Properties

Label 245.2.b.a
Level $245$
Weight $2$
Character orbit 245.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
2.00000i 1.00000i −2.00000 2.00000 1.00000i −2.00000 0 0 2.00000 −2.00000 4.00000i
99.2 2.00000i 1.00000i −2.00000 2.00000 + 1.00000i −2.00000 0 0 2.00000 −2.00000 + 4.00000i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.b.a 2
3.b odd 2 1 2205.2.d.b 2
5.b even 2 1 inner 245.2.b.a 2
5.c odd 4 1 1225.2.a.a 1
5.c odd 4 1 1225.2.a.i 1
7.b odd 2 1 35.2.b.a 2
7.c even 3 2 245.2.j.d 4
7.d odd 6 2 245.2.j.e 4
15.d odd 2 1 2205.2.d.b 2
21.c even 2 1 315.2.d.a 2
28.d even 2 1 560.2.g.b 2
35.c odd 2 1 35.2.b.a 2
35.f even 4 1 175.2.a.a 1
35.f even 4 1 175.2.a.c 1
35.i odd 6 2 245.2.j.e 4
35.j even 6 2 245.2.j.d 4
56.e even 2 1 2240.2.g.g 2
56.h odd 2 1 2240.2.g.h 2
84.h odd 2 1 5040.2.t.p 2
105.g even 2 1 315.2.d.a 2
105.k odd 4 1 1575.2.a.a 1
105.k odd 4 1 1575.2.a.k 1
140.c even 2 1 560.2.g.b 2
140.j odd 4 1 2800.2.a.l 1
140.j odd 4 1 2800.2.a.w 1
280.c odd 2 1 2240.2.g.h 2
280.n even 2 1 2240.2.g.g 2
420.o odd 2 1 5040.2.t.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 7.b odd 2 1
35.2.b.a 2 35.c odd 2 1
175.2.a.a 1 35.f even 4 1
175.2.a.c 1 35.f even 4 1
245.2.b.a 2 1.a even 1 1 trivial
245.2.b.a 2 5.b even 2 1 inner
245.2.j.d 4 7.c even 3 2
245.2.j.d 4 35.j even 6 2
245.2.j.e 4 7.d odd 6 2
245.2.j.e 4 35.i odd 6 2
315.2.d.a 2 21.c even 2 1
315.2.d.a 2 105.g even 2 1
560.2.g.b 2 28.d even 2 1
560.2.g.b 2 140.c even 2 1
1225.2.a.a 1 5.c odd 4 1
1225.2.a.i 1 5.c odd 4 1
1575.2.a.a 1 105.k odd 4 1
1575.2.a.k 1 105.k odd 4 1
2205.2.d.b 2 3.b odd 2 1
2205.2.d.b 2 15.d odd 2 1
2240.2.g.g 2 56.e even 2 1
2240.2.g.g 2 280.n even 2 1
2240.2.g.h 2 56.h odd 2 1
2240.2.g.h 2 280.c odd 2 1
2800.2.a.l 1 140.j odd 4 1
2800.2.a.w 1 140.j odd 4 1
5040.2.t.p 2 84.h odd 2 1
5040.2.t.p 2 420.o odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 49 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 10)^{2} \) Copy content Toggle raw display
$61$ \( (T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T - 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 49 \) Copy content Toggle raw display
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