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Properties

Label 1050.3.l.h
Level $1050$
Weight $3$
Character orbit 1050.l
Analytic conductor $28.610$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,3,Mod(43,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.l (of order \(4\), 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: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 32 x^{14} + 152 x^{13} + 1954 x^{12} - 12664 x^{11} + 50336 x^{10} + 231896 x^{9} + \cdots + 2560000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + \beta_{4} q^{3} + 2 \beta_{2} q^{4} + (\beta_{4} + \beta_{3}) q^{6} + \beta_{7} q^{7} + (2 \beta_{2} - 2) q^{8} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + \beta_{4} q^{3} + 2 \beta_{2} q^{4} + (\beta_{4} + \beta_{3}) q^{6} + \beta_{7} q^{7} + (2 \beta_{2} - 2) q^{8} - 3 \beta_{2} q^{9} + ( - \beta_{12} + 2 \beta_{8} + \cdots + 1) q^{11}+ \cdots + (3 \beta_{9} + 6 \beta_{8} + \cdots - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} - 32 q^{8} + 8 q^{11} + 32 q^{13} - 64 q^{16} - 56 q^{17} + 48 q^{18} + 8 q^{22} - 24 q^{23} + 64 q^{26} - 112 q^{31} - 64 q^{32} - 24 q^{33} + 96 q^{36} + 152 q^{37} - 48 q^{46} - 80 q^{47} - 72 q^{51} + 64 q^{52} - 48 q^{53} - 24 q^{57} - 96 q^{58} + 96 q^{61} - 112 q^{62} - 48 q^{66} + 80 q^{67} + 112 q^{68} + 536 q^{71} + 96 q^{72} + 168 q^{77} + 48 q^{78} - 144 q^{81} + 256 q^{83} + 144 q^{87} - 16 q^{88} - 48 q^{92} - 192 q^{93} - 688 q^{97} - 112 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 32 x^{14} + 152 x^{13} + 1954 x^{12} - 12664 x^{11} + 50336 x^{10} + 231896 x^{9} + \cdots + 2560000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 124150896048631 \nu^{15} + 759860784649304 \nu^{14} - 962633185803656 \nu^{13} + \cdots - 12\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 50\!\cdots\!87 \nu^{15} + \cdots + 23\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 73\!\cdots\!87 \nu^{15} + \cdots + 22\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 38\!\cdots\!54 \nu^{15} + \cdots + 74\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 84\!\cdots\!11 \nu^{15} + \cdots - 79\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27\!\cdots\!26 \nu^{15} + \cdots + 43\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!47 \nu^{15} + \cdots - 33\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 60\!\cdots\!14 \nu^{15} + \cdots - 11\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\!\cdots\!25 \nu^{15} + \cdots + 28\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 96\!\cdots\!06 \nu^{15} + \cdots + 13\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!86 \nu^{15} + \cdots + 79\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 34\!\cdots\!88 \nu^{15} + \cdots + 23\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2097730881 \nu^{15} + 14409577452 \nu^{14} - 47557870248 \nu^{13} + \cdots + 10\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 55\!\cdots\!55 \nu^{15} + \cdots - 25\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 30\!\cdots\!39 \nu^{15} + \cdots - 11\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{13} - 3 \beta_{12} - \beta_{11} - 2 \beta_{10} + 3 \beta_{9} - \beta_{7} + 4 \beta_{6} + \cdots + 7 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{15} + 6 \beta_{14} - \beta_{13} + 2 \beta_{11} - 3 \beta_{10} + 4 \beta_{9} + \cdots - 226 \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 82 \beta_{15} + 164 \beta_{14} - 163 \beta_{13} + 83 \beta_{12} - 41 \beta_{11} + 83 \beta_{9} + \cdots - 507 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 43 \beta_{15} + 102 \beta_{14} - 43 \beta_{13} + 24 \beta_{12} + 43 \beta_{10} - 360 \beta_{8} + \cdots - 1854 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5949 \beta_{13} + 2547 \beta_{12} + 1749 \beta_{11} + 3798 \beta_{10} - 2547 \beta_{9} - 9200 \beta_{8} + \cdots - 31743 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 14507 \beta_{15} - 36214 \beta_{14} + 64769 \beta_{13} + 5462 \beta_{11} + 14507 \beta_{10} + \cdots + 461594 \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 205098 \beta_{15} - 485396 \beta_{14} + 601307 \beta_{13} - 84587 \beta_{12} + 83849 \beta_{11} + \cdots + 2033723 ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 192619 \beta_{15} - 488582 \beta_{14} + 192619 \beta_{13} - 39232 \beta_{12} - 192619 \beta_{10} + \cdots + 5233270 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 25728301 \beta_{13} - 3142203 \beta_{12} - 4516101 \beta_{11} - 12183502 \beta_{10} + \cdots + 132013007 ) / 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 63463483 \beta_{15} + 161696566 \beta_{14} - 333391961 \beta_{13} - 37338078 \beta_{11} + \cdots - 1596905586 \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 761028322 \beta_{15} + 1906866244 \beta_{14} - 2426251123 \beta_{13} + 135164643 \beta_{12} + \cdots - 8610221147 ) / 10 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 832972299 \beta_{15} + 2124161894 \beta_{14} - 832972299 \beta_{13} + 160514152 \beta_{12} + \cdots - 20219529966 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 107997866749 \beta_{13} + 6752694947 \beta_{12} + 16340771349 \beta_{11} + 48694529798 \beta_{10} + \cdots - 562349185343 ) / 10 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 272692226027 \beta_{15} - 695324153654 \beta_{14} + 1478556161809 \beta_{13} + \cdots + 6509919300234 \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3150306189658 \beta_{15} - 7997847164916 \beta_{14} + 10171665397547 \beta_{13} - 379132207227 \beta_{12} + \cdots + 36735471689883 ) / 10 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−1.37832 − 1.37832i
3.60306 + 3.60306i
5.71348 + 5.71348i
−3.48873 − 3.48873i
−0.394902 − 0.394902i
0.170157 + 0.170157i
−3.99135 − 3.99135i
3.76660 + 3.76660i
−1.37832 + 1.37832i
3.60306 − 3.60306i
5.71348 − 5.71348i
−3.48873 + 3.48873i
−0.394902 + 0.394902i
0.170157 − 0.170157i
−3.99135 + 3.99135i
3.76660 − 3.76660i
1.00000 − 1.00000i −1.22474 − 1.22474i − 2.00000i 0 −2.44949 −1.87083 + 1.87083i −2.00000 − 2.00000i 3.00000i 0
43.2 1.00000 − 1.00000i −1.22474 − 1.22474i − 2.00000i 0 −2.44949 −1.87083 + 1.87083i −2.00000 − 2.00000i 3.00000i 0
43.3 1.00000 − 1.00000i −1.22474 − 1.22474i − 2.00000i 0 −2.44949 1.87083 − 1.87083i −2.00000 − 2.00000i 3.00000i 0
43.4 1.00000 − 1.00000i −1.22474 − 1.22474i − 2.00000i 0 −2.44949 1.87083 − 1.87083i −2.00000 − 2.00000i 3.00000i 0
43.5 1.00000 − 1.00000i 1.22474 + 1.22474i − 2.00000i 0 2.44949 −1.87083 + 1.87083i −2.00000 − 2.00000i 3.00000i 0
43.6 1.00000 − 1.00000i 1.22474 + 1.22474i − 2.00000i 0 2.44949 −1.87083 + 1.87083i −2.00000 − 2.00000i 3.00000i 0
43.7 1.00000 − 1.00000i 1.22474 + 1.22474i − 2.00000i 0 2.44949 1.87083 − 1.87083i −2.00000 − 2.00000i 3.00000i 0
43.8 1.00000 − 1.00000i 1.22474 + 1.22474i − 2.00000i 0 2.44949 1.87083 − 1.87083i −2.00000 − 2.00000i 3.00000i 0
757.1 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 −1.87083 − 1.87083i −2.00000 + 2.00000i − 3.00000i 0
757.2 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 −1.87083 − 1.87083i −2.00000 + 2.00000i − 3.00000i 0
757.3 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i − 3.00000i 0
757.4 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i − 3.00000i 0
757.5 1.00000 + 1.00000i 1.22474 − 1.22474i 2.00000i 0 2.44949 −1.87083 − 1.87083i −2.00000 + 2.00000i − 3.00000i 0
757.6 1.00000 + 1.00000i 1.22474 − 1.22474i 2.00000i 0 2.44949 −1.87083 − 1.87083i −2.00000 + 2.00000i − 3.00000i 0
757.7 1.00000 + 1.00000i 1.22474 − 1.22474i 2.00000i 0 2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i − 3.00000i 0
757.8 1.00000 + 1.00000i 1.22474 − 1.22474i 2.00000i 0 2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i − 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.l.h 16
5.b even 2 1 210.3.l.b ✓ 16
5.c odd 4 1 210.3.l.b ✓ 16
5.c odd 4 1 inner 1050.3.l.h 16
15.d odd 2 1 630.3.o.f 16
15.e even 4 1 630.3.o.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.l.b ✓ 16 5.b even 2 1
210.3.l.b ✓ 16 5.c odd 4 1
630.3.o.f 16 15.d odd 2 1
630.3.o.f 16 15.e even 4 1
1050.3.l.h 16 1.a even 1 1 trivial
1050.3.l.h 16 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{8} - 4 T_{11}^{7} - 540 T_{11}^{6} + 368 T_{11}^{5} + 94224 T_{11}^{4} + 257024 T_{11}^{3} + \cdots - 50723840 \) Copy content Toggle raw display
\( T_{13}^{16} - 32 T_{13}^{15} + 512 T_{13}^{14} - 7216 T_{13}^{13} + 417256 T_{13}^{12} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} + 49)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 4 T^{7} + \cdots - 50723840)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + 56 T^{7} + \cdots - 58191200000)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} - 5196 T^{6} + \cdots + 262151483392)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} - 48 T^{7} + \cdots - 3514265600)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 39499497333760)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
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