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: \( x^{2} - x + 1 \) 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 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.

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

Character values

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

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

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 \(S_{2}^{\mathrm{new}}(245, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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