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Properties

Label 1850.2.c.i
Level $1850$
Weight $2$
Character orbit 1850.c
Analytic conductor $14.772$
Analytic rank $0$
Dimension $6$
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] = [1850,2,Mod(1849,1850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1850.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.c (of order \(2\), degree \(1\), not 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: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 370)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{5} q^{3} + q^{4} - \beta_{5} q^{6} + ( - \beta_{4} + \beta_{2}) q^{7} - q^{8} + ( - \beta_{3} + \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_{5} q^{3} + q^{4} - \beta_{5} q^{6} + ( - \beta_{4} + \beta_{2}) q^{7} - q^{8} + ( - \beta_{3} + \beta_1 - 2) q^{9} - \beta_{3} q^{11} + \beta_{5} q^{12} + (\beta_{3} - \beta_1 - 1) q^{13} + (\beta_{4} - \beta_{2}) q^{14} + q^{16} + ( - \beta_{3} + 2 \beta_1 - 2) q^{17} + (\beta_{3} - \beta_1 + 2) q^{18} + (\beta_{5} + \beta_{2}) q^{19} + ( - 2 \beta_{3} - 2) q^{21} + \beta_{3} q^{22} + (\beta_{3} - \beta_1 - 1) q^{23} - \beta_{5} q^{24} + ( - \beta_{3} + \beta_1 + 1) q^{26} + (4 \beta_{4} - 2 \beta_{2}) q^{27} + ( - \beta_{4} + \beta_{2}) q^{28} + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{29} + (2 \beta_{5} + \beta_{4}) q^{31} - q^{32} + (4 \beta_{4} - 4 \beta_{2}) q^{33} + (\beta_{3} - 2 \beta_1 + 2) q^{34} + ( - \beta_{3} + \beta_1 - 2) q^{36} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - 1) q^{37}+ \cdots + (5 \beta_{3} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 14 q^{9} - 2 q^{11} - 4 q^{13} + 6 q^{16} - 14 q^{17} + 14 q^{18} - 16 q^{21} + 2 q^{22} - 4 q^{23} + 4 q^{26} - 6 q^{32} + 14 q^{34} - 14 q^{36} - 8 q^{37} + 2 q^{41} + 16 q^{42} + 2 q^{43} - 2 q^{44} + 4 q^{46} + 8 q^{49} - 4 q^{52} - 40 q^{57} + 6 q^{64} - 14 q^{68} - 20 q^{71} + 14 q^{72} + 8 q^{74} + 6 q^{81} - 2 q^{82} - 16 q^{84} - 2 q^{86} + 48 q^{87} + 2 q^{88} - 4 q^{92} - 56 q^{93} + 46 q^{97} - 8 q^{98} + 58 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - \nu^{3} + 5\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{5} + 2\nu^{4} - 5\nu^{3} + 6\nu^{2} - 2\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{5} - 4\nu^{4} + 11\nu^{3} - 16\nu^{2} + 14\nu - 28 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} + 2\beta_{3} - \beta_{2} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} + \beta_{3} + 3\beta_{2} + 3\beta _1 + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} + 2\beta_{3} + 5\beta_{2} - 4\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{5} - 6\beta_{4} + 3\beta_{3} - 3\beta_{2} - 7\beta _1 + 7 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.40680 − 0.144584i
0.264658 − 1.38923i
−0.671462 − 1.24464i
−0.671462 + 1.24464i
0.264658 + 1.38923i
1.40680 + 0.144584i
−1.00000 − 3.10278i 1.00000 0 3.10278i − 3.81361i −1.00000 −6.62721 0
1849.2 −1.00000 − 2.24914i 1.00000 0 2.24914i 1.52932i −1.00000 −2.05863 0
1849.3 −1.00000 − 1.14637i 1.00000 0 1.14637i 0.342923i −1.00000 1.68585 0
1849.4 −1.00000 1.14637i 1.00000 0 − 1.14637i − 0.342923i −1.00000 1.68585 0
1849.5 −1.00000 2.24914i 1.00000 0 − 2.24914i − 1.52932i −1.00000 −2.05863 0
1849.6 −1.00000 3.10278i 1.00000 0 − 3.10278i 3.81361i −1.00000 −6.62721 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1850.2.c.i 6
5.b even 2 1 1850.2.c.j 6
5.c odd 4 1 370.2.d.c ✓ 6
5.c odd 4 1 1850.2.d.f 6
15.e even 4 1 3330.2.h.n 6
20.e even 4 1 2960.2.p.g 6
37.b even 2 1 1850.2.c.j 6
185.d even 2 1 inner 1850.2.c.i 6
185.h odd 4 1 370.2.d.c ✓ 6
185.h odd 4 1 1850.2.d.f 6
555.n even 4 1 3330.2.h.n 6
740.m even 4 1 2960.2.p.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.d.c ✓ 6 5.c odd 4 1
370.2.d.c ✓ 6 185.h odd 4 1
1850.2.c.i 6 1.a even 1 1 trivial
1850.2.c.i 6 185.d even 2 1 inner
1850.2.c.j 6 5.b even 2 1
1850.2.c.j 6 37.b even 2 1
1850.2.d.f 6 5.c odd 4 1
1850.2.d.f 6 185.h odd 4 1
2960.2.p.g 6 20.e even 4 1
2960.2.p.g 6 740.m even 4 1
3330.2.h.n 6 15.e even 4 1
3330.2.h.n 6 555.n even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1850, [\chi])\):

\( T_{3}^{6} + 16T_{3}^{4} + 68T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{6} + 17T_{7}^{4} + 36T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 16T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{3} + 2T_{13}^{2} - 16T_{13} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 16 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 17 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 2 T^{2} - 16 T - 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 7 T^{2} + \cdots - 128)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 36 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T^{3} + 2 T^{2} - 16 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 49 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( T^{6} + 57 T^{4} + \cdots + 6724 \) Copy content Toggle raw display
$37$ \( T^{6} + 8 T^{5} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( (T^{3} - T^{2} - 56 T + 172)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - T^{2} - 16 T - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 32 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{6} + 209 T^{4} + \cdots + 123904 \) Copy content Toggle raw display
$59$ \( T^{6} + 96 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( T^{6} + 121 T^{4} + \cdots + 15376 \) Copy content Toggle raw display
$67$ \( T^{6} + 228 T^{4} + \cdots + 15376 \) Copy content Toggle raw display
$71$ \( (T^{3} + 10 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 236 T^{4} + \cdots + 118336 \) Copy content Toggle raw display
$79$ \( T^{6} + 80 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$83$ \( T^{6} + 224 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$89$ \( T^{6} + 256 T^{4} + \cdots + 262144 \) Copy content Toggle raw display
$97$ \( (T^{3} - 23 T^{2} + \cdots - 172)^{2} \) Copy content Toggle raw display
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