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Properties

Label 825.2.n.c
Level $825$
Weight $2$
Character orbit 825.n
Analytic conductor $6.588$
Analytic rank $1$
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] = [825,2,Mod(301,825)]
 
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([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.n (of order \(5\), 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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + \cdots - 1) q^{2}+ \cdots + (\zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + \cdots - 1) q^{2}+ \cdots + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{3} - 9 q^{4} + q^{6} - q^{7} - 13 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - q^{3} - 9 q^{4} + q^{6} - q^{7} - 13 q^{8} - q^{9} - 11 q^{11} + 6 q^{12} - 7 q^{13} - 4 q^{14} + q^{16} - 12 q^{17} - 4 q^{18} - 10 q^{19} + 4 q^{21} - 4 q^{22} + 8 q^{23} + 7 q^{24} + 2 q^{26} - q^{27} + 6 q^{28} + 6 q^{29} - 12 q^{31} + 30 q^{32} + 9 q^{33} + 12 q^{34} - 9 q^{36} - 9 q^{37} + 10 q^{38} - 7 q^{39} - 3 q^{41} + q^{42} + 36 q^{44} - 3 q^{46} - 17 q^{47} - 14 q^{48} + 6 q^{49} + 3 q^{51} - 3 q^{52} - 4 q^{53} + 6 q^{54} + 12 q^{56} + 5 q^{57} + 24 q^{58} - 6 q^{59} - 21 q^{61} + 27 q^{62} - q^{63} + 13 q^{64} - 4 q^{66} + 6 q^{67} + 27 q^{68} + 3 q^{69} + 15 q^{71} + 7 q^{72} - 14 q^{73} - 26 q^{74} + 60 q^{76} - q^{77} + 2 q^{78} - 11 q^{79} - q^{81} + 3 q^{82} - 13 q^{83} - 9 q^{84} + 15 q^{86} - 24 q^{87} + 37 q^{88} - 24 q^{89} + 3 q^{91} - 3 q^{92} - 12 q^{93} + 17 q^{94} - 15 q^{96} - 3 q^{97} - 36 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1
0.809017 − 0.587785i
−0.309017 − 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.309017 − 0.224514i 0.309017 − 0.951057i −0.572949 − 1.76336i 0 −0.309017 + 0.224514i 0.309017 + 0.951057i −0.454915 + 1.40008i −0.809017 − 0.587785i 0
526.1 0.809017 − 2.48990i −0.809017 + 0.587785i −3.92705 − 2.85317i 0 0.809017 + 2.48990i −0.809017 − 0.587785i −6.04508 + 4.39201i 0.309017 − 0.951057i 0
676.1 0.809017 + 2.48990i −0.809017 − 0.587785i −3.92705 + 2.85317i 0 0.809017 − 2.48990i −0.809017 + 0.587785i −6.04508 − 4.39201i 0.309017 + 0.951057i 0
751.1 −0.309017 + 0.224514i 0.309017 + 0.951057i −0.572949 + 1.76336i 0 −0.309017 − 0.224514i 0.309017 − 0.951057i −0.454915 − 1.40008i −0.809017 + 0.587785i 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
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.n.c 4
5.b even 2 1 33.2.e.b ✓ 4
5.c odd 4 2 825.2.bx.d 8
11.c even 5 1 inner 825.2.n.c 4
11.c even 5 1 9075.2.a.cb 2
11.d odd 10 1 9075.2.a.u 2
15.d odd 2 1 99.2.f.a 4
20.d odd 2 1 528.2.y.b 4
45.h odd 6 2 891.2.n.b 8
45.j even 6 2 891.2.n.c 8
55.d odd 2 1 363.2.e.f 4
55.h odd 10 1 363.2.a.i 2
55.h odd 10 2 363.2.e.b 4
55.h odd 10 1 363.2.e.f 4
55.j even 10 1 33.2.e.b ✓ 4
55.j even 10 1 363.2.a.d 2
55.j even 10 2 363.2.e.k 4
55.k odd 20 2 825.2.bx.d 8
165.o odd 10 1 99.2.f.a 4
165.o odd 10 1 1089.2.a.t 2
165.r even 10 1 1089.2.a.l 2
220.n odd 10 1 528.2.y.b 4
220.n odd 10 1 5808.2.a.cj 2
220.o even 10 1 5808.2.a.ci 2
495.bl even 30 2 891.2.n.c 8
495.bp odd 30 2 891.2.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b ✓ 4 5.b even 2 1
33.2.e.b ✓ 4 55.j even 10 1
99.2.f.a 4 15.d odd 2 1
99.2.f.a 4 165.o odd 10 1
363.2.a.d 2 55.j even 10 1
363.2.a.i 2 55.h odd 10 1
363.2.e.b 4 55.h odd 10 2
363.2.e.f 4 55.d odd 2 1
363.2.e.f 4 55.h odd 10 1
363.2.e.k 4 55.j even 10 2
528.2.y.b 4 20.d odd 2 1
528.2.y.b 4 220.n odd 10 1
825.2.n.c 4 1.a even 1 1 trivial
825.2.n.c 4 11.c even 5 1 inner
825.2.bx.d 8 5.c odd 4 2
825.2.bx.d 8 55.k odd 20 2
891.2.n.b 8 45.h odd 6 2
891.2.n.b 8 495.bp odd 30 2
891.2.n.c 8 45.j even 6 2
891.2.n.c 8 495.bl even 30 2
1089.2.a.l 2 165.r even 10 1
1089.2.a.t 2 165.o odd 10 1
5808.2.a.ci 2 220.o even 10 1
5808.2.a.cj 2 220.n odd 10 1
9075.2.a.u 2 11.d odd 10 1
9075.2.a.cb 2 11.c even 5 1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} + 7T_{13}^{3} + 19T_{13}^{2} + 3T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + 6 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 11 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + 7 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{4} + 10 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( (T^{2} - 4 T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$37$ \( T^{4} + 9 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$41$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 17 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$61$ \( T^{4} + 21 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$67$ \( (T^{2} - 3 T - 9)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 15 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$79$ \( T^{4} + 11 T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$83$ \( T^{4} + 13 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T + 31)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
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