Properties

Label 3775.2.a.p
Level $3775$
Weight $2$
Character orbit 3775.a
Self dual yes
Analytic conductor $30.144$
Analytic rank $0$
Dimension $6$
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] = [3775,2,Mod(1,3775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3775, 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("3775.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3775 = 5^{2} \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3775.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: \(30.1435267630\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.4838537.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 7x^{4} + 3x^{3} + 13x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 151)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} - \beta_{3} + \beta_{2} + 1) q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{6} + (\beta_{5} + \beta_{4} + 2 \beta_{2} - 1) q^{7} + ( - \beta_{3} - 1) q^{8} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{4} - \beta_{3} + \beta_{2} + 1) q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{6} + (\beta_{5} + \beta_{4} + 2 \beta_{2} - 1) q^{7} + ( - \beta_{3} - 1) q^{8} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots + 4) q^{9}+ \cdots + (4 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 5 q^{3} + 3 q^{4} - 2 q^{6} - 3 q^{7} - 9 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} + 5 q^{3} + 3 q^{4} - 2 q^{6} - 3 q^{7} - 9 q^{8} + 15 q^{9} + 8 q^{11} + 11 q^{12} + q^{13} + 6 q^{14} - 3 q^{16} - 9 q^{17} + 16 q^{18} - 6 q^{19} + 13 q^{21} + 12 q^{22} + 4 q^{23} + q^{24} - 7 q^{26} + 2 q^{27} + 24 q^{28} - 2 q^{29} - 8 q^{31} + 11 q^{32} - 3 q^{33} - 9 q^{34} - 25 q^{36} + 12 q^{37} + 3 q^{38} - 22 q^{39} + 41 q^{41} + 24 q^{42} - q^{43} - 17 q^{46} - 28 q^{47} - 9 q^{48} + 33 q^{49} - 31 q^{51} - 15 q^{52} - 14 q^{53} + 27 q^{54} - 9 q^{56} + 28 q^{57} - q^{58} + 12 q^{59} + 5 q^{61} + 9 q^{62} - 4 q^{63} - 27 q^{64} + 8 q^{66} + 15 q^{67} - 11 q^{68} + 33 q^{69} - 2 q^{71} - q^{72} + 7 q^{73} + 53 q^{74} - 3 q^{76} + 15 q^{77} - 16 q^{78} - 9 q^{79} + 66 q^{81} - 10 q^{82} + 11 q^{83} + 49 q^{84} - 26 q^{86} + 26 q^{87} - 24 q^{88} + 36 q^{89} - 35 q^{91} + 38 q^{92} + q^{93} + 42 q^{94} - 34 q^{96} - 11 q^{97} - 15 q^{98} + 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 7x^{4} + 3x^{3} + 13x^{2} + 3x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - \nu^{4} - 6\nu^{3} + 3\nu^{2} + 9\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + 2\nu^{4} + 6\nu^{3} - 8\nu^{2} - 10\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{4} + 5\beta_{2} + 6\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 2\beta_{4} + 6\beta_{3} + 2\beta_{2} + 18\beta _1 + 7 \) 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
2.34838
2.05089
0.183668
−0.503289
−1.18639
−1.89325
−2.34838 1.59233 3.51488 0 −3.73938 3.82411 −3.55751 −0.464497 0
1.2 −2.05089 0.642853 2.20615 0 −1.31842 −4.07943 −0.422789 −2.58674 0
1.3 −0.183668 3.29466 −1.96627 0 −0.605125 −3.65107 0.728478 7.85477 0
1.4 0.503289 −3.23034 −1.74670 0 −1.62580 −2.18587 −1.88567 7.43510 0
1.5 1.18639 −0.248968 −0.592471 0 −0.295374 −1.68227 −3.07569 −2.93801 0
1.6 1.89325 2.94947 1.58441 0 5.58410 4.77452 −0.786819 5.69938 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(151\) \( -1 \)

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 3775.2.a.p 6
5.b even 2 1 151.2.a.c 6
15.d odd 2 1 1359.2.a.i 6
20.d odd 2 1 2416.2.a.o 6
35.c odd 2 1 7399.2.a.e 6
40.e odd 2 1 9664.2.a.bc 6
40.f even 2 1 9664.2.a.bh 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
151.2.a.c 6 5.b even 2 1
1359.2.a.i 6 15.d odd 2 1
2416.2.a.o 6 20.d odd 2 1
3775.2.a.p 6 1.a even 1 1 trivial
7399.2.a.e 6 35.c odd 2 1
9664.2.a.bc 6 40.e odd 2 1
9664.2.a.bh 6 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3775))\):

\( T_{2}^{6} + T_{2}^{5} - 7T_{2}^{4} - 3T_{2}^{3} + 13T_{2}^{2} - 3T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{6} - 5T_{3}^{5} - 4T_{3}^{4} + 51T_{3}^{3} - 68T_{3}^{2} + 12T_{3} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} - 7 T^{4} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 5 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} + \cdots + 1000 \) Copy content Toggle raw display
$11$ \( T^{6} - 8 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$13$ \( T^{6} - T^{5} + \cdots - 328 \) Copy content Toggle raw display
$17$ \( T^{6} + 9 T^{5} + \cdots + 253 \) Copy content Toggle raw display
$19$ \( T^{6} + 6 T^{5} + \cdots + 115 \) Copy content Toggle raw display
$23$ \( T^{6} - 4 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$29$ \( T^{6} + 2 T^{5} + \cdots - 5 \) Copy content Toggle raw display
$31$ \( T^{6} + 8 T^{5} + \cdots + 271 \) Copy content Toggle raw display
$37$ \( T^{6} - 12 T^{5} + \cdots + 56789 \) Copy content Toggle raw display
$41$ \( T^{6} - 41 T^{5} + \cdots + 73432 \) Copy content Toggle raw display
$43$ \( T^{6} + T^{5} + \cdots - 11425 \) Copy content Toggle raw display
$47$ \( T^{6} + 28 T^{5} + \cdots - 65843 \) Copy content Toggle raw display
$53$ \( T^{6} + 14 T^{5} + \cdots + 24664 \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + \cdots - 5 \) Copy content Toggle raw display
$61$ \( T^{6} - 5 T^{5} + \cdots - 16984 \) Copy content Toggle raw display
$67$ \( T^{6} - 15 T^{5} + \cdots + 14696 \) Copy content Toggle raw display
$71$ \( T^{6} + 2 T^{5} + \cdots - 4024 \) Copy content Toggle raw display
$73$ \( T^{6} - 7 T^{5} + \cdots - 135872 \) Copy content Toggle raw display
$79$ \( T^{6} + 9 T^{5} + \cdots - 195080 \) Copy content Toggle raw display
$83$ \( T^{6} - 11 T^{5} + \cdots - 260696 \) Copy content Toggle raw display
$89$ \( T^{6} - 36 T^{5} + \cdots + 64000 \) Copy content Toggle raw display
$97$ \( T^{6} + 11 T^{5} + \cdots - 193 \) Copy content Toggle raw display
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