Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} -2 \zeta_{6} q^{12} -5 q^{13} - q^{15} -4 \zeta_{6} q^{16} + ( 3 - 3 \zeta_{6} ) q^{17} + 2 \zeta_{6} q^{19} -2 q^{20} + 6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 5 q^{27} + 3 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} -3 \zeta_{6} q^{33} + 4 q^{36} -2 \zeta_{6} q^{37} + ( -5 + 5 \zeta_{6} ) q^{39} + 12 q^{41} -10 q^{43} -6 \zeta_{6} q^{44} + ( 2 - 2 \zeta_{6} ) q^{45} + 9 \zeta_{6} q^{47} -4 q^{48} -3 \zeta_{6} q^{51} + ( -10 + 10 \zeta_{6} ) q^{52} + ( -12 + 12 \zeta_{6} ) q^{53} -3 q^{55} + 2 q^{57} + ( -2 + 2 \zeta_{6} ) q^{60} + 8 \zeta_{6} q^{61} -8 q^{64} + 5 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} + 6 q^{69} + ( 2 - 2 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + 4 q^{76} + \zeta_{6} q^{79} + ( -4 + 4 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -12 q^{83} -3 q^{85} + ( 3 - 3 \zeta_{6} ) q^{87} -12 \zeta_{6} q^{89} + 12 q^{92} + 4 \zeta_{6} q^{93} + ( 2 - 2 \zeta_{6} ) q^{95} + q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{4} - q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{4} - q^{5} + 2q^{9} + 3q^{11} - 2q^{12} - 10q^{13} - 2q^{15} - 4q^{16} + 3q^{17} + 2q^{19} - 4q^{20} + 6q^{23} - q^{25} + 10q^{27} + 6q^{29} - 4q^{31} - 3q^{33} + 8q^{36} - 2q^{37} - 5q^{39} + 24q^{41} - 20q^{43} - 6q^{44} + 2q^{45} + 9q^{47} - 8q^{48} - 3q^{51} - 10q^{52} - 12q^{53} - 6q^{55} + 4q^{57} - 2q^{60} + 8q^{61} - 16q^{64} + 5q^{65} + 4q^{67} - 6q^{68} + 12q^{69} + 2q^{73} + q^{75} + 8q^{76} + q^{79} - 4q^{80} - q^{81} - 24q^{83} - 6q^{85} + 3q^{87} - 12q^{89} + 24q^{92} + 4q^{93} + 2q^{95} + 2q^{97} + 12q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 1.00000 1.73205i −0.500000 0.866025i 0 0 0 1.00000 + 1.73205i 0
226.1 0 0.500000 + 0.866025i 1.00000 + 1.73205i −0.500000 + 0.866025i 0 0 0 1.00000 1.73205i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.e.b 2
7.b odd 2 1 245.2.e.a 2
7.c even 3 1 245.2.a.c 1
7.c even 3 1 inner 245.2.e.b 2
7.d odd 6 1 35.2.a.a 1
7.d odd 6 1 245.2.e.a 2
21.g even 6 1 315.2.a.b 1
21.h odd 6 1 2205.2.a.e 1
28.f even 6 1 560.2.a.b 1
28.g odd 6 1 3920.2.a.ba 1
35.i odd 6 1 175.2.a.b 1
35.j even 6 1 1225.2.a.e 1
35.k even 12 2 175.2.b.a 2
35.l odd 12 2 1225.2.b.d 2
56.j odd 6 1 2240.2.a.k 1
56.m even 6 1 2240.2.a.u 1
77.i even 6 1 4235.2.a.c 1
84.j odd 6 1 5040.2.a.v 1
91.s odd 6 1 5915.2.a.f 1
105.p even 6 1 1575.2.a.f 1
105.w odd 12 2 1575.2.d.c 2
140.s even 6 1 2800.2.a.z 1
140.x odd 12 2 2800.2.g.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 7.d odd 6 1
175.2.a.b 1 35.i odd 6 1
175.2.b.a 2 35.k even 12 2
245.2.a.c 1 7.c even 3 1
245.2.e.a 2 7.b odd 2 1
245.2.e.a 2 7.d odd 6 1
245.2.e.b 2 1.a even 1 1 trivial
245.2.e.b 2 7.c even 3 1 inner
315.2.a.b 1 21.g even 6 1
560.2.a.b 1 28.f even 6 1
1225.2.a.e 1 35.j even 6 1
1225.2.b.d 2 35.l odd 12 2
1575.2.a.f 1 105.p even 6 1
1575.2.d.c 2 105.w odd 12 2
2205.2.a.e 1 21.h odd 6 1
2240.2.a.k 1 56.j odd 6 1
2240.2.a.u 1 56.m even 6 1
2800.2.a.z 1 140.s even 6 1
2800.2.g.l 2 140.x odd 12 2
3920.2.a.ba 1 28.g odd 6 1
4235.2.a.c 1 77.i even 6 1
5040.2.a.v 1 84.j odd 6 1
5915.2.a.f 1 91.s odd 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( ( 5 + T )^{2} \)
$17$ \( 9 - 3 T + T^{2} \)
$19$ \( 4 - 2 T + T^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( ( -12 + T )^{2} \)
$43$ \( ( 10 + T )^{2} \)
$47$ \( 81 - 9 T + T^{2} \)
$53$ \( 144 + 12 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( 16 - 4 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 - 2 T + T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( 144 + 12 T + T^{2} \)
$97$ \( ( -1 + T )^{2} \)
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