Properties

Label 825.2.bi.c
Level $825$
Weight $2$
Character orbit 825.bi
Analytic conductor $6.588$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(101,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.bi (of order \(10\), degree \(4\), 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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{20}^{7} - \zeta_{20}) q^{2} + (\zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20} + 1) q^{3} + ( - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{4} + ( - \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{5} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2}) q^{6} + (2 \zeta_{20}^{4} + 2 \zeta_{20}^{2}) q^{7} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{8} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} - \zeta_{20}^{4} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{2} + 2 \zeta_{20}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{20}^{7} - \zeta_{20}) q^{2} + (\zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20} + 1) q^{3} + ( - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{4} + ( - \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{5} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2}) q^{6} + (2 \zeta_{20}^{4} + 2 \zeta_{20}^{2}) q^{7} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{8} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} - \zeta_{20}^{4} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{2} + 2 \zeta_{20}) q^{9} + (2 \zeta_{20}^{7} - \zeta_{20}^{5} + 4 \zeta_{20}^{3} - 2 \zeta_{20}) q^{11} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} + \zeta_{20}^{3} + 1) q^{12} + ( - 2 \zeta_{20}^{7} - 4 \zeta_{20}^{3} + 4 \zeta_{20}) q^{14} + (3 \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{16} + ( - 4 \zeta_{20}^{5} + 4 \zeta_{20}^{3} - 2 \zeta_{20}) q^{17} + ( - 4 \zeta_{20}^{7} - 2 \zeta_{20}^{6} + 3 \zeta_{20}^{5} - 2 \zeta_{20}^{3} + \zeta_{20} - 2) q^{18} + ( - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{4} - \zeta_{20}^{2} + 4) q^{19} + ( - 2 \zeta_{20}^{7} + 4 \zeta_{20}^{6} + 2 \zeta_{20}^{3} + 4 \zeta_{20}^{2} - 2) q^{21} + (\zeta_{20}^{6} - 2 \zeta_{20}^{4} + \zeta_{20}^{2} + 4) q^{22} + (5 \zeta_{20}^{7} + \zeta_{20}^{5} + 5 \zeta_{20}^{3}) q^{23} + ( - \zeta_{20}^{7} + \zeta_{20}^{6} - 2 \zeta_{20}^{5} - \zeta_{20}^{4} - 2 \zeta_{20}^{3} + \zeta_{20}^{2} - \zeta_{20} - 2) q^{24} + ( - \zeta_{20}^{7} + 3 \zeta_{20}^{6} + \zeta_{20}^{5} - \zeta_{20}^{2} + 4 \zeta_{20} + 1) q^{27} + ( - 2 \zeta_{20}^{6} + 4 \zeta_{20}^{4} - 2 \zeta_{20}^{2} + 2) q^{28} + ( - 4 \zeta_{20}^{7} - 3 \zeta_{20}^{5} - 2 \zeta_{20}^{3} + 7 \zeta_{20}) q^{29} - 5 \zeta_{20}^{4} q^{31} + (\zeta_{20}^{7} - 3 \zeta_{20}^{5} + 5 \zeta_{20}^{3} - 6 \zeta_{20}) q^{32} + (5 \zeta_{20}^{7} - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{5} + 3 \zeta_{20}^{4} + 3 \zeta_{20}^{3} + \zeta_{20}^{2} - 3 \zeta_{20} - 1) q^{33} + (6 \zeta_{20}^{6} - 6 \zeta_{20}^{4} + 2) q^{34} + ( - \zeta_{20}^{4} - 2 \zeta_{20}^{3} + 3 \zeta_{20}^{2} + 2 \zeta_{20} - 1) q^{36} + (5 \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} - 5) q^{37} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} - 7 \zeta_{20}) q^{38} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{41} + ( - 6 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 4 \zeta_{20}^{5} - 6 \zeta_{20}^{4} - 4 \zeta_{20}^{3} + 2 \zeta_{20}^{2} + 6 \zeta_{20}) q^{42} + ( - \zeta_{20}^{6} - 5 \zeta_{20}^{4} + 4 \zeta_{20}^{2} - 2) q^{43} + ( - 2 \zeta_{20}^{7} + 3 \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}) q^{44} + ( - 6 \zeta_{20}^{6} + 5 \zeta_{20}^{4} - 4 \zeta_{20}^{2} + 10) q^{46} + ( - 3 \zeta_{20}^{7} - 5 \zeta_{20}^{3} + 5 \zeta_{20}) q^{47} + (2 \zeta_{20}^{6} + 4 \zeta_{20}^{5} + \zeta_{20}^{4} - 5 \zeta_{20}^{3} - \zeta_{20}^{2} + 4 \zeta_{20} - 2) q^{48} + (5 \zeta_{20}^{6} + 4 \zeta_{20}^{2} - 4) q^{49} + ( - 4 \zeta_{20}^{6} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} + 2 \zeta_{20} - 4) q^{51} + (6 \zeta_{20}^{7} - 7 \zeta_{20}^{5} + 7 \zeta_{20}^{3} - 6 \zeta_{20}) q^{53} + ( - 3 \zeta_{20}^{7} - 4 \zeta_{20}^{6} - \zeta_{20}^{5} + 2 \zeta_{20}^{4} + 5 \zeta_{20}^{3} - 6 \zeta_{20}^{2} + \cdots + 3) q^{54} + \cdots + (3 \zeta_{20}^{7} + 4 \zeta_{20}^{6} - 3 \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 5 \zeta_{20}^{3} + 8 \zeta_{20}^{2} + \cdots - 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} + 4 q^{4} - 10 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} + 4 q^{4} - 10 q^{6} + 10 q^{9} + 8 q^{12} + 12 q^{16} - 20 q^{18} + 30 q^{19} + 40 q^{22} - 10 q^{24} + 12 q^{27} + 10 q^{31} - 16 q^{33} + 40 q^{34} - 34 q^{37} + 20 q^{42} + 50 q^{46} - 16 q^{48} - 14 q^{49} - 40 q^{51} + 40 q^{57} - 20 q^{58} - 10 q^{61} - 40 q^{63} + 26 q^{64} + 30 q^{66} - 36 q^{67} + 14 q^{69} - 20 q^{72} - 10 q^{73} + 10 q^{79} - 2 q^{81} + 30 q^{88} + 20 q^{93} - 50 q^{94} - 50 q^{96} + 34 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.951057 0.309017i
−0.951057 + 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
0.587785 0.809017i
−0.587785 + 0.809017i
0.951057 + 0.309017i
−0.951057 0.309017i
−0.363271 + 1.11803i 1.67229 0.451057i 0.500000 + 0.363271i 0 −0.103198 + 2.03353i 2.23607 3.07768i −2.48990 + 1.80902i 2.59310 1.50859i 0
101.2 0.363271 1.11803i 0.945746 1.45106i 0.500000 + 0.363271i 0 −1.27877 1.58450i 2.23607 3.07768i 2.48990 1.80902i −1.21113 2.74466i 0
326.1 −1.53884 1.11803i 1.72982 0.0877853i 0.500000 + 1.53884i 0 −2.76007 1.79892i −2.23607 + 0.726543i −0.224514 + 0.690983i 2.98459 0.303706i 0
326.2 1.53884 + 1.11803i −1.34786 1.08779i 0.500000 + 1.53884i 0 −0.857960 3.18088i −2.23607 + 0.726543i 0.224514 0.690983i 0.633446 + 2.93236i 0
701.1 −1.53884 + 1.11803i 1.72982 + 0.0877853i 0.500000 1.53884i 0 −2.76007 + 1.79892i −2.23607 0.726543i −0.224514 0.690983i 2.98459 + 0.303706i 0
701.2 1.53884 1.11803i −1.34786 + 1.08779i 0.500000 1.53884i 0 −0.857960 + 3.18088i −2.23607 0.726543i 0.224514 + 0.690983i 0.633446 2.93236i 0
776.1 −0.363271 1.11803i 1.67229 + 0.451057i 0.500000 0.363271i 0 −0.103198 2.03353i 2.23607 + 3.07768i −2.48990 1.80902i 2.59310 + 1.50859i 0
776.2 0.363271 + 1.11803i 0.945746 + 1.45106i 0.500000 0.363271i 0 −1.27877 + 1.58450i 2.23607 + 3.07768i 2.48990 + 1.80902i −1.21113 + 2.74466i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.bi.c yes 8
3.b odd 2 1 inner 825.2.bi.c yes 8
5.b even 2 1 825.2.bi.a 8
5.c odd 4 1 825.2.bs.b 8
5.c odd 4 1 825.2.bs.c 8
11.d odd 10 1 inner 825.2.bi.c yes 8
15.d odd 2 1 825.2.bi.a 8
15.e even 4 1 825.2.bs.b 8
15.e even 4 1 825.2.bs.c 8
33.f even 10 1 inner 825.2.bi.c yes 8
55.h odd 10 1 825.2.bi.a 8
55.l even 20 1 825.2.bs.b 8
55.l even 20 1 825.2.bs.c 8
165.r even 10 1 825.2.bi.a 8
165.u odd 20 1 825.2.bs.b 8
165.u odd 20 1 825.2.bs.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.bi.a 8 5.b even 2 1
825.2.bi.a 8 15.d odd 2 1
825.2.bi.a 8 55.h odd 10 1
825.2.bi.a 8 165.r even 10 1
825.2.bi.c yes 8 1.a even 1 1 trivial
825.2.bi.c yes 8 3.b odd 2 1 inner
825.2.bi.c yes 8 11.d odd 10 1 inner
825.2.bi.c yes 8 33.f even 10 1 inner
825.2.bs.b 8 5.c odd 4 1
825.2.bs.b 8 15.e even 4 1
825.2.bs.b 8 55.l even 20 1
825.2.bs.b 8 165.u odd 20 1
825.2.bs.c 8 5.c odd 4 1
825.2.bs.c 8 15.e even 4 1
825.2.bs.c 8 55.l even 20 1
825.2.bs.c 8 165.u odd 20 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{8} + 10T_{2}^{4} + 25T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{4} + 40T_{7} + 80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + 13 T^{6} - 10 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 40 T + 80)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{6} - 74 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 20 T^{6} + 1360 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$19$ \( (T^{4} - 15 T^{3} + 105 T^{2} - 385 T + 605)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 87 T^{2} + 361)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 25 T^{6} + 6610 T^{4} + \cdots + 23088025 \) Copy content Toggle raw display
$31$ \( (T^{4} - 5 T^{3} + 25 T^{2} - 125 T + 625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 17 T^{3} + 139 T^{2} + 533 T + 1681)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 5 T^{6} + 85 T^{4} + 75 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$43$ \( (T^{4} + 65 T^{2} + 605)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 19 T^{6} + 1086 T^{4} + \cdots + 923521 \) Copy content Toggle raw display
$53$ \( T^{8} - 11 T^{6} + 2526 T^{4} + \cdots + 2825761 \) Copy content Toggle raw display
$59$ \( T^{8} - 41 T^{6} + 681 T^{4} + \cdots + 130321 \) Copy content Toggle raw display
$61$ \( (T^{4} + 5 T^{3} - 15 T^{2} + 435 T + 4205)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 9 T - 41)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + 80 T^{6} + 18400 T^{4} + \cdots + 160000 \) Copy content Toggle raw display
$73$ \( (T^{4} + 5 T^{3} + 15 T^{2} + 15 T + 5)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 5 T^{3} + 125)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 160 T^{6} + 67360 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$89$ \( (T^{4} + 35 T^{2} + 25)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 17 T^{3} + 289 T^{2} + \cdots + 83521)^{2} \) Copy content Toggle raw display
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