 \beta_{5} - \beta_{4}) q^{4} + ( - \beta_{11} - \beta_{9}) q^{5} + ( - \beta_{10} + \beta_{2}) q^{6} + ( - \beta_{12} - \beta_{7} + 4 \beta_{5} + \cdots - 2) q^{8}+ \cdots + (\beta_{7} - \beta_{4} - 2 \beta_{3} + \cdots - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{9} q^{3} + ( - \beta_{8} + 3 \beta_{5} - \beta_{4}) q^{4} + ( - \beta_{11} - \beta_{9}) q^{5} + ( - \beta_{10} + \beta_{2}) q^{6} + ( - \beta_{12} - \beta_{7} + 4 \beta_{5} + \cdots - 2) q^{8}+ \cdots + ( - 6 \beta_{12} + 6 \beta_{8} + \cdots + 29) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} - 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 20 q^{4} - 66 q^{9} + 126 q^{11} - 42 q^{15} - 124 q^{16} - 114 q^{18} - 56 q^{22} - 108 q^{23} + 70 q^{25} - 54 q^{29} + 120 q^{30} - 306 q^{32} - 240 q^{36} + 84 q^{37} - 96 q^{39} + 60 q^{43} - 328 q^{46} + 270 q^{50} + 12 q^{51} + 24 q^{57} - 10 q^{58} + 96 q^{60} + 124 q^{64} - 486 q^{65} + 322 q^{67} - 66 q^{72} + 324 q^{74} + 1488 q^{78} + 98 q^{79} + 726 q^{81} + 198 q^{85} + 1458 q^{86} - 408 q^{88} - 90 q^{92} - 654 q^{93} - 432 q^{95} + 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 30x^{14} + 699x^{12} + 5328x^{10} + 29790x^{8} + 65691x^{6} + 106920x^{4} + 28431x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 21282484 \nu^{14} + 804608211 \nu^{12} + 20583395502 \nu^{10} + 244848110778 \nu^{8} + \cdots - 16574313025098 ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25408145 \nu^{15} + 368916351 \nu^{13} + 5397186864 \nu^{11} - 155765402118 \nu^{9} + \cdots - 54703903774725 \nu ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 65563387 \nu^{14} - 2078101884 \nu^{12} - 49233800652 \nu^{10} - 427260977505 \nu^{8} + \cdots - 5672088198792 ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 170837740 \nu^{14} - 5067470046 \nu^{12} - 117347878302 \nu^{10} - 860933175282 \nu^{8} + \cdots + 2065344345864 ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9104111 \nu^{14} - 270218880 \nu^{12} - 6282065280 \nu^{10} - 46627480908 \nu^{8} + \cdots - 34104136320 ) / 212504811471 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 225560263 \nu^{15} - 7143676722 \nu^{13} - 168456107625 \nu^{11} + \cdots - 30395289498354 \nu ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 395998759 \nu^{14} + 13011964563 \nu^{12} + 310136238771 \nu^{10} + 2878954356330 \nu^{8} + \cdots + 43019718213789 ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25713893 \nu^{14} - 759898914 \nu^{12} - 17678635986 \nu^{10} - 130360980789 \nu^{8} + \cdots - 169011884880 ) / 120610838943 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1096139321 \nu^{15} - 32371666410 \nu^{13} - 750998975241 \nu^{11} + \cdots + 4613913244629 \nu ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1296291439 \nu^{15} - 39146426781 \nu^{13} - 914057896002 \nu^{11} + \cdots - 67097476905777 \nu ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 469378687 \nu^{15} - 13890156838 \nu^{13} - 322385937462 \nu^{11} + \cdots + 10622180893239 \nu ) / 1487533680297 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1466984245 \nu^{14} + 43310985330 \nu^{12} + 1003079833071 \nu^{10} + 7290823118904 \nu^{8} + \cdots - 15222494380671 ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1791818123 \nu^{15} - 54977990757 \nu^{13} - 1287846205536 \nu^{11} + \cdots - 112110494969529 \nu ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 611596542 \nu^{15} - 18097719049 \nu^{13} - 419884986432 \nu^{11} + \cdots - 6195946171653 \nu ) / 1487533680297 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2181847172 \nu^{15} - 65469020325 \nu^{13} - 1525986845787 \nu^{11} + \cdots - 55381211181522 \nu ) / 4462601040891 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - \beta_{9} - \beta_{6} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 8\beta_{5} + 3\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{15} - 6\beta_{14} - 6\beta_{13} + 16\beta_{11} + 5\beta_{10} + 13\beta_{9} + 32\beta_{6} + 14\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{12} - 18\beta_{8} - 4\beta_{7} + 132\beta_{5} - 47\beta_{4} - 76\beta_{3} - 18\beta _1 - 114 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 87\beta_{15} + 51\beta_{13} - 114\beta_{11} - 151\beta_{10} + 91\beta_{9} - 110\beta_{6} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -51\beta_{12} + 51\beta_{7} - 669\beta_{4} + 1026\beta_{3} + 357\beta _1 + 2274 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -83\beta_{15} + 1128\beta_{14} + 344\beta_{11} + 2398\beta_{10} - 4405\beta_{9} - 3086\beta_{6} - 1697\beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2256\beta_{12} + 7443\beta_{8} + 1128\beta_{7} - 55125\beta_{5} + 36627\beta_{4} + 14592\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 35922 \beta_{15} - 24291 \beta_{14} - 24291 \beta_{13} + 43365 \beta_{11} + 16668 \beta_{10} + \cdots + 37524 \beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 24291 \beta_{12} - 157770 \beta_{8} - 48582 \beta_{7} + 1170603 \beta_{5} - 471159 \beta_{4} + \cdots - 1012833 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 798099 \beta_{15} + 519741 \beta_{13} - 1087197 \beta_{11} - 1442505 \beta_{10} + 896322 \beta_{9} + \cdots - 33105 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -519741\beta_{12} + 519741\beta_{7} - 6701148\beta_{4} + 10062054\beta_{3} + 3360906\beta _1 + 21589848 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 699876 \beta_{15} + 11101536 \beta_{14} + 3520377 \beta_{11} + 23199786 \beta_{10} + \cdots - 16318503 \beta_{2} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 22203072 \beta_{12} + 71698986 \beta_{8} + 11101536 \beta_{7} - 532367181 \beta_{5} + \cdots + 143110854 \beta_{3} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 348262461 \beta_{15} - 237012912 \beta_{14} - 237012912 \beta_{13} + 419961447 \beta_{11} + \cdots + 363160215 \beta_{2} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1
−1.23319 − 2.13594i
1.23319 + 2.13594i
−0.260383 − 0.450996i
0.260383 + 0.450996i
−0.758328 − 1.31346i
0.758328 + 1.31346i
−2.31006 − 4.00114i
2.31006 + 4.00114i
−1.23319 + 2.13594i
1.23319 − 2.13594i
−0.260383 + 0.450996i
0.260383 − 0.450996i
−0.758328 + 1.31346i
0.758328 − 1.31346i
−2.31006 + 4.00114i
2.31006 − 4.00114i
−2.84159 + 1.64059i −0.608180 − 2.93771i 3.38310 − 5.85970i −1.87503 − 1.08255i 6.54778 + 7.34999i 0 9.07642i −8.26023 + 3.57331i 7.10408
50.2 −2.84159 + 1.64059i 0.608180 + 2.93771i 3.38310 − 5.85970i 1.87503 + 1.08255i −6.54778 − 7.34999i 0 9.07642i −8.26023 + 3.57331i −7.10408
50.3 −0.876086 + 0.505809i −2.88037 − 0.838718i −1.48832 + 2.57784i 7.85997 + 4.53796i 2.94769 − 0.722129i 0 − 7.05768i 7.59311 + 4.83164i −9.18135
50.4 −0.876086 + 0.505809i 2.88037 + 0.838718i −1.48832 + 2.57784i −7.85997 − 4.53796i −2.94769 + 0.722129i 0 − 7.05768i 7.59311 + 4.83164i 9.18135
50.5 0.490034 − 0.282921i −0.989018 + 2.83229i −1.83991 + 3.18682i 0.692070 + 0.399567i 0.316661 + 1.66773i 0 4.34557i −7.04369 − 5.60236i 0.452183
50.6 0.490034 − 0.282921i 0.989018 − 2.83229i −1.83991 + 3.18682i −0.692070 − 0.399567i −0.316661 − 1.66773i 0 4.34557i −7.04369 − 5.60236i −0.452183
50.7 3.22764 − 1.86348i −0.324667 + 2.98238i 4.94513 − 8.56521i −5.95619 − 3.43881i 4.50970 + 10.2311i 0 − 21.9528i −8.78918 − 1.93656i −25.6326
50.8 3.22764 − 1.86348i 0.324667 − 2.98238i 4.94513 − 8.56521i 5.95619 + 3.43881i −4.50970 − 10.2311i 0 − 21.9528i −8.78918 − 1.93656i 25.6326
344.1 −2.84159 − 1.64059i −0.608180 + 2.93771i 3.38310 + 5.85970i −1.87503 + 1.08255i 6.54778 − 7.34999i 0 − 9.07642i −8.26023 − 3.57331i 7.10408
344.2 −2.84159 − 1.64059i 0.608180 − 2.93771i 3.38310 + 5.85970i 1.87503 − 1.08255i −6.54778 + 7.34999i 0 − 9.07642i −8.26023 − 3.57331i −7.10408
344.3 −0.876086 − 0.505809i −2.88037 + 0.838718i −1.48832 − 2.57784i 7.85997 − 4.53796i 2.94769 + 0.722129i 0 7.05768i 7.59311 − 4.83164i −9.18135
344.4 −0.876086 − 0.505809i 2.88037 − 0.838718i −1.48832 − 2.57784i −7.85997 + 4.53796i −2.94769 − 0.722129i 0 7.05768i 7.59311 − 4.83164i 9.18135
344.5 0.490034 + 0.282921i −0.989018 − 2.83229i −1.83991 − 3.18682i 0.692070 − 0.399567i 0.316661 − 1.66773i 0 − 4.34557i −7.04369 + 5.60236i 0.452183
344.6 0.490034 + 0.282921i 0.989018 + 2.83229i −1.83991 − 3.18682i −0.692070 + 0.399567i −0.316661 + 1.66773i 0 − 4.34557i −7.04369 + 5.60236i −0.452183
344.7 3.22764 + 1.86348i −0.324667 − 2.98238i 4.94513 + 8.56521i −5.95619 + 3.43881i 4.50970 − 10.2311i 0 21.9528i −8.78918 + 1.93656i −25.6326
344.8 3.22764 + 1.86348i 0.324667 + 2.98238i 4.94513 + 8.56521i 5.95619 − 3.43881i −4.50970 + 10.2311i 0 21.9528i −8.78918 + 1.93656i 25.6326
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.r.e ✓ 16
7.b odd 2 1 inner 441.3.r.e ✓ 16
7.c even 3 1 441.3.j.e 16
7.c even 3 1 441.3.n.e 16
7.d odd 6 1 441.3.j.e 16
7.d odd 6 1 441.3.n.e 16
9.d odd 6 1 inner 441.3.r.e ✓ 16
63.i even 6 1 441.3.n.e 16
63.j odd 6 1 441.3.n.e 16
63.n odd 6 1 441.3.j.e 16
63.o even 6 1 inner 441.3.r.e ✓ 16
63.s even 6 1 441.3.j.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.3.j.e 16 7.c even 3 1
441.3.j.e 16 7.d odd 6 1
441.3.j.e 16 63.n odd 6 1
441.3.j.e 16 63.s even 6 1
441.3.n.e 16 7.c even 3 1
441.3.n.e 16 7.d odd 6 1
441.3.n.e 16 63.i even 6 1
441.3.n.e 16 63.j odd 6 1
441.3.r.e ✓ 16 1.a even 1 1 trivial
441.3.r.e ✓ 16 7.b odd 2 1 inner
441.3.r.e ✓ 16 9.d odd 6 1 inner
441.3.r.e ✓ 16 63.o even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{8} - 13T_{2}^{6} + 162T_{2}^{4} + 117T_{2}^{3} - 64T_{2}^{2} - 63T_{2} + 49 \) Copy content Toggle raw display
\( T_{5}^{16} - 135 T_{5}^{14} + 13635 T_{5}^{12} - 577368 T_{5}^{10} + 18202401 T_{5}^{8} + \cdots + 136048896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 13 T^{6} + \cdots + 49)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 33 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 136048896 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 63 T^{7} + \cdots + 13344409)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( (T^{8} + 1035 T^{6} + \cdots + 172896201)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 1818 T^{6} + \cdots + 4924249929)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 54 T^{7} + \cdots + 77123524)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 27 T^{7} + \cdots + 310953447424)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{4} - 21 T^{3} + \cdots + 843408)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 20\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( (T^{8} - 30 T^{7} + \cdots + 160551674721)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{8} + 734 T^{6} + \cdots + 1882384)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 32\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 6699591959449)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 78136795608004)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 4327968462129)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 83599422865984)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 42\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 92\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 23\!\cdots\!61 \) Copy content Toggle raw display
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