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Properties

Label 1400.3.f.d
Level $1400$
Weight $3$
Character orbit 1400.f
Analytic conductor $38.147$
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] = [1400,3,Mod(601,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1400.601");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1400.f (of order \(2\), degree \(1\), 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: \(38.1472370104\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 104 x^{14} + 4298 x^{12} + 90444 x^{10} + 1029673 x^{8} + 6233716 x^{6} + 18628948 x^{4} + \cdots + 7268416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{8} + 1) q^{7} + (\beta_{2} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{8} + 1) q^{7} + (\beta_{2} - 4) q^{9} + ( - \beta_{5} - 1) q^{11} + ( - \beta_{13} - \beta_{10} + \beta_1) q^{13} + ( - \beta_{14} - \beta_{13} + \cdots - \beta_{7}) q^{17}+ \cdots + (3 \beta_{12} - 3 \beta_{11} + \cdots + 34) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{7} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{7} - 64 q^{9} - 8 q^{11} + 60 q^{21} - 136 q^{29} + 144 q^{37} - 192 q^{39} - 272 q^{43} + 68 q^{49} - 144 q^{51} - 240 q^{53} + 48 q^{57} - 104 q^{63} - 48 q^{67} + 344 q^{71} - 360 q^{77} + 8 q^{79} + 120 q^{81} - 280 q^{91} + 208 q^{93} + 568 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 104 x^{14} + 4298 x^{12} + 90444 x^{10} + 1029673 x^{8} + 6233716 x^{6} + 18628948 x^{4} + \cdots + 7268416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 200373 \nu^{14} - 19881695 \nu^{12} - 772576599 \nu^{10} - 15009048937 \nu^{8} + \cdots - 1478678618176 ) / 58072774144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 769 \nu^{14} - 71943 \nu^{12} - 2611555 \nu^{10} - 46875049 \nu^{8} - 434127184 \nu^{6} + \cdots - 1308552128 ) / 89068672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1030503 \nu^{14} - 95418485 \nu^{12} - 3440674269 \nu^{10} - 62313788195 \nu^{8} + \cdots - 4236073562816 ) / 116145548288 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 456677 \nu^{14} + 44454583 \nu^{12} + 1704185559 \nu^{10} + 32873011841 \nu^{8} + \cdots + 619371738432 ) / 29036387072 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27451454 \nu^{15} + 332353781 \nu^{14} - 3264320618 \nu^{13} + 32136616223 \nu^{12} + \cdots + 317517760550976 ) / 39141049773056 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27451454 \nu^{15} + 332353781 \nu^{14} + 3264320618 \nu^{13} + 32136616223 \nu^{12} + \cdots + 317517760550976 ) / 39141049773056 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 48859409 \nu^{15} - 4801618323 \nu^{13} - 169044739947 \nu^{11} + \cdots + 29\!\cdots\!04 \nu ) / 39141049773056 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 60671 \nu^{15} - 6050631 \nu^{13} - 236519167 \nu^{11} - 4607233889 \nu^{9} + \cdots - 270779831584 \nu ) / 15008071232 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 181406308 \nu^{15} - 381045563 \nu^{14} + 18066497068 \nu^{13} - 43254645905 \nu^{12} + \cdots - 13\!\cdots\!52 ) / 39141049773056 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 181406308 \nu^{15} + 381045563 \nu^{14} + 18066497068 \nu^{13} + 43254645905 \nu^{12} + \cdots + 13\!\cdots\!52 ) / 39141049773056 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 295255241 \nu^{15} + 30930767003 \nu^{13} + 1273180825075 \nu^{11} + \cdots + 895548684391744 \nu ) / 39141049773056 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 161735775 \nu^{15} - 16922112957 \nu^{13} - 695556558309 \nu^{11} + \cdots - 14\!\cdots\!12 \nu ) / 19570524886528 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 223101357 \nu^{15} + 22500443079 \nu^{13} + 889569087247 \nu^{11} + \cdots + 781973914425152 \nu ) / 19570524886528 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} - \beta_{12} - \beta_{11} + 3\beta_{10} - \beta_{8} + \beta_{7} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{11} - 4 \beta_{8} - 4 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 8 \beta_{4} + \cdots + 275 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 74 \beta_{15} + 2 \beta_{14} - 12 \beta_{13} + 27 \beta_{12} + 27 \beta_{11} - 151 \beta_{10} + \cdots + 515 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 51 \beta_{12} + 51 \beta_{11} + 184 \beta_{8} + 184 \beta_{7} + 98 \beta_{6} + 122 \beta_{5} + \cdots - 6767 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2378 \beta_{15} - 148 \beta_{14} + 726 \beta_{13} - 721 \beta_{12} - 721 \beta_{11} + 5727 \beta_{10} + \cdots - 13591 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2007 \beta_{12} - 2007 \beta_{11} - 6748 \beta_{8} - 6748 \beta_{7} - 3830 \beta_{6} - 5470 \beta_{5} + \cdots + 180447 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 74570 \beta_{15} + 6454 \beta_{14} - 31868 \beta_{13} + 20671 \beta_{12} + 20671 \beta_{11} + \cdots + 377831 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 71671 \beta_{12} + 71671 \beta_{11} + 228752 \beta_{8} + 228752 \beta_{7} + 138306 \beta_{6} + \cdots - 5085143 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2335370 \beta_{15} - 234216 \beta_{14} + 1226106 \beta_{13} - 625901 \beta_{12} - 625901 \beta_{11} + \cdots - 10934159 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2444687 \beta_{12} - 2444687 \beta_{11} - 7500196 \beta_{8} - 7500196 \beta_{7} - 4787582 \beta_{6} + \cdots + 149139311 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 73430026 \beta_{15} + 7863466 \beta_{14} - 43978008 \beta_{13} + 19543547 \beta_{12} + \cdots + 326184567 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 81382183 \beta_{12} + 81382183 \beta_{11} + 242367768 \beta_{8} + 242367768 \beta_{7} + \cdots - 4499878967 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2319326410 \beta_{15} - 255038556 \beta_{14} + 1513370534 \beta_{13} - 619439401 \beta_{12} + \cdots - 9948461903 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
− 5.65681i
− 4.62081i
− 4.50217i
− 3.94841i
− 2.84878i
− 1.93287i
− 1.57863i
− 0.667489i
0.667489i
1.57863i
1.93287i
2.84878i
3.94841i
4.50217i
4.62081i
5.65681i
0 − 5.65681i 0 0 0 3.22019 + 6.21533i 0 −22.9995 0
601.2 0 − 4.62081i 0 0 0 0.801563 − 6.95396i 0 −12.3519 0
601.3 0 − 4.50217i 0 0 0 5.97360 + 3.64913i 0 −11.2695 0
601.4 0 − 3.94841i 0 0 0 −6.97315 − 0.612556i 0 −6.58992 0
601.5 0 − 2.84878i 0 0 0 0.106789 + 6.99919i 0 0.884449 0
601.6 0 − 1.93287i 0 0 0 −6.91630 − 1.07927i 0 5.26400 0
601.7 0 − 1.57863i 0 0 0 5.46800 − 4.37047i 0 6.50793 0
601.8 0 − 0.667489i 0 0 0 6.31931 + 3.01104i 0 8.55446 0
601.9 0 0.667489i 0 0 0 6.31931 − 3.01104i 0 8.55446 0
601.10 0 1.57863i 0 0 0 5.46800 + 4.37047i 0 6.50793 0
601.11 0 1.93287i 0 0 0 −6.91630 + 1.07927i 0 5.26400 0
601.12 0 2.84878i 0 0 0 0.106789 − 6.99919i 0 0.884449 0
601.13 0 3.94841i 0 0 0 −6.97315 + 0.612556i 0 −6.58992 0
601.14 0 4.50217i 0 0 0 5.97360 − 3.64913i 0 −11.2695 0
601.15 0 4.62081i 0 0 0 0.801563 + 6.95396i 0 −12.3519 0
601.16 0 5.65681i 0 0 0 3.22019 − 6.21533i 0 −22.9995 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.3.f.d 16
5.b even 2 1 280.3.f.a ✓ 16
5.c odd 4 2 1400.3.p.c 32
7.b odd 2 1 inner 1400.3.f.d 16
15.d odd 2 1 2520.3.s.a 16
20.d odd 2 1 560.3.f.e 16
35.c odd 2 1 280.3.f.a ✓ 16
35.f even 4 2 1400.3.p.c 32
105.g even 2 1 2520.3.s.a 16
140.c even 2 1 560.3.f.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.3.f.a ✓ 16 5.b even 2 1
280.3.f.a ✓ 16 35.c odd 2 1
560.3.f.e 16 20.d odd 2 1
560.3.f.e 16 140.c even 2 1
1400.3.f.d 16 1.a even 1 1 trivial
1400.3.f.d 16 7.b odd 2 1 inner
1400.3.p.c 32 5.c odd 4 2
1400.3.p.c 32 35.f even 4 2
2520.3.s.a 16 15.d odd 2 1
2520.3.s.a 16 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{16} + 104 T_{3}^{14} + 4298 T_{3}^{12} + 90444 T_{3}^{10} + 1029673 T_{3}^{8} + 6233716 T_{3}^{6} + \cdots + 7268416 \) Copy content Toggle raw display
\( T_{23}^{8} - 2034 T_{23}^{6} + 27144 T_{23}^{5} + 731768 T_{23}^{4} - 17982208 T_{23}^{3} + \cdots + 229505024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 104 T^{14} + \cdots + 7268416 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{8} + 4 T^{7} + \cdots - 27768256)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 87\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{8} - 2034 T^{6} + \cdots + 229505024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 68 T^{7} + \cdots - 39728327024)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{8} - 72 T^{7} + \cdots + 24362911744)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + 136 T^{7} + \cdots + 717172439936)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 4410764216576)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 358067025241216)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 30\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 414805590588704)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 54\!\cdots\!96 \) Copy content Toggle raw display
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