Properties

Label 1850.2.a.bc
Level $1850$
Weight $2$
Character orbit 1850.a
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1850,2,Mod(1,1850)]
 
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([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

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: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + (\beta_1 + 1) q^{7} + q^{8} + (\beta_{2} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + (\beta_1 + 1) q^{7} + q^{8} + (\beta_{2} + \beta_1 + 2) q^{9} + ( - \beta_{2} - 2) q^{11} + \beta_1 q^{12} + ( - \beta_1 + 3) q^{13} + (\beta_1 + 1) q^{14} + q^{16} + ( - \beta_1 - 2) q^{17} + (\beta_{2} + \beta_1 + 2) q^{18} + ( - \beta_{2} + \beta_1 - 1) q^{19} + (\beta_{2} + 2 \beta_1 + 5) q^{21} + ( - \beta_{2} - 2) q^{22} + ( - 2 \beta_1 + 2) q^{23} + \beta_1 q^{24} + ( - \beta_1 + 3) q^{26} + (\beta_{2} + 2 \beta_1 + 4) q^{27} + (\beta_1 + 1) q^{28} + ( - 2 \beta_1 - 4) q^{29} + ( - \beta_{2} + 2 \beta_1 + 1) q^{31} + q^{32} + ( - 4 \beta_1 + 1) q^{33} + ( - \beta_1 - 2) q^{34} + (\beta_{2} + \beta_1 + 2) q^{36} + q^{37} + ( - \beta_{2} + \beta_1 - 1) q^{38} + ( - \beta_{2} + 2 \beta_1 - 5) q^{39} + (2 \beta_{2} + 7) q^{41} + (\beta_{2} + 2 \beta_1 + 5) q^{42} + ( - \beta_{2} - 3 \beta_1 + 6) q^{43} + ( - \beta_{2} - 2) q^{44} + ( - 2 \beta_1 + 2) q^{46} + \beta_1 q^{48} + (\beta_{2} + 3 \beta_1 - 1) q^{49} + ( - \beta_{2} - 3 \beta_1 - 5) q^{51} + ( - \beta_1 + 3) q^{52} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{53} + (\beta_{2} + 2 \beta_1 + 4) q^{54} + (\beta_1 + 1) q^{56} + (\beta_{2} - 2 \beta_1 + 6) q^{57} + ( - 2 \beta_1 - 4) q^{58} + ( - \beta_{2} - 3 \beta_1 - 2) q^{59} + ( - 3 \beta_1 + 5) q^{61} + ( - \beta_{2} + 2 \beta_1 + 1) q^{62} + (2 \beta_{2} + 6 \beta_1 + 6) q^{63} + q^{64} + ( - 4 \beta_1 + 1) q^{66} + ( - 2 \beta_{2} + \beta_1) q^{67} + ( - \beta_1 - 2) q^{68} + ( - 2 \beta_{2} - 10) q^{69} + (2 \beta_{2} + \beta_1 + 3) q^{71} + (\beta_{2} + \beta_1 + 2) q^{72} + (\beta_{2} - \beta_1 + 5) q^{73} + q^{74} + ( - \beta_{2} + \beta_1 - 1) q^{76} + ( - \beta_{2} - 4 \beta_1 - 1) q^{77} + ( - \beta_{2} + 2 \beta_1 - 5) q^{78} + (2 \beta_{2} - 2 \beta_1 + 2) q^{79} + ( - \beta_{2} + 5 \beta_1 + 3) q^{81} + (2 \beta_{2} + 7) q^{82} + ( - \beta_{2} - 2 \beta_1 - 6) q^{83} + (\beta_{2} + 2 \beta_1 + 5) q^{84} + ( - \beta_{2} - 3 \beta_1 + 6) q^{86} + ( - 2 \beta_{2} - 6 \beta_1 - 10) q^{87} + ( - \beta_{2} - 2) q^{88} + (3 \beta_{2} - 2 \beta_1 + 2) q^{89} + ( - \beta_{2} + \beta_1 - 2) q^{91} + ( - 2 \beta_1 + 2) q^{92} + (2 \beta_{2} + \beta_1 + 11) q^{93} + \beta_1 q^{96} + ( - 2 \beta_{2} + 4) q^{97} + (\beta_{2} + 3 \beta_1 - 1) q^{98} + ( - \beta_{2} - 3 \beta_1 - 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9} - 5 q^{11} + q^{12} + 8 q^{13} + 4 q^{14} + 3 q^{16} - 7 q^{17} + 6 q^{18} - q^{19} + 16 q^{21} - 5 q^{22} + 4 q^{23} + q^{24} + 8 q^{26} + 13 q^{27} + 4 q^{28} - 14 q^{29} + 6 q^{31} + 3 q^{32} - q^{33} - 7 q^{34} + 6 q^{36} + 3 q^{37} - q^{38} - 12 q^{39} + 19 q^{41} + 16 q^{42} + 16 q^{43} - 5 q^{44} + 4 q^{46} + q^{48} - q^{49} - 17 q^{51} + 8 q^{52} - 6 q^{53} + 13 q^{54} + 4 q^{56} + 15 q^{57} - 14 q^{58} - 8 q^{59} + 12 q^{61} + 6 q^{62} + 22 q^{63} + 3 q^{64} - q^{66} + 3 q^{67} - 7 q^{68} - 28 q^{69} + 8 q^{71} + 6 q^{72} + 13 q^{73} + 3 q^{74} - q^{76} - 6 q^{77} - 12 q^{78} + 2 q^{79} + 15 q^{81} + 19 q^{82} - 19 q^{83} + 16 q^{84} + 16 q^{86} - 34 q^{87} - 5 q^{88} + q^{89} - 4 q^{91} + 4 q^{92} + 32 q^{93} + q^{96} + 14 q^{97} - q^{98} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.27307
0.140435
3.13264
1.00000 −2.27307 1.00000 0 −2.27307 −1.27307 1.00000 2.16686 0
1.2 1.00000 0.140435 1.00000 0 0.140435 1.14044 1.00000 −2.98028 0
1.3 1.00000 3.13264 1.00000 0 3.13264 4.13264 1.00000 6.81342 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(37\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1850.2.a.bc yes 3
5.b even 2 1 1850.2.a.y 3
5.c odd 4 2 1850.2.b.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.y 3 5.b even 2 1
1850.2.a.bc yes 3 1.a even 1 1 trivial
1850.2.b.p 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\):

\( T_{3}^{3} - T_{3}^{2} - 7T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{3} - 4T_{7}^{2} - 2T_{7} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 7T + 1 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 4 T^{2} - 2 T + 6 \) Copy content Toggle raw display
$11$ \( T^{3} + 5 T^{2} - 9 T - 51 \) Copy content Toggle raw display
$13$ \( T^{3} - 8 T^{2} + 14 T + 2 \) Copy content Toggle raw display
$17$ \( T^{3} + 7 T^{2} + 9 T - 3 \) Copy content Toggle raw display
$19$ \( T^{3} + T^{2} - 25 T + 11 \) Copy content Toggle raw display
$23$ \( T^{3} - 4 T^{2} - 24 T + 48 \) Copy content Toggle raw display
$29$ \( T^{3} + 14 T^{2} + 36 T - 24 \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 36 T + 214 \) Copy content Toggle raw display
$37$ \( (T - 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 19 T^{2} + 51 T + 399 \) Copy content Toggle raw display
$43$ \( T^{3} - 16 T^{2} + 4 T + 564 \) Copy content Toggle raw display
$47$ \( T^{3} \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 84 T - 216 \) Copy content Toggle raw display
$59$ \( T^{3} + 8 T^{2} - 60 T + 84 \) Copy content Toggle raw display
$61$ \( T^{3} - 12 T^{2} - 18 T + 238 \) Copy content Toggle raw display
$67$ \( T^{3} - 3 T^{2} - 75 T - 17 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 54 T + 378 \) Copy content Toggle raw display
$73$ \( T^{3} - 13 T^{2} + 31 T + 9 \) Copy content Toggle raw display
$79$ \( T^{3} - 2 T^{2} - 100 T - 88 \) Copy content Toggle raw display
$83$ \( T^{3} + 19 T^{2} + 75 T + 63 \) Copy content Toggle raw display
$89$ \( T^{3} - T^{2} - 189 T + 147 \) Copy content Toggle raw display
$97$ \( T^{3} - 14 T^{2} - 4 T + 8 \) Copy content Toggle raw display
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