Properties

Label 245.2.j.e
Level $245$
Weight $2$
Character orbit 245.j
Analytic conductor $1.956$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{2} q^{4} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} ) q^{5} + 2 q^{6} + ( -2 + 2 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + 2 \zeta_{12} q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{2} q^{4} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} ) q^{5} + 2 q^{6} + ( -2 + 2 \zeta_{12}^{2} ) q^{9} + ( 4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{10} + 3 \zeta_{12}^{2} q^{11} + 2 \zeta_{12} q^{12} + \zeta_{12}^{3} q^{13} + ( -1 - 2 \zeta_{12}^{3} ) q^{15} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{17} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{18} + ( 4 - 2 \zeta_{12}^{3} ) q^{20} + 6 \zeta_{12}^{3} q^{22} -6 \zeta_{12} q^{23} + ( -4 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{25} + ( -2 + 2 \zeta_{12}^{2} ) q^{26} + 5 \zeta_{12}^{3} q^{27} + 5 q^{29} + ( 4 - 2 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{30} -2 \zeta_{12}^{2} q^{31} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{32} + 3 \zeta_{12} q^{33} -14 q^{34} -4 q^{36} + 2 \zeta_{12} q^{37} + \zeta_{12}^{2} q^{39} + 2 q^{41} -4 \zeta_{12}^{3} q^{43} + ( -6 + 6 \zeta_{12}^{2} ) q^{44} + ( 2 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{45} -12 \zeta_{12}^{2} q^{46} -3 \zeta_{12} q^{47} -4 \zeta_{12}^{3} q^{48} + ( -8 - 6 \zeta_{12}^{3} ) q^{50} + ( -7 + 7 \zeta_{12}^{2} ) q^{51} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{52} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{53} + ( -10 + 10 \zeta_{12}^{2} ) q^{54} + ( 6 - 3 \zeta_{12}^{3} ) q^{55} + 10 \zeta_{12} q^{58} + 10 \zeta_{12}^{2} q^{59} + ( 4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{60} + ( 8 - 8 \zeta_{12}^{2} ) q^{61} -4 \zeta_{12}^{3} q^{62} + 8 q^{64} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{65} + 6 \zeta_{12}^{2} q^{66} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{67} -14 \zeta_{12} q^{68} -6 q^{69} -8 q^{71} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{73} + 4 \zeta_{12}^{2} q^{74} + ( -4 - 3 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{75} + 2 \zeta_{12}^{3} q^{78} + ( -5 + 5 \zeta_{12}^{2} ) q^{79} + ( -4 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{80} -\zeta_{12}^{2} q^{81} + 4 \zeta_{12} q^{82} -4 \zeta_{12}^{3} q^{83} + ( 7 + 14 \zeta_{12}^{3} ) q^{85} + ( 8 - 8 \zeta_{12}^{2} ) q^{86} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{87} + ( 4 + 8 \zeta_{12}^{3} ) q^{90} -12 \zeta_{12}^{3} q^{92} -2 \zeta_{12} q^{93} -6 \zeta_{12}^{2} q^{94} + ( 8 - 8 \zeta_{12}^{2} ) q^{96} -7 \zeta_{12}^{3} q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 4 q^{5} + 8 q^{6} - 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{4} + 4 q^{5} + 8 q^{6} - 4 q^{9} - 4 q^{10} + 6 q^{11} - 4 q^{15} + 8 q^{16} + 16 q^{20} - 6 q^{25} - 4 q^{26} + 20 q^{29} + 8 q^{30} - 4 q^{31} - 56 q^{34} - 16 q^{36} + 2 q^{39} + 8 q^{41} - 12 q^{44} + 8 q^{45} - 24 q^{46} - 32 q^{50} - 14 q^{51} - 20 q^{54} + 24 q^{55} + 20 q^{59} - 4 q^{60} + 16 q^{61} + 32 q^{64} + 2 q^{65} + 12 q^{66} - 24 q^{69} - 32 q^{71} + 8 q^{74} - 8 q^{75} - 10 q^{79} - 16 q^{80} - 2 q^{81} + 28 q^{85} + 16 q^{86} + 16 q^{90} - 12 q^{94} + 16 q^{96} - 24 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−1.73205 1.00000i −0.866025 + 0.500000i 1.00000 + 1.73205i 1.86603 1.23205i 2.00000 0 0 −1.00000 + 1.73205i −4.46410 + 0.267949i
79.2 1.73205 + 1.00000i 0.866025 0.500000i 1.00000 + 1.73205i 0.133975 2.23205i 2.00000 0 0 −1.00000 + 1.73205i 2.46410 3.73205i
214.1 −1.73205 + 1.00000i −0.866025 0.500000i 1.00000 1.73205i 1.86603 + 1.23205i 2.00000 0 0 −1.00000 1.73205i −4.46410 0.267949i
214.2 1.73205 1.00000i 0.866025 + 0.500000i 1.00000 1.73205i 0.133975 + 2.23205i 2.00000 0 0 −1.00000 1.73205i 2.46410 + 3.73205i
\(n\): 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
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.j.e 4
5.b even 2 1 inner 245.2.j.e 4
7.b odd 2 1 245.2.j.d 4
7.c even 3 1 35.2.b.a 2
7.c even 3 1 inner 245.2.j.e 4
7.d odd 6 1 245.2.b.a 2
7.d odd 6 1 245.2.j.d 4
21.g even 6 1 2205.2.d.b 2
21.h odd 6 1 315.2.d.a 2
28.g odd 6 1 560.2.g.b 2
35.c odd 2 1 245.2.j.d 4
35.i odd 6 1 245.2.b.a 2
35.i odd 6 1 245.2.j.d 4
35.j even 6 1 35.2.b.a 2
35.j even 6 1 inner 245.2.j.e 4
35.k even 12 1 1225.2.a.a 1
35.k even 12 1 1225.2.a.i 1
35.l odd 12 1 175.2.a.a 1
35.l odd 12 1 175.2.a.c 1
56.k odd 6 1 2240.2.g.g 2
56.p even 6 1 2240.2.g.h 2
84.n even 6 1 5040.2.t.p 2
105.o odd 6 1 315.2.d.a 2
105.p even 6 1 2205.2.d.b 2
105.x even 12 1 1575.2.a.a 1
105.x even 12 1 1575.2.a.k 1
140.p odd 6 1 560.2.g.b 2
140.w even 12 1 2800.2.a.l 1
140.w even 12 1 2800.2.a.w 1
280.bf even 6 1 2240.2.g.h 2
280.bi odd 6 1 2240.2.g.g 2
420.ba even 6 1 5040.2.t.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 7.c even 3 1
35.2.b.a 2 35.j even 6 1
175.2.a.a 1 35.l odd 12 1
175.2.a.c 1 35.l odd 12 1
245.2.b.a 2 7.d odd 6 1
245.2.b.a 2 35.i odd 6 1
245.2.j.d 4 7.b odd 2 1
245.2.j.d 4 7.d odd 6 1
245.2.j.d 4 35.c odd 2 1
245.2.j.d 4 35.i odd 6 1
245.2.j.e 4 1.a even 1 1 trivial
245.2.j.e 4 5.b even 2 1 inner
245.2.j.e 4 7.c even 3 1 inner
245.2.j.e 4 35.j even 6 1 inner
315.2.d.a 2 21.h odd 6 1
315.2.d.a 2 105.o odd 6 1
560.2.g.b 2 28.g odd 6 1
560.2.g.b 2 140.p odd 6 1
1225.2.a.a 1 35.k even 12 1
1225.2.a.i 1 35.k even 12 1
1575.2.a.a 1 105.x even 12 1
1575.2.a.k 1 105.x even 12 1
2205.2.d.b 2 21.g even 6 1
2205.2.d.b 2 105.p even 6 1
2240.2.g.g 2 56.k odd 6 1
2240.2.g.g 2 280.bi odd 6 1
2240.2.g.h 2 56.p even 6 1
2240.2.g.h 2 280.bf even 6 1
2800.2.a.l 1 140.w even 12 1
2800.2.a.w 1 140.w even 12 1
5040.2.t.p 2 84.n even 6 1
5040.2.t.p 2 420.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{4} - 4 T_{2}^{2} + 16 \)
\( T_{3}^{4} - T_{3}^{2} + 1 \)
\( T_{31}^{2} + 2 T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 4 T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( 25 - 20 T + 11 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 9 - 3 T + T^{2} )^{2} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( 2401 - 49 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( 1296 - 36 T^{2} + T^{4} \)
$29$ \( ( -5 + T )^{4} \)
$31$ \( ( 4 + 2 T + T^{2} )^{2} \)
$37$ \( 16 - 4 T^{2} + T^{4} \)
$41$ \( ( -2 + T )^{4} \)
$43$ \( ( 16 + T^{2} )^{2} \)
$47$ \( 81 - 9 T^{2} + T^{4} \)
$53$ \( 1296 - 36 T^{2} + T^{4} \)
$59$ \( ( 100 - 10 T + T^{2} )^{2} \)
$61$ \( ( 64 - 8 T + T^{2} )^{2} \)
$67$ \( 16 - 4 T^{2} + T^{4} \)
$71$ \( ( 8 + T )^{4} \)
$73$ \( 1296 - 36 T^{2} + T^{4} \)
$79$ \( ( 25 + 5 T + T^{2} )^{2} \)
$83$ \( ( 16 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( 49 + T^{2} )^{2} \)
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