Properties

Label 245.2.l.a
Level $245$
Weight $2$
Character orbit 245.l
Analytic conductor $1.956$
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] = [245,2,Mod(68,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.l (of order \(12\), degree \(4\), 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: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12} - 1) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2}) q^{3} + (\zeta_{12}^{2} + 1) q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{5} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{6} + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 1) q^{8} + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12} - 1) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2}) q^{3} + (\zeta_{12}^{2} + 1) q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{5} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{6} + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 1) q^{8} + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 2) q^{9} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 1) q^{10} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 1) q^{11} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} - 1) q^{12} + ( - 2 \zeta_{12}^{3} - 2) q^{13} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{15} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12}) q^{16} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2}) q^{17} + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{18} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{19} + ( - \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{20} + (\zeta_{12}^{3} + 1) q^{22} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + \zeta_{12} - 3) q^{23} + (\zeta_{12}^{2} - 1) q^{24} + (3 \zeta_{12}^{2} + 4 \zeta_{12} - 3) q^{25} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{26} + (\zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 1) q^{27} - 3 \zeta_{12}^{3} q^{29} + (3 \zeta_{12}^{3} - 5 \zeta_{12} + 4) q^{30} + (2 \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{31} + ( - \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{32} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{33} + 2 q^{34} + (\zeta_{12}^{3} - 2 \zeta_{12} - 3) q^{36} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{37} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12} - 1) q^{38} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{39} + (\zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{40} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{41} + ( - \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 1) q^{43} + ( - 3 \zeta_{12}^{3} - \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{44} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{45} + (8 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - 4 \zeta_{12} + 7) q^{46} + (\zeta_{12}^{3} - \zeta_{12}^{2} - 5 \zeta_{12} - 4) q^{47} + (5 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 5) q^{48} + (3 \zeta_{12}^{3} + \zeta_{12}^{2} - 7 \zeta_{12} + 3) q^{50} - 2 \zeta_{12}^{2} q^{51} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{52} + ( - 5 \zeta_{12}^{2} - 5 \zeta_{12} + 5) q^{53} + (2 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 2 \zeta_{12}) q^{54} + ( - 3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12} + 3) q^{55} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 3) q^{57} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3) q^{58} + (6 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{59} + ( - \zeta_{12}^{3} + 5 \zeta_{12} - 3) q^{60} + (5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 5 \zeta_{12} + 4) q^{61} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{62} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{64} + ( - 2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12}) q^{65} + (\zeta_{12}^{2} - \zeta_{12} + 1) q^{66} + (5 \zeta_{12}^{3} + \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{67} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{68} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} - 7) q^{69} + (\zeta_{12}^{3} - 2 \zeta_{12} + 3) q^{71} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} + 2) q^{72} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 8 \zeta_{12} + 8) q^{73} + ( - 2 \zeta_{12}^{2} + 6 \zeta_{12} - 2) q^{74} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12} + 1) q^{75} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{76} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{78} + (5 \zeta_{12}^{3} - \zeta_{12}^{2} - 5 \zeta_{12} + 2) q^{79} + (8 \zeta_{12}^{3} - 5 \zeta_{12} + 4) q^{80} + ( - 10 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 5 \zeta_{12} - 2) q^{81} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{82} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 2) q^{83} + ( - 4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{85} + ( - 4 \zeta_{12}^{3} + 9 \zeta_{12}^{2} - 4 \zeta_{12}) q^{86} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12} - 3) q^{87} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + \zeta_{12} - 1) q^{88} + ( - 5 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 5 \zeta_{12}) q^{89} + ( - 3 \zeta_{12}^{3} + 1) q^{90} + ( - 7 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 7) q^{92} + (2 \zeta_{12} + 2) q^{93} + ( - 2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} + 3) q^{94} + (4 \zeta_{12}^{3} - \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{95} + ( - 6 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 6 \zeta_{12} - 10) q^{96} + (3 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12} - 3) q^{97} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 2 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 2 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{8} - 6 q^{9} + 2 q^{11} - 8 q^{13} - 6 q^{15} - 2 q^{16} - 4 q^{17} + 2 q^{18} - 2 q^{19} + 4 q^{22} - 4 q^{23} - 2 q^{24} - 6 q^{25} + 12 q^{26} + 2 q^{27} + 16 q^{30} + 12 q^{31} - 6 q^{32} - 2 q^{33} + 8 q^{34} - 12 q^{36} + 12 q^{37} + 2 q^{38} - 12 q^{39} - 10 q^{40} - 6 q^{43} + 6 q^{44} - 14 q^{45} + 14 q^{46} - 18 q^{47} + 14 q^{48} + 14 q^{50} - 4 q^{51} - 12 q^{52} + 10 q^{53} - 10 q^{54} + 8 q^{55} + 8 q^{57} + 6 q^{58} - 6 q^{59} - 12 q^{60} + 12 q^{61} - 4 q^{62} - 12 q^{65} + 6 q^{66} - 22 q^{67} - 28 q^{69} + 12 q^{71} + 2 q^{72} + 24 q^{73} - 12 q^{74} - 4 q^{75} + 16 q^{78} + 6 q^{79} + 16 q^{80} - 4 q^{81} + 6 q^{82} + 2 q^{83} + 4 q^{85} + 18 q^{86} - 12 q^{87} + 2 q^{88} + 16 q^{89} + 4 q^{90} - 18 q^{92} + 8 q^{93} + 6 q^{94} + 10 q^{95} - 30 q^{96} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(\zeta_{12}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.133975 0.500000i 0.500000 + 0.133975i 1.50000 0.866025i 1.86603 1.23205i 0.267949i 0 −1.36603 1.36603i −2.36603 1.36603i −0.866025 0.767949i
117.1 −1.86603 + 0.500000i 0.500000 1.86603i 1.50000 0.866025i 0.133975 2.23205i 3.73205i 0 0.366025 0.366025i −0.633975 0.366025i 0.866025 + 4.23205i
178.1 −1.86603 0.500000i 0.500000 + 1.86603i 1.50000 + 0.866025i 0.133975 + 2.23205i 3.73205i 0 0.366025 + 0.366025i −0.633975 + 0.366025i 0.866025 4.23205i
227.1 −0.133975 + 0.500000i 0.500000 0.133975i 1.50000 + 0.866025i 1.86603 + 1.23205i 0.267949i 0 −1.36603 + 1.36603i −2.36603 + 1.36603i −0.866025 + 0.767949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.l.a 4
5.c odd 4 1 245.2.l.b 4
7.b odd 2 1 35.2.k.a 4
7.c even 3 1 35.2.k.b yes 4
7.c even 3 1 245.2.f.b 4
7.d odd 6 1 245.2.f.a 4
7.d odd 6 1 245.2.l.b 4
21.c even 2 1 315.2.bz.b 4
21.h odd 6 1 315.2.bz.a 4
28.d even 2 1 560.2.ci.a 4
28.g odd 6 1 560.2.ci.b 4
35.c odd 2 1 175.2.o.b 4
35.f even 4 1 35.2.k.b yes 4
35.f even 4 1 175.2.o.a 4
35.j even 6 1 175.2.o.a 4
35.k even 12 1 245.2.f.b 4
35.k even 12 1 inner 245.2.l.a 4
35.l odd 12 1 35.2.k.a 4
35.l odd 12 1 175.2.o.b 4
35.l odd 12 1 245.2.f.a 4
105.k odd 4 1 315.2.bz.a 4
105.x even 12 1 315.2.bz.b 4
140.j odd 4 1 560.2.ci.b 4
140.w even 12 1 560.2.ci.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.k.a 4 7.b odd 2 1
35.2.k.a 4 35.l odd 12 1
35.2.k.b yes 4 7.c even 3 1
35.2.k.b yes 4 35.f even 4 1
175.2.o.a 4 35.f even 4 1
175.2.o.a 4 35.j even 6 1
175.2.o.b 4 35.c odd 2 1
175.2.o.b 4 35.l odd 12 1
245.2.f.a 4 7.d odd 6 1
245.2.f.a 4 35.l odd 12 1
245.2.f.b 4 7.c even 3 1
245.2.f.b 4 35.k even 12 1
245.2.l.a 4 1.a even 1 1 trivial
245.2.l.a 4 35.k even 12 1 inner
245.2.l.b 4 5.c odd 4 1
245.2.l.b 4 7.d odd 6 1
315.2.bz.a 4 21.h odd 6 1
315.2.bz.a 4 105.k odd 4 1
315.2.bz.b 4 21.c even 2 1
315.2.bz.b 4 105.x even 12 1
560.2.ci.a 4 28.d even 2 1
560.2.ci.a 4 140.w even 12 1
560.2.ci.b 4 28.g odd 6 1
560.2.ci.b 4 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 4T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 5 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + 11 T^{2} - 20 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + 20 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + 53 T^{2} + 14 T + 1 \) Copy content Toggle raw display
$29$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} + 44 T^{2} + 48 T + 16 \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$41$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$43$ \( T^{4} + 6 T^{3} + 18 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$47$ \( T^{4} + 18 T^{3} + 90 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$53$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + 35 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$67$ \( T^{4} + 22 T^{3} + 137 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + 144 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} - 10 T^{2} + 132 T + 484 \) Copy content Toggle raw display
$83$ \( T^{4} - 2 T^{3} + 2 T^{2} + 26 T + 169 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} + 267 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + 8 T^{2} + 376 T + 8836 \) Copy content Toggle raw display
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