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Properties

Label 1050.3.p.h
Level $1050$
Weight $3$
Character orbit 1050.p
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(451,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.451");
 
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.p (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: \(28.6104277578\)
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: \( x^{16} - 4 x^{15} + 88 x^{13} + 156 x^{12} + 564 x^{11} + 10360 x^{10} + 11692 x^{9} + 24765 x^{8} + \cdots + 6620929 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 210)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{8} + \beta_{3}) q^{2} + ( - \beta_{7} + 1) q^{3} + ( - 2 \beta_{7} - 2) q^{4} + ( - 2 \beta_{8} + \beta_{3}) q^{6} + (\beta_{11} - \beta_{10} + \cdots + \beta_1) q^{7}+ \cdots - 3 \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} + \beta_{3}) q^{2} + ( - \beta_{7} + 1) q^{3} + ( - 2 \beta_{7} - 2) q^{4} + ( - 2 \beta_{8} + \beta_{3}) q^{6} + (\beta_{11} - \beta_{10} + \cdots + \beta_1) q^{7}+ \cdots + ( - 3 \beta_{14} + 3 \beta_{12} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 24 q^{3} - 16 q^{4} - 12 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 24 q^{3} - 16 q^{4} - 12 q^{7} + 24 q^{9} + 24 q^{11} - 48 q^{12} + 8 q^{14} - 32 q^{16} - 12 q^{17} - 24 q^{19} - 12 q^{21} + 36 q^{23} + 48 q^{26} + 24 q^{28} - 88 q^{29} - 24 q^{31} + 72 q^{33} - 96 q^{36} - 48 q^{37} - 24 q^{38} + 36 q^{39} + 48 q^{44} + 16 q^{46} + 228 q^{47} + 136 q^{49} - 12 q^{51} + 72 q^{52} - 240 q^{53} - 32 q^{56} - 48 q^{57} - 372 q^{59} - 336 q^{61} + 128 q^{64} - 348 q^{67} + 24 q^{68} - 72 q^{71} - 360 q^{73} + 208 q^{74} + 564 q^{77} + 96 q^{78} - 64 q^{79} - 72 q^{81} + 168 q^{82} + 48 q^{84} + 152 q^{86} - 132 q^{87} + 24 q^{89} + 488 q^{91} - 144 q^{92} - 24 q^{93} - 264 q^{94} - 192 q^{98} + 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 88 x^{13} + 156 x^{12} + 564 x^{11} + 10360 x^{10} + 11692 x^{9} + 24765 x^{8} + \cdots + 6620929 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 37\!\cdots\!00 \nu^{15} + \cdots + 22\!\cdots\!50 ) / 92\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!94 \nu^{15} + \cdots + 91\!\cdots\!98 ) / 92\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 69\!\cdots\!52 \nu^{15} + \cdots + 18\!\cdots\!60 ) / 37\!\cdots\!07 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 49\!\cdots\!56 \nu^{15} + \cdots + 13\!\cdots\!36 ) / 21\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 72\!\cdots\!39 \nu^{15} + \cdots + 83\!\cdots\!33 ) / 21\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 91\!\cdots\!86 \nu^{15} + \cdots + 76\!\cdots\!88 ) / 21\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 59\!\cdots\!84 \nu^{15} + \cdots + 10\!\cdots\!75 ) / 12\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 63\!\cdots\!51 \nu^{15} + \cdots - 24\!\cdots\!96 ) / 61\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!01 \nu^{15} + \cdots - 21\!\cdots\!61 ) / 21\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 35\!\cdots\!20 \nu^{15} + \cdots - 19\!\cdots\!79 ) / 21\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39\!\cdots\!29 \nu^{15} + \cdots + 13\!\cdots\!48 ) / 21\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 50\!\cdots\!28 \nu^{15} + \cdots + 24\!\cdots\!69 ) / 21\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 53\!\cdots\!91 \nu^{15} + \cdots + 13\!\cdots\!06 ) / 21\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 74\!\cdots\!12 \nu^{15} + \cdots + 31\!\cdots\!47 ) / 21\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 82\!\cdots\!90 \nu^{15} + \cdots + 30\!\cdots\!67 ) / 21\!\cdots\!55 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 3 \beta_{15} + \beta_{14} + 5 \beta_{13} - 2 \beta_{12} - 3 \beta_{11} - \beta_{9} + 5 \beta_{7} + \cdots + 3 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 15 \beta_{15} + 19 \beta_{14} - 3 \beta_{13} + 4 \beta_{12} + 6 \beta_{11} + 21 \beta_{10} - 5 \beta_{9} + \cdots + 8 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 132 \beta_{15} + 184 \beta_{14} - 109 \beta_{13} - 46 \beta_{12} + 8 \beta_{11} + 189 \beta_{10} + \cdots + 48 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 366 \beta_{15} + 262 \beta_{14} - 76 \beta_{13} + 64 \beta_{12} + 243 \beta_{11} + 273 \beta_{10} + \cdots - 628 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 849 \beta_{15} - 1508 \beta_{14} - 2143 \beta_{13} + 2211 \beta_{12} + 1850 \beta_{11} - 1841 \beta_{10} + \cdots - 8843 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11329 \beta_{15} - 13301 \beta_{14} + 499 \beta_{13} + 7590 \beta_{12} + 3664 \beta_{11} - 12488 \beta_{10} + \cdots - 52048 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 221393 \beta_{15} - 225490 \beta_{14} + 25562 \beta_{13} + 83116 \beta_{12} - 14780 \beta_{11} + \cdots - 454493 ) / 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 677158 \beta_{15} - 671218 \beta_{14} + 216248 \beta_{13} + 66966 \beta_{12} - 198654 \beta_{11} + \cdots - 556023 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5476206 \beta_{15} - 5282877 \beta_{14} + 3239680 \beta_{13} - 874479 \beta_{12} - 3214259 \beta_{11} + \cdots + 5892732 ) / 14 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3948901 \beta_{15} - 2230687 \beta_{14} + 7734574 \beta_{13} - 8011716 \beta_{12} - 10003730 \beta_{11} + \cdots + 48850958 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 130331675 \beta_{15} + 154566654 \beta_{14} + 46421689 \beta_{13} - 123111570 \beta_{12} + \cdots + 739172355 ) / 14 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 778391415 \beta_{15} + 844742828 \beta_{14} - 77449188 \beta_{13} - 336542981 \beta_{12} + \cdots + 2051445967 ) / 7 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 11107787491 \beta_{15} + 11418870583 \beta_{14} - 2942477197 \beta_{13} - 2314183860 \beta_{12} + \cdots + 13650766853 ) / 14 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 27320523364 \beta_{15} + 26630550441 \beta_{14} - 12476337765 \beta_{13} + 839795968 \beta_{12} + \cdots - 4613422421 ) / 7 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 145737362644 \beta_{15} + 120814254178 \beta_{14} - 141702820571 \beta_{13} + 98711164618 \beta_{12} + \cdots - 600646770984 ) / 14 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(-\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
−3.59405 − 1.80085i
1.30130 + 3.45110i
5.03303 − 3.32967i
−1.74029 − 0.0526321i
2.35212 − 0.923363i
−1.73912 + 2.06010i
−0.252343 + 0.632697i
0.639346 − 3.50149i
−3.59405 + 1.80085i
1.30130 − 3.45110i
5.03303 + 3.32967i
−1.74029 + 0.0526321i
2.35212 + 0.923363i
−1.73912 − 2.06010i
−0.252343 − 0.632697i
0.639346 + 3.50149i
−0.707107 − 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 − 2.44949i −6.99767 + 0.180689i 2.82843 1.50000 + 2.59808i 0
451.2 −0.707107 − 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 − 2.44949i −3.78961 + 5.88548i 2.82843 1.50000 + 2.59808i 0
451.3 −0.707107 − 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 − 2.44949i −1.95772 − 6.72066i 2.82843 1.50000 + 2.59808i 0
451.4 −0.707107 − 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 − 2.44949i 6.91657 − 1.07756i 2.82843 1.50000 + 2.59808i 0
451.5 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i −6.96630 + 0.686090i −2.82843 1.50000 + 2.59808i 0
451.6 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i −3.28288 − 6.18245i −2.82843 1.50000 + 2.59808i 0
451.7 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i 3.47127 + 6.07868i −2.82843 1.50000 + 2.59808i 0
451.8 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i 6.60634 − 2.31437i −2.82843 1.50000 + 2.59808i 0
901.1 −0.707107 + 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 2.44949i −6.99767 − 0.180689i 2.82843 1.50000 − 2.59808i 0
901.2 −0.707107 + 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 2.44949i −3.78961 − 5.88548i 2.82843 1.50000 − 2.59808i 0
901.3 −0.707107 + 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 2.44949i −1.95772 + 6.72066i 2.82843 1.50000 − 2.59808i 0
901.4 −0.707107 + 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 2.44949i 6.91657 + 1.07756i 2.82843 1.50000 − 2.59808i 0
901.5 0.707107 − 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 − 2.44949i −6.96630 − 0.686090i −2.82843 1.50000 − 2.59808i 0
901.6 0.707107 − 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 − 2.44949i −3.28288 + 6.18245i −2.82843 1.50000 − 2.59808i 0
901.7 0.707107 − 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 − 2.44949i 3.47127 − 6.07868i −2.82843 1.50000 − 2.59808i 0
901.8 0.707107 − 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 − 2.44949i 6.60634 + 2.31437i −2.82843 1.50000 − 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.p.h 16
5.b even 2 1 1050.3.p.g 16
5.c odd 4 2 210.3.p.a ✓ 32
7.d odd 6 1 inner 1050.3.p.h 16
15.e even 4 2 630.3.bc.b 32
35.i odd 6 1 1050.3.p.g 16
35.k even 12 2 210.3.p.a ✓ 32
35.k even 12 2 1470.3.h.a 32
35.l odd 12 2 1470.3.h.a 32
105.w odd 12 2 630.3.bc.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.p.a ✓ 32 5.c odd 4 2
210.3.p.a ✓ 32 35.k even 12 2
630.3.bc.b 32 15.e even 4 2
630.3.bc.b 32 105.w odd 12 2
1050.3.p.g 16 5.b even 2 1
1050.3.p.g 16 35.i odd 6 1
1050.3.p.h 16 1.a even 1 1 trivial
1050.3.p.h 16 7.d odd 6 1 inner
1470.3.h.a 32 35.k even 12 2
1470.3.h.a 32 35.l odd 12 2

Hecke kernels

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

\( T_{11}^{16} - 24 T_{11}^{15} + 916 T_{11}^{14} - 10152 T_{11}^{13} + 318160 T_{11}^{12} + \cdots + 5302648351504 \) Copy content Toggle raw display
\( T_{17}^{16} + 12 T_{17}^{15} - 684 T_{17}^{14} - 8784 T_{17}^{13} + 359540 T_{17}^{12} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 5302648351504 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 12\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 756122020414929 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{8} + 44 T^{7} + \cdots - 116294442236)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 17\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 38\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 71\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{8} - 6328 T^{6} + \cdots - 681757959159)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 68\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 654301540016068)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 42\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 86\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
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