Properties

Label 775.2.b.d
Level $775$
Weight $2$
Character orbit 775.b
Analytic conductor $6.188$
Analytic rank $0$
Dimension $4$
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] = [775,2,Mod(249,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.b (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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_1 q^{3} + (\beta_{2} + 1) q^{4} + (2 \beta_{2} - 2) q^{6} + (3 \beta_{3} + 2 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + (4 \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_1 q^{3} + (\beta_{2} + 1) q^{4} + (2 \beta_{2} - 2) q^{6} + (3 \beta_{3} + 2 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + (4 \beta_{2} - 1) q^{9} + 2 q^{11} + 2 \beta_{3} q^{12} + 2 \beta_1 q^{13} + ( - \beta_{2} - 2) q^{14} + 3 \beta_{2} q^{16} + ( - 4 \beta_{3} - 2 \beta_1) q^{17} + (4 \beta_{3} - 5 \beta_1) q^{18} + ( - 2 \beta_{2} - 1) q^{19} + ( - 2 \beta_{2} - 4) q^{21} + 2 \beta_1 q^{22} + ( - 4 \beta_{3} - 6 \beta_1) q^{23} + (2 \beta_{2} - 4) q^{24} + (2 \beta_{2} - 2) q^{26} + (8 \beta_{3} - 4 \beta_1) q^{27} + (5 \beta_{3} + 3 \beta_1) q^{28} + ( - 2 \beta_{2} - 6) q^{29} + q^{31} + (5 \beta_{3} + \beta_1) q^{32} + 4 \beta_1 q^{33} + (2 \beta_{2} + 2) q^{34} + ( - \beta_{2} + 3) q^{36} + 2 \beta_{3} q^{37} + ( - 2 \beta_{3} + \beta_1) q^{38} + (4 \beta_{2} - 4) q^{39} + 7 q^{41} + ( - 2 \beta_{3} - 2 \beta_1) q^{42} + ( - 2 \beta_{3} - 2 \beta_1) q^{43} + (2 \beta_{2} + 2) q^{44} + ( - 2 \beta_{2} + 6) q^{46} + (4 \beta_{3} + 4 \beta_1) q^{47} + (6 \beta_{3} - 6 \beta_1) q^{48} + ( - 8 \beta_{2} - 6) q^{49} + (4 \beta_{2} + 4) q^{51} + 2 \beta_{3} q^{52} + ( - 4 \beta_{3} + 4 \beta_1) q^{53} + ( - 12 \beta_{2} + 4) q^{54} + ( - 4 \beta_{2} - 7) q^{56} + ( - 4 \beta_{3} + 2 \beta_1) q^{57} + ( - 2 \beta_{3} - 4 \beta_1) q^{58} + (2 \beta_{2} + 1) q^{59} + ( - 10 \beta_{2} - 8) q^{61} + \beta_1 q^{62} + (5 \beta_{3} + 2 \beta_1) q^{63} + (2 \beta_{2} - 1) q^{64} + (4 \beta_{2} - 4) q^{66} - 8 \beta_{3} q^{67} + ( - 6 \beta_{3} - 4 \beta_1) q^{68} + ( - 4 \beta_{2} + 12) q^{69} + (10 \beta_{2} + 7) q^{71} + (7 \beta_{3} - 6 \beta_1) q^{72} + (2 \beta_{3} - 4 \beta_1) q^{73} - 2 \beta_{2} q^{74} + ( - \beta_{2} - 3) q^{76} + (6 \beta_{3} + 4 \beta_1) q^{77} + (4 \beta_{3} - 8 \beta_1) q^{78} + ( - 6 \beta_{2} + 2) q^{79} + ( - 12 \beta_{2} + 5) q^{81} + 7 \beta_1 q^{82} + ( - 2 \beta_{3} + 8 \beta_1) q^{83} + ( - 4 \beta_{2} - 6) q^{84} + 2 q^{86} + ( - 4 \beta_{3} - 8 \beta_1) q^{87} + (2 \beta_{3} + 4 \beta_1) q^{88} + (6 \beta_{2} - 2) q^{89} + ( - 2 \beta_{2} - 4) q^{91} + ( - 10 \beta_{3} - 4 \beta_1) q^{92} + 2 \beta_1 q^{93} - 4 q^{94} + ( - 8 \beta_{2} - 2) q^{96} + (3 \beta_{3} - 8 \beta_1) q^{97} + ( - 8 \beta_{3} + 2 \beta_1) q^{98} + (8 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 12 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 12 q^{6} - 12 q^{9} + 8 q^{11} - 6 q^{14} - 6 q^{16} - 12 q^{21} - 20 q^{24} - 12 q^{26} - 20 q^{29} + 4 q^{31} + 4 q^{34} + 14 q^{36} - 24 q^{39} + 28 q^{41} + 4 q^{44} + 28 q^{46} - 8 q^{49} + 8 q^{51} + 40 q^{54} - 20 q^{56} - 12 q^{61} - 8 q^{64} - 24 q^{66} + 56 q^{69} + 8 q^{71} + 4 q^{74} - 10 q^{76} + 20 q^{79} + 44 q^{81} - 16 q^{84} + 8 q^{86} - 20 q^{89} - 12 q^{91} - 16 q^{94} + 8 q^{96} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(652\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 3.23607i −0.618034 0 −5.23607 0.236068i 2.23607i −7.47214 0
249.2 0.618034i 1.23607i 1.61803 0 −0.763932 4.23607i 2.23607i 1.47214 0
249.3 0.618034i 1.23607i 1.61803 0 −0.763932 4.23607i 2.23607i 1.47214 0
249.4 1.61803i 3.23607i −0.618034 0 −5.23607 0.236068i 2.23607i −7.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.b.d 4
5.b even 2 1 inner 775.2.b.d 4
5.c odd 4 1 31.2.a.a 2
5.c odd 4 1 775.2.a.d 2
15.e even 4 1 279.2.a.a 2
15.e even 4 1 6975.2.a.y 2
20.e even 4 1 496.2.a.i 2
35.f even 4 1 1519.2.a.a 2
40.i odd 4 1 1984.2.a.r 2
40.k even 4 1 1984.2.a.n 2
55.e even 4 1 3751.2.a.b 2
60.l odd 4 1 4464.2.a.bf 2
65.h odd 4 1 5239.2.a.f 2
85.g odd 4 1 8959.2.a.b 2
155.f even 4 1 961.2.a.f 2
155.o odd 12 2 961.2.c.e 4
155.p even 12 2 961.2.c.c 4
155.r even 20 2 961.2.d.a 4
155.r even 20 2 961.2.d.g 4
155.s odd 20 2 961.2.d.c 4
155.s odd 20 2 961.2.d.d 4
155.w odd 60 4 961.2.g.a 8
155.w odd 60 4 961.2.g.h 8
155.x even 60 4 961.2.g.d 8
155.x even 60 4 961.2.g.e 8
465.m odd 4 1 8649.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.a.a 2 5.c odd 4 1
279.2.a.a 2 15.e even 4 1
496.2.a.i 2 20.e even 4 1
775.2.a.d 2 5.c odd 4 1
775.2.b.d 4 1.a even 1 1 trivial
775.2.b.d 4 5.b even 2 1 inner
961.2.a.f 2 155.f even 4 1
961.2.c.c 4 155.p even 12 2
961.2.c.e 4 155.o odd 12 2
961.2.d.a 4 155.r even 20 2
961.2.d.c 4 155.s odd 20 2
961.2.d.d 4 155.s odd 20 2
961.2.d.g 4 155.r even 20 2
961.2.g.a 8 155.w odd 60 4
961.2.g.d 8 155.x even 60 4
961.2.g.e 8 155.x even 60 4
961.2.g.h 8 155.w odd 60 4
1519.2.a.a 2 35.f even 4 1
1984.2.a.n 2 40.k even 4 1
1984.2.a.r 2 40.i odd 4 1
3751.2.a.b 2 55.e even 4 1
4464.2.a.bf 2 60.l odd 4 1
5239.2.a.f 2 65.h odd 4 1
6975.2.a.y 2 15.e even 4 1
8649.2.a.c 2 465.m odd 4 1
8959.2.a.b 2 85.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 18T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 2)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 92T^{2} + 1936 \) Copy content Toggle raw display
$29$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T - 7)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 112T^{2} + 256 \) Copy content Toggle raw display
$59$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T - 116)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 121)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} - 10 T - 20)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 232T^{2} + 1936 \) Copy content Toggle raw display
$89$ \( (T^{2} + 10 T - 20)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 258T^{2} + 961 \) Copy content Toggle raw display
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