Properties

Label 3075.2.a.t
Level $3075$
Weight $2$
Character orbit 3075.a
Self dual yes
Analytic conductor $24.554$
Analytic rank $1$
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] = [3075,2,Mod(1,3075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3075 = 3 \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3075.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: \(24.5539986215\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 123)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_1 q^{6} + (\beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{2} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_1 q^{6} + (\beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{2} - 1) q^{8} + q^{9} + ( - \beta_1 - 1) q^{11} + (\beta_{2} + 1) q^{12} + ( - \beta_{2} + \beta_1 - 3) q^{13} + ( - 2 \beta_{2} - 4) q^{14} + ( - \beta_{2} + 2 \beta_1 - 1) q^{16} + ( - 2 \beta_{2} + \beta_1 - 1) q^{17} - \beta_1 q^{18} + (\beta_{2} - \beta_1 + 1) q^{19} + (\beta_{2} + \beta_1 - 1) q^{21} + (\beta_{2} + \beta_1 + 3) q^{22} + ( - \beta_{2} + \beta_1 + 3) q^{23} + ( - \beta_{2} - 1) q^{24} + (4 \beta_1 - 2) q^{26} + q^{27} + (4 \beta_1 + 4) q^{28} + ( - 3 \beta_1 - 1) q^{29} + (\beta_{2} + 4 \beta_1 - 2) q^{31} + (\beta_{2} + 2 \beta_1 - 3) q^{32} + ( - \beta_1 - 1) q^{33} + (\beta_{2} + 3 \beta_1 - 1) q^{34} + (\beta_{2} + 1) q^{36} + (\beta_{2} - 2 \beta_1 - 6) q^{37} + ( - 2 \beta_1 + 2) q^{38} + ( - \beta_{2} + \beta_1 - 3) q^{39} + q^{41} + ( - 2 \beta_{2} - 4) q^{42} + ( - 5 \beta_{2} + 2 \beta_1 - 4) q^{43} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{44} + ( - 2 \beta_1 - 2) q^{46} + ( - 2 \beta_{2} - \beta_1 - 1) q^{47} + ( - \beta_{2} + 2 \beta_1 - 1) q^{48} + (2 \beta_1 + 3) q^{49} + ( - 2 \beta_{2} + \beta_1 - 1) q^{51} + ( - 2 \beta_{2} - 6) q^{52} + ( - 2 \beta_{2} + 4 \beta_1 - 6) q^{53} - \beta_1 q^{54} + ( - 4 \beta_1 - 4) q^{56} + (\beta_{2} - \beta_1 + 1) q^{57} + (3 \beta_{2} + \beta_1 + 9) q^{58} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{59} + (\beta_{2} - 2 \beta_1 - 2) q^{61} + ( - 5 \beta_{2} + \beta_1 - 13) q^{62} + (\beta_{2} + \beta_1 - 1) q^{63} + ( - \beta_{2} - 2 \beta_1 - 5) q^{64} + (\beta_{2} + \beta_1 + 3) q^{66} + (4 \beta_{2} - 6 \beta_1 - 2) q^{67} + ( - 2 \beta_1 - 8) q^{68} + ( - \beta_{2} + \beta_1 + 3) q^{69} + (\beta_1 - 11) q^{71} + ( - \beta_{2} - 1) q^{72} + (3 \beta_{2} + 2 \beta_1 - 2) q^{73} + (\beta_{2} + 5 \beta_1 + 5) q^{74} + 4 q^{76} + ( - 3 \beta_{2} - \beta_1 - 3) q^{77} + (4 \beta_1 - 2) q^{78} + ( - 2 \beta_{2} + 4 \beta_1 - 8) q^{79} + q^{81} - \beta_1 q^{82} + (3 \beta_{2} - \beta_1 + 5) q^{83} + (4 \beta_1 + 4) q^{84} + (3 \beta_{2} + 9 \beta_1 - 1) q^{86} + ( - 3 \beta_1 - 1) q^{87} + (2 \beta_{2} + 2 \beta_1 + 2) q^{88} + ( - 4 \beta_1 + 6) q^{89} + ( - 6 \beta_1 + 2) q^{91} + 4 \beta_{2} q^{92} + (\beta_{2} + 4 \beta_1 - 2) q^{93} + (3 \beta_{2} + 3 \beta_1 + 5) q^{94} + (\beta_{2} + 2 \beta_1 - 3) q^{96} + ( - \beta_{2} + 3 \beta_1 + 3) q^{97} + ( - 2 \beta_{2} - 3 \beta_1 - 6) q^{98} + ( - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{3} + 3 q^{4} - q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{3} + 3 q^{4} - q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9} - 4 q^{11} + 3 q^{12} - 8 q^{13} - 12 q^{14} - q^{16} - 2 q^{17} - q^{18} + 2 q^{19} - 2 q^{21} + 10 q^{22} + 10 q^{23} - 3 q^{24} - 2 q^{26} + 3 q^{27} + 16 q^{28} - 6 q^{29} - 2 q^{31} - 7 q^{32} - 4 q^{33} + 3 q^{36} - 20 q^{37} + 4 q^{38} - 8 q^{39} + 3 q^{41} - 12 q^{42} - 10 q^{43} - 8 q^{44} - 8 q^{46} - 4 q^{47} - q^{48} + 11 q^{49} - 2 q^{51} - 18 q^{52} - 14 q^{53} - q^{54} - 16 q^{56} + 2 q^{57} + 28 q^{58} - 8 q^{59} - 8 q^{61} - 38 q^{62} - 2 q^{63} - 17 q^{64} + 10 q^{66} - 12 q^{67} - 26 q^{68} + 10 q^{69} - 32 q^{71} - 3 q^{72} - 4 q^{73} + 20 q^{74} + 12 q^{76} - 10 q^{77} - 2 q^{78} - 20 q^{79} + 3 q^{81} - q^{82} + 14 q^{83} + 16 q^{84} + 6 q^{86} - 6 q^{87} + 8 q^{88} + 14 q^{89} - 2 q^{93} + 18 q^{94} - 7 q^{96} + 12 q^{97} - 21 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
2.34292
0.470683
−1.81361
−2.34292 1.00000 3.48929 0 −2.34292 3.83221 −3.48929 1.00000 0
1.2 −0.470683 1.00000 −1.77846 0 −0.470683 −3.30777 1.77846 1.00000 0
1.3 1.81361 1.00000 1.28917 0 1.81361 −2.52444 −1.28917 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(41\) \(-1\)

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 3075.2.a.t 3
3.b odd 2 1 9225.2.a.bx 3
5.b even 2 1 123.2.a.d 3
15.d odd 2 1 369.2.a.e 3
20.d odd 2 1 1968.2.a.w 3
35.c odd 2 1 6027.2.a.s 3
40.e odd 2 1 7872.2.a.bs 3
40.f even 2 1 7872.2.a.bx 3
60.h even 2 1 5904.2.a.bd 3
205.c even 2 1 5043.2.a.n 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
123.2.a.d 3 5.b even 2 1
369.2.a.e 3 15.d odd 2 1
1968.2.a.w 3 20.d odd 2 1
3075.2.a.t 3 1.a even 1 1 trivial
5043.2.a.n 3 205.c even 2 1
5904.2.a.bd 3 60.h even 2 1
6027.2.a.s 3 35.c odd 2 1
7872.2.a.bs 3 40.e odd 2 1
7872.2.a.bx 3 40.f even 2 1
9225.2.a.bx 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3075))\):

\( T_{2}^{3} + T_{2}^{2} - 4T_{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 14T_{7} - 32 \) Copy content Toggle raw display
\( T_{13}^{3} + 8T_{13}^{2} + 14T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 4T - 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 14 T - 32 \) Copy content Toggle raw display
$11$ \( T^{3} + 4T^{2} + T - 4 \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + 14 T - 4 \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} - 23 T - 62 \) Copy content Toggle raw display
$19$ \( T^{3} - 2 T^{2} - 6 T + 8 \) Copy content Toggle raw display
$23$ \( T^{3} - 10 T^{2} + 26 T - 16 \) Copy content Toggle raw display
$29$ \( T^{3} + 6 T^{2} - 27 T - 86 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 91 T - 256 \) Copy content Toggle raw display
$37$ \( T^{3} + 20 T^{2} + 117 T + 166 \) Copy content Toggle raw display
$41$ \( (T - 1)^{3} \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} - 119 T - 1156 \) Copy content Toggle raw display
$47$ \( T^{3} + 4 T^{2} - 35 T + 8 \) Copy content Toggle raw display
$53$ \( T^{3} + 14T^{2} - 32 \) Copy content Toggle raw display
$59$ \( T^{3} + 8 T^{2} - 40 T + 32 \) Copy content Toggle raw display
$61$ \( T^{3} + 8 T^{2} + 5 T - 46 \) Copy content Toggle raw display
$67$ \( T^{3} + 12 T^{2} - 124 T - 976 \) Copy content Toggle raw display
$71$ \( T^{3} + 32 T^{2} + 337 T + 1168 \) Copy content Toggle raw display
$73$ \( T^{3} + 4 T^{2} - 99 T - 454 \) Copy content Toggle raw display
$79$ \( T^{3} + 20 T^{2} + 68 T + 32 \) Copy content Toggle raw display
$83$ \( T^{3} - 14 T^{2} + 10 T + 296 \) Copy content Toggle raw display
$89$ \( T^{3} - 14 T^{2} - 4 T + 184 \) Copy content Toggle raw display
$97$ \( T^{3} - 12 T^{2} + 14 T + 148 \) Copy content Toggle raw display
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