Properties

Label 1775.2.a.f
Level $1775$
Weight $2$
Character orbit 1775.a
Self dual yes
Analytic conductor $14.173$
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] = [1775,2,Mod(1,1775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1775, 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("1775.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1775 = 5^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1775.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: \(14.1734463588\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 71)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_1 q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} - 3) q^{6} - 2 \beta_1 q^{7} + \beta_{2} q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_1 q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} - 3) q^{6} - 2 \beta_1 q^{7} + \beta_{2} q^{8} + \beta_{2} q^{9} + 2 \beta_{2} q^{11} + ( - \beta_{2} - 2 \beta_1) q^{12} + (2 \beta_{2} + 2) q^{13} + ( - 2 \beta_{2} - 6) q^{14} + ( - \beta_{2} + \beta_1 - 2) q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + (\beta_{2} + \beta_1) q^{18} + (\beta_{2} - 2 \beta_1 + 1) q^{19} + (2 \beta_{2} + 6) q^{21} + (2 \beta_{2} + 2 \beta_1) q^{22} + 4 q^{23} + ( - \beta_{2} - \beta_1) q^{24} + (2 \beta_{2} + 4 \beta_1) q^{26} + ( - \beta_{2} + 2 \beta_1) q^{27} + ( - 2 \beta_{2} - 4 \beta_1) q^{28} + ( - 2 \beta_{2} - \beta_1 + 4) q^{29} + 4 q^{31} + ( - 2 \beta_{2} - 3 \beta_1 + 3) q^{32} + ( - 2 \beta_{2} - 2 \beta_1) q^{33} + ( - 2 \beta_1 + 6) q^{34} + (\beta_1 + 3) q^{36} + (\beta_{2} + 5) q^{37} + ( - \beta_{2} + 2 \beta_1 - 6) q^{38} + ( - 2 \beta_{2} - 4 \beta_1) q^{39} + (4 \beta_1 - 2) q^{41} + (2 \beta_{2} + 8 \beta_1) q^{42} + (\beta_{2} - \beta_1 - 4) q^{43} + (2 \beta_1 + 6) q^{44} + 4 \beta_1 q^{46} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{47} + (3 \beta_1 - 3) q^{48} + (4 \beta_{2} + 5) q^{49} + (2 \beta_1 - 6) q^{51} + (2 \beta_{2} + 2 \beta_1 + 8) q^{52} + (4 \beta_{2} + 6) q^{53} + (\beta_{2} - \beta_1 + 6) q^{54} + ( - 2 \beta_{2} - 2 \beta_1) q^{56} + (\beta_{2} - 2 \beta_1 + 6) q^{57} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{58} + (2 \beta_{2} + 2 \beta_1 - 2) q^{59} + (4 \beta_1 + 4) q^{61} + 4 \beta_1 q^{62} + ( - 2 \beta_{2} - 2 \beta_1) q^{63} + ( - 3 \beta_{2} - \beta_1 - 5) q^{64} + ( - 4 \beta_{2} - 2 \beta_1 - 6) q^{66} + (2 \beta_{2} + 4) q^{67} + (2 \beta_{2} + 2 \beta_1 - 6) q^{68} - 4 \beta_1 q^{69} + q^{71} + ( - \beta_{2} + \beta_1 + 3) q^{72} + ( - \beta_{2} + 3 \beta_1 - 10) q^{73} + (\beta_{2} + 6 \beta_1) q^{74} + ( - \beta_{2} - 3 \beta_1 + 4) q^{76} + ( - 4 \beta_{2} - 4 \beta_1) q^{77} + ( - 6 \beta_{2} - 2 \beta_1 - 12) q^{78} + ( - \beta_{2} - 3 \beta_1) q^{79} + ( - 4 \beta_{2} + \beta_1 - 6) q^{81} + (4 \beta_{2} - 2 \beta_1 + 12) q^{82} + ( - \beta_{2} - 2 \beta_1 + 7) q^{83} + (6 \beta_{2} + 2 \beta_1 + 12) q^{84} + ( - 3 \beta_1 - 3) q^{86} + (3 \beta_{2} - 2 \beta_1 + 3) q^{87} + ( - 2 \beta_{2} + 2 \beta_1 + 6) q^{88} + ( - 2 \beta_{2} + \beta_1) q^{89} + ( - 4 \beta_{2} - 8 \beta_1) q^{91} + (4 \beta_{2} + 4) q^{92} - 4 \beta_1 q^{93} + ( - 4 \beta_1 + 6) q^{94} + (5 \beta_{2} - \beta_1 + 9) q^{96} + (2 \beta_1 - 8) q^{97} + (4 \beta_{2} + 9 \beta_1) q^{98} + ( - 2 \beta_{2} + 2 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - q^{3} + 3 q^{4} - 9 q^{6} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - q^{3} + 3 q^{4} - 9 q^{6} - 2 q^{7} - 2 q^{12} + 6 q^{13} - 18 q^{14} - 5 q^{16} + 2 q^{17} + q^{18} + q^{19} + 18 q^{21} + 2 q^{22} + 12 q^{23} - q^{24} + 4 q^{26} + 2 q^{27} - 4 q^{28} + 11 q^{29} + 12 q^{31} + 6 q^{32} - 2 q^{33} + 16 q^{34} + 10 q^{36} + 15 q^{37} - 16 q^{38} - 4 q^{39} - 2 q^{41} + 8 q^{42} - 13 q^{43} + 20 q^{44} + 4 q^{46} - 4 q^{47} - 6 q^{48} + 15 q^{49} - 16 q^{51} + 26 q^{52} + 18 q^{53} + 17 q^{54} - 2 q^{56} + 16 q^{57} - 7 q^{58} - 4 q^{59} + 16 q^{61} + 4 q^{62} - 2 q^{63} - 16 q^{64} - 20 q^{66} + 12 q^{67} - 16 q^{68} - 4 q^{69} + 3 q^{71} + 10 q^{72} - 27 q^{73} + 6 q^{74} + 9 q^{76} - 4 q^{77} - 38 q^{78} - 3 q^{79} - 17 q^{81} + 34 q^{82} + 19 q^{83} + 38 q^{84} - 12 q^{86} + 7 q^{87} + 20 q^{88} + q^{89} - 8 q^{91} + 12 q^{92} - 4 q^{93} + 14 q^{94} + 26 q^{96} - 22 q^{97} + 9 q^{98} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 3 \) : 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
−1.91223
0.713538
2.19869
−1.91223 1.91223 1.65662 0 −3.65662 3.82446 0.656620 0.656620 0
1.2 0.713538 −0.713538 −1.49086 0 −0.509136 −1.42708 −2.49086 −2.49086 0
1.3 2.19869 −2.19869 2.83424 0 −4.83424 −4.39738 1.83424 1.83424 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(71\) \(-1\)

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 1775.2.a.f 3
5.b even 2 1 71.2.a.a 3
15.d odd 2 1 639.2.a.h 3
20.d odd 2 1 1136.2.a.h 3
35.c odd 2 1 3479.2.a.k 3
40.e odd 2 1 4544.2.a.u 3
40.f even 2 1 4544.2.a.r 3
55.d odd 2 1 8591.2.a.g 3
355.c odd 2 1 5041.2.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.2.a.a 3 5.b even 2 1
639.2.a.h 3 15.d odd 2 1
1136.2.a.h 3 20.d odd 2 1
1775.2.a.f 3 1.a even 1 1 trivial
3479.2.a.k 3 35.c odd 2 1
4544.2.a.r 3 40.f even 2 1
4544.2.a.u 3 40.e odd 2 1
5041.2.a.a 3 355.c odd 2 1
8591.2.a.g 3 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 4T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1775))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 4T + 3 \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 4T - 3 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 16 T - 24 \) Copy content Toggle raw display
$11$ \( T^{3} - 20T + 24 \) Copy content Toggle raw display
$13$ \( T^{3} - 6 T^{2} - 8 T + 56 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 32 T + 24 \) Copy content Toggle raw display
$19$ \( T^{3} - T^{2} - 20 T - 25 \) Copy content Toggle raw display
$23$ \( (T - 4)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 11 T^{2} + 14 T + 71 \) Copy content Toggle raw display
$31$ \( (T - 4)^{3} \) Copy content Toggle raw display
$37$ \( T^{3} - 15 T^{2} + 70 T - 97 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} - 68 T + 56 \) Copy content Toggle raw display
$43$ \( T^{3} + 13 T^{2} + 48 T + 45 \) Copy content Toggle raw display
$47$ \( T^{3} + 4 T^{2} - 28 T - 40 \) Copy content Toggle raw display
$53$ \( T^{3} - 18 T^{2} + 28 T + 456 \) Copy content Toggle raw display
$59$ \( T^{3} + 4 T^{2} - 36 T - 152 \) Copy content Toggle raw display
$61$ \( T^{3} - 16 T^{2} + 16 T + 320 \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} + 28 T + 40 \) Copy content Toggle raw display
$71$ \( (T - 1)^{3} \) Copy content Toggle raw display
$73$ \( T^{3} + 27 T^{2} + 202 T + 461 \) Copy content Toggle raw display
$79$ \( T^{3} + 3 T^{2} - 44 T + 15 \) Copy content Toggle raw display
$83$ \( T^{3} - 19 T^{2} + 96 T - 63 \) Copy content Toggle raw display
$89$ \( T^{3} - T^{2} - 22 T - 27 \) Copy content Toggle raw display
$97$ \( T^{3} + 22 T^{2} + 144 T + 280 \) Copy content Toggle raw display
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