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Properties

Label 1050.3.p.f
Level $1050$
Weight $3$
Character orbit 1050.p
Analytic conductor $28.610$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 56 x^{10} + 300 x^{9} + 1007 x^{8} + 12456 x^{7} + 209990 x^{6} - 250384 x^{5} + \cdots + 6882692292 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4}) q^{2} + ( - \beta_{3} + 2) q^{3} - 2 \beta_{3} q^{4} + ( - \beta_{5} + 2 \beta_{4}) q^{6} + ( - \beta_{11} + \beta_{7} + \cdots + \beta_{3}) q^{7}+ \cdots + ( - 3 \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4}) q^{2} + ( - \beta_{3} + 2) q^{3} - 2 \beta_{3} q^{4} + ( - \beta_{5} + 2 \beta_{4}) q^{6} + ( - \beta_{11} + \beta_{7} + \cdots + \beta_{3}) q^{7}+ \cdots + ( - 3 \beta_{11} - 3 \beta_{9} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 18 q^{3} - 12 q^{4} + 8 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 18 q^{3} - 12 q^{4} + 8 q^{7} + 18 q^{9} - 4 q^{11} - 36 q^{12} + 8 q^{14} - 24 q^{16} - 24 q^{17} + 12 q^{19} + 18 q^{21} + 24 q^{22} - 60 q^{23} - 24 q^{26} + 4 q^{28} - 24 q^{29} - 198 q^{31} - 12 q^{33} - 72 q^{36} + 70 q^{37} - 60 q^{38} - 36 q^{39} + 36 q^{42} - 84 q^{43} - 8 q^{44} + 32 q^{46} - 60 q^{47} + 28 q^{49} - 24 q^{51} - 72 q^{52} + 44 q^{53} + 40 q^{56} + 24 q^{57} + 8 q^{58} - 48 q^{59} + 186 q^{61} + 30 q^{63} + 96 q^{64} + 36 q^{66} + 152 q^{67} + 48 q^{68} - 136 q^{71} + 18 q^{73} - 64 q^{74} + 132 q^{77} - 48 q^{78} - 70 q^{79} - 54 q^{81} - 84 q^{82} - 12 q^{84} - 208 q^{86} - 36 q^{87} - 24 q^{88} + 168 q^{89} + 292 q^{91} + 240 q^{92} - 198 q^{93} - 204 q^{94} + 48 q^{98} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 56 x^{10} + 300 x^{9} + 1007 x^{8} + 12456 x^{7} + 209990 x^{6} - 250384 x^{5} + \cdots + 6882692292 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 73\!\cdots\!47 \nu^{11} + \cdots + 15\!\cdots\!04 ) / 21\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 97053122943645 \nu^{11} + 103711575797434 \nu^{10} + \cdots - 12\!\cdots\!32 ) / 24\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24\!\cdots\!93 \nu^{11} + \cdots - 24\!\cdots\!48 ) / 43\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 173144824834609 \nu^{11} + 327589951576233 \nu^{10} + \cdots - 44\!\cdots\!64 ) / 16\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 284500915977146 \nu^{11} + \cdots + 66\!\cdots\!40 ) / 24\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34\!\cdots\!22 \nu^{11} + \cdots + 10\!\cdots\!40 ) / 14\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!40 \nu^{11} + \cdots - 46\!\cdots\!48 ) / 43\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 55\!\cdots\!61 \nu^{11} + \cdots - 12\!\cdots\!68 ) / 14\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 57\!\cdots\!01 \nu^{11} + \cdots - 10\!\cdots\!40 ) / 14\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 42\!\cdots\!00 \nu^{11} + \cdots - 56\!\cdots\!48 ) / 91\!\cdots\!94 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} - \beta_{8} + 6\beta_{7} - \beta_{6} - 6\beta_{5} + 17\beta_{4} + 4\beta_{3} + \beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 15 \beta_{10} + 20 \beta_{8} - 8 \beta_{7} + 6 \beta_{6} + 11 \beta_{5} + 9 \beta_{4} - 266 \beta_{3} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 14 \beta_{11} - 26 \beta_{10} - 105 \beta_{9} + 45 \beta_{8} - 103 \beta_{7} - 238 \beta_{6} + \cdots - 377 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 1778 \beta_{11} + 1152 \beta_{10} - 224 \beta_{9} - 732 \beta_{8} - 451 \beta_{7} - 29 \beta_{6} + \cdots - 220 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1344 \beta_{11} - 1574 \beta_{10} + 6643 \beta_{9} - 7355 \beta_{8} - 6859 \beta_{7} + 3450 \beta_{6} + \cdots - 47505 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 49756 \beta_{11} - 12586 \beta_{10} - 12782 \beta_{9} - 21518 \beta_{8} + 16583 \beta_{7} + \cdots + 81928 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 129948 \beta_{11} - 46600 \beta_{10} - 128499 \beta_{9} + 451639 \beta_{8} - 164431 \beta_{7} + \cdots + 2699555 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1889356 \beta_{11} + 2356528 \beta_{10} - 760130 \beta_{9} - 53642 \beta_{8} + 5400723 \beta_{7} + \cdots - 9816566 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2920680 \beta_{11} + 11033388 \beta_{10} + 21939785 \beta_{9} - 2908245 \beta_{8} + 37427371 \beta_{7} + \cdots + 59634639 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 45693032 \beta_{11} - 168135152 \beta_{10} + 124062764 \beta_{9} + 66719712 \beta_{8} + \cdots + 585995148 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(1 - \beta_{3}\) \(1\)

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
3.11049 − 6.59331i
−5.81071 + 4.13641i
4.40732 + 3.68164i
−0.0663422 − 7.58000i
−4.82374 + 1.03135i
5.18297 + 5.32390i
3.11049 + 6.59331i
−5.81071 − 4.13641i
4.40732 − 3.68164i
−0.0663422 + 7.58000i
−4.82374 − 1.03135i
5.18297 − 5.32390i
−0.707107 − 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 − 2.44949i −2.94762 + 6.34914i 2.82843 1.50000 + 2.59808i 0
451.2 −0.707107 − 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 − 2.44949i 1.88399 − 6.74171i 2.82843 1.50000 + 2.59808i 0
451.3 −0.707107 − 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 − 2.44949i 6.59916 + 2.33475i 2.82843 1.50000 + 2.59808i 0
451.4 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i −6.80475 + 1.64177i −2.82843 1.50000 + 2.59808i 0
451.5 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i −1.72580 − 6.78392i −2.82843 1.50000 + 2.59808i 0
451.6 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i 6.99501 − 0.264136i −2.82843 1.50000 + 2.59808i 0
901.1 −0.707107 + 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 2.44949i −2.94762 − 6.34914i 2.82843 1.50000 − 2.59808i 0
901.2 −0.707107 + 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 2.44949i 1.88399 + 6.74171i 2.82843 1.50000 − 2.59808i 0
901.3 −0.707107 + 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 2.44949i 6.59916 − 2.33475i 2.82843 1.50000 − 2.59808i 0
901.4 0.707107 − 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 − 2.44949i −6.80475 − 1.64177i −2.82843 1.50000 − 2.59808i 0
901.5 0.707107 − 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 − 2.44949i −1.72580 + 6.78392i −2.82843 1.50000 − 2.59808i 0
901.6 0.707107 − 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 0 − 2.44949i 6.99501 + 0.264136i −2.82843 1.50000 − 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.6
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.f yes 12
5.b even 2 1 1050.3.p.e ✓ 12
5.c odd 4 2 1050.3.q.d 24
7.d odd 6 1 inner 1050.3.p.f yes 12
35.i odd 6 1 1050.3.p.e ✓ 12
35.k even 12 2 1050.3.q.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.p.e ✓ 12 5.b even 2 1
1050.3.p.e ✓ 12 35.i odd 6 1
1050.3.p.f yes 12 1.a even 1 1 trivial
1050.3.p.f yes 12 7.d odd 6 1 inner
1050.3.q.d 24 5.c odd 4 2
1050.3.q.d 24 35.k even 12 2

Hecke kernels

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

\( T_{11}^{12} + 4 T_{11}^{11} + 562 T_{11}^{10} - 2336 T_{11}^{9} + 227242 T_{11}^{8} + \cdots + 463630257216 \) Copy content Toggle raw display
\( T_{17}^{12} + 24 T_{17}^{11} - 530 T_{17}^{10} - 17328 T_{17}^{9} + 311466 T_{17}^{8} + \cdots + 4222235916864 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 463630257216 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 28563704064 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 4222235916864 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 78337292544 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 876231988678656 \) Copy content Toggle raw display
$29$ \( (T^{6} + 12 T^{5} + \cdots - 5447232)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 85\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 10\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{6} + 42 T^{5} + \cdots - 4767344)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 47\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 32\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{6} + 68 T^{5} + \cdots - 19178846208)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 133904078886249 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 35\!\cdots\!49 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 50\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 66\!\cdots\!01 \) Copy content Toggle raw display
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