Properties

Label 245.2.e.e
Level $245$
Weight $2$
Character orbit 245.e
Analytic conductor $1.956$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−1.20711 2.09077i −0.207107 + 0.358719i −1.91421 + 3.31552i 0.500000 + 0.866025i 1.00000 0 4.41421 1.41421 + 2.44949i 1.20711 2.09077i
116.2 0.207107 + 0.358719i 1.20711 2.09077i 0.914214 1.58346i 0.500000 + 0.866025i 1.00000 0 1.58579 −1.41421 2.44949i −0.207107 + 0.358719i
226.1 −1.20711 + 2.09077i −0.207107 0.358719i −1.91421 3.31552i 0.500000 0.866025i 1.00000 0 4.41421 1.41421 2.44949i 1.20711 + 2.09077i
226.2 0.207107 0.358719i 1.20711 + 2.09077i 0.914214 + 1.58346i 0.500000 0.866025i 1.00000 0 1.58579 −1.41421 + 2.44949i −0.207107 0.358719i
\(n\): 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.e 4
7.b odd 2 1 35.2.e.a 4
7.c even 3 1 245.2.a.g 2
7.c even 3 1 inner 245.2.e.e 4
7.d odd 6 1 35.2.e.a 4
7.d odd 6 1 245.2.a.h 2
21.c even 2 1 315.2.j.e 4
21.g even 6 1 315.2.j.e 4
21.g even 6 1 2205.2.a.n 2
21.h odd 6 1 2205.2.a.q 2
28.d even 2 1 560.2.q.k 4
28.f even 6 1 560.2.q.k 4
28.f even 6 1 3920.2.a.bq 2
28.g odd 6 1 3920.2.a.bv 2
35.c odd 2 1 175.2.e.c 4
35.f even 4 2 175.2.k.a 8
35.i odd 6 1 175.2.e.c 4
35.i odd 6 1 1225.2.a.k 2
35.j even 6 1 1225.2.a.m 2
35.k even 12 2 175.2.k.a 8
35.k even 12 2 1225.2.b.g 4
35.l odd 12 2 1225.2.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 7.b odd 2 1
35.2.e.a 4 7.d odd 6 1
175.2.e.c 4 35.c odd 2 1
175.2.e.c 4 35.i odd 6 1
175.2.k.a 8 35.f even 4 2
175.2.k.a 8 35.k even 12 2
245.2.a.g 2 7.c even 3 1
245.2.a.h 2 7.d odd 6 1
245.2.e.e 4 1.a even 1 1 trivial
245.2.e.e 4 7.c even 3 1 inner
315.2.j.e 4 21.c even 2 1
315.2.j.e 4 21.g even 6 1
560.2.q.k 4 28.d even 2 1
560.2.q.k 4 28.f even 6 1
1225.2.a.k 2 35.i odd 6 1
1225.2.a.m 2 35.j even 6 1
1225.2.b.g 4 35.k even 12 2
1225.2.b.h 4 35.l odd 12 2
2205.2.a.n 2 21.g even 6 1
2205.2.a.q 2 21.h odd 6 1
3920.2.a.bq 2 28.f even 6 1
3920.2.a.bv 2 28.g odd 6 1

Hecke kernels

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

\( T_{2}^{4} + 2 T_{2}^{3} + 5 T_{2}^{2} - 2 T_{2} + 1 \)
\( T_{3}^{4} - 2 T_{3}^{3} + 5 T_{3}^{2} + 2 T_{3} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( 1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4} \)
$13$ \( ( -4 - 4 T + T^{2} )^{2} \)
$17$ \( 16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( 64 + 8 T^{2} + T^{4} \)
$23$ \( 1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( ( 1 + T )^{4} \)
$31$ \( ( 36 + 6 T + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( ( 17 - 10 T + T^{2} )^{2} \)
$43$ \( ( 23 - 10 T + T^{2} )^{2} \)
$47$ \( ( 4 - 2 T + T^{2} )^{2} \)
$53$ \( 64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4} \)
$59$ \( 3136 - 448 T + 120 T^{2} + 8 T^{3} + T^{4} \)
$61$ \( 3969 - 378 T + 99 T^{2} + 6 T^{3} + T^{4} \)
$67$ \( 14161 + 2618 T + 365 T^{2} + 22 T^{3} + T^{4} \)
$71$ \( ( -56 + 8 T + T^{2} )^{2} \)
$73$ \( 16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4} \)
$79$ \( 18496 + 3264 T + 440 T^{2} + 24 T^{3} + T^{4} \)
$83$ \( ( -161 + 2 T + T^{2} )^{2} \)
$89$ \( 529 - 138 T + 59 T^{2} + 6 T^{3} + T^{4} \)
$97$ \( ( 4 + 12 T + T^{2} )^{2} \)
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