Properties

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

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + \beta_{2} q^{5} + 4 q^{6} + (\beta_{3} - 4) q^{8} + ( - 2 \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + \beta_{2} q^{5} + 4 q^{6} + (\beta_{3} - 4) q^{8} + ( - 2 \beta_{2} + \beta_1) q^{9} + (\beta_{3} + \beta_1) q^{10} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{11} + (2 \beta_{2} + 2 \beta_1) q^{12} + ( - \beta_{3} - 3) q^{13} + ( - \beta_{3} - 1) q^{15} - 3 \beta_1 q^{16} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{17} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 - 4) q^{18} + ( - 2 \beta_{2} - 2 \beta_1) q^{19} + (\beta_{3} - 2) q^{20} + 4 q^{22} + (2 \beta_{2} - 2 \beta_1) q^{23} + (4 \beta_{3} + 4 \beta_1) q^{24} + (\beta_{2} - 1) q^{25} + (4 \beta_{2} - 2 \beta_1) q^{26} + ( - \beta_{3} + 3) q^{27} + ( - 3 \beta_{3} - 1) q^{29} + 4 \beta_{2} q^{30} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{32} + ( - 5 \beta_{2} + \beta_1) q^{33} + (2 \beta_{3} + 4) q^{34} + \beta_{3} q^{36} - 6 \beta_{2} q^{37} + ( - 4 \beta_{3} - 8 \beta_{2} - 4 \beta_1 + 8) q^{38} + (3 \beta_{3} - 7 \beta_{2} + 3 \beta_1 + 7) q^{39} + ( - 4 \beta_{2} - \beta_1) q^{40} + 2 \beta_{3} q^{41} + (2 \beta_{3} + 6) q^{43} + (2 \beta_{2} + 2 \beta_1) q^{44} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{45} + ( - 8 \beta_{2} + 8) q^{46} + ( - \beta_{2} - 3 \beta_1) q^{47} - 12 q^{48} + \beta_{3} q^{50} + ( - 7 \beta_{2} + 3 \beta_1) q^{51} + ( - 2 \beta_{2} + 2) q^{52} + ( - 2 \beta_{3} - 2 \beta_1) q^{53} + (4 \beta_{2} + 4 \beta_1) q^{54} + ( - \beta_{3} - 1) q^{55} + (2 \beta_{3} - 6) q^{57} + (12 \beta_{2} + 2 \beta_1) q^{58} + (4 \beta_{2} - 4) q^{59} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{60} + 6 \beta_1 q^{61} + (\beta_{3} + 4) q^{64} + ( - 3 \beta_{2} + \beta_1) q^{65} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 4) q^{66} + (4 \beta_{3} + 4 \beta_1) q^{67} - 2 \beta_{2} q^{68} + ( - 2 \beta_{3} - 10) q^{69} + 8 q^{71} + (4 \beta_{2} - 3 \beta_1) q^{72} + (4 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{73} + ( - 6 \beta_{3} - 6 \beta_1) q^{74} + ( - \beta_{2} + \beta_1) q^{75} + ( - 8 \beta_{3} + 12) q^{76} + ( - 4 \beta_{3} - 12) q^{78} + (5 \beta_{2} - \beta_1) q^{79} + ( - 3 \beta_{3} - 3 \beta_1) q^{80} + ( - 7 \beta_{2} + 7) q^{81} + ( - 8 \beta_{2} - 2 \beta_1) q^{82} - 4 q^{83} + ( - \beta_{3} - 3) q^{85} + ( - 8 \beta_{2} + 4 \beta_1) q^{86} + (\beta_{3} - 13 \beta_{2} + \beta_1 + 13) q^{87} + (4 \beta_{3} + 4 \beta_1) q^{88} + (4 \beta_{2} - 2 \beta_1) q^{89} + ( - \beta_{3} - 4) q^{90} + ( - 4 \beta_{3} + 4) q^{92} + ( - 4 \beta_{3} - 12 \beta_{2} - 4 \beta_1 + 12) q^{94} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{95} - 4 \beta_1 q^{96} + (5 \beta_{3} + 7) q^{97} + (2 \beta_{3} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{3} - 5 q^{4} + 2 q^{5} + 16 q^{6} - 18 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - q^{3} - 5 q^{4} + 2 q^{5} + 16 q^{6} - 18 q^{8} - 3 q^{9} - q^{10} - q^{11} + 6 q^{12} - 10 q^{13} - 2 q^{15} - 3 q^{16} - 5 q^{17} - 7 q^{18} - 6 q^{19} - 10 q^{20} + 16 q^{22} + 2 q^{23} - 4 q^{24} - 2 q^{25} + 6 q^{26} + 14 q^{27} + 2 q^{29} + 8 q^{30} + 9 q^{32} - 9 q^{33} + 12 q^{34} - 2 q^{36} - 12 q^{37} + 20 q^{38} + 11 q^{39} - 9 q^{40} - 4 q^{41} + 20 q^{43} + 6 q^{44} + 3 q^{45} + 16 q^{46} - 5 q^{47} - 48 q^{48} - 2 q^{50} - 11 q^{51} + 4 q^{52} + 2 q^{53} + 12 q^{54} - 2 q^{55} - 28 q^{57} + 26 q^{58} - 8 q^{59} - 6 q^{60} + 6 q^{61} + 14 q^{64} - 5 q^{65} - 4 q^{66} - 4 q^{67} - 4 q^{68} - 36 q^{69} + 32 q^{71} + 5 q^{72} - 8 q^{73} + 6 q^{74} - q^{75} + 64 q^{76} - 40 q^{78} + 9 q^{79} + 3 q^{80} + 14 q^{81} - 18 q^{82} - 16 q^{83} - 10 q^{85} - 12 q^{86} + 25 q^{87} - 4 q^{88} + 6 q^{89} - 14 q^{90} + 24 q^{92} + 28 q^{94} + 6 q^{95} - 4 q^{96} + 18 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Character values

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

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

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
−0.780776 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i
1.28078 2.21837i
−0.780776 1.35234i −1.28078 + 2.21837i −0.219224 + 0.379706i 0.500000 + 0.866025i 4.00000 0 −2.43845 −1.78078 3.08440i 0.780776 1.35234i
116.2 1.28078 + 2.21837i 0.780776 1.35234i −2.28078 + 3.95042i 0.500000 + 0.866025i 4.00000 0 −6.56155 0.280776 + 0.486319i −1.28078 + 2.21837i
226.1 −0.780776 + 1.35234i −1.28078 2.21837i −0.219224 0.379706i 0.500000 0.866025i 4.00000 0 −2.43845 −1.78078 + 3.08440i 0.780776 + 1.35234i
226.2 1.28078 2.21837i 0.780776 + 1.35234i −2.28078 3.95042i 0.500000 0.866025i 4.00000 0 −6.56155 0.280776 0.486319i −1.28078 2.21837i
\(n\): 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.h 4
7.b odd 2 1 245.2.e.i 4
7.c even 3 1 245.2.a.d 2
7.c even 3 1 inner 245.2.e.h 4
7.d odd 6 1 35.2.a.b 2
7.d odd 6 1 245.2.e.i 4
21.g even 6 1 315.2.a.e 2
21.h odd 6 1 2205.2.a.x 2
28.f even 6 1 560.2.a.i 2
28.g odd 6 1 3920.2.a.bs 2
35.i odd 6 1 175.2.a.f 2
35.j even 6 1 1225.2.a.s 2
35.k even 12 2 175.2.b.b 4
35.l odd 12 2 1225.2.b.f 4
56.j odd 6 1 2240.2.a.bh 2
56.m even 6 1 2240.2.a.bd 2
77.i even 6 1 4235.2.a.m 2
84.j odd 6 1 5040.2.a.bt 2
91.s odd 6 1 5915.2.a.l 2
105.p even 6 1 1575.2.a.p 2
105.w odd 12 2 1575.2.d.e 4
140.s even 6 1 2800.2.a.bi 2
140.x odd 12 2 2800.2.g.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 7.d odd 6 1
175.2.a.f 2 35.i odd 6 1
175.2.b.b 4 35.k even 12 2
245.2.a.d 2 7.c even 3 1
245.2.e.h 4 1.a even 1 1 trivial
245.2.e.h 4 7.c even 3 1 inner
245.2.e.i 4 7.b odd 2 1
245.2.e.i 4 7.d odd 6 1
315.2.a.e 2 21.g even 6 1
560.2.a.i 2 28.f even 6 1
1225.2.a.s 2 35.j even 6 1
1225.2.b.f 4 35.l odd 12 2
1575.2.a.p 2 105.p even 6 1
1575.2.d.e 4 105.w odd 12 2
2205.2.a.x 2 21.h odd 6 1
2240.2.a.bd 2 56.m even 6 1
2240.2.a.bh 2 56.j odd 6 1
2800.2.a.bi 2 140.s even 6 1
2800.2.g.t 4 140.x odd 12 2
3920.2.a.bs 2 28.g odd 6 1
4235.2.a.m 2 77.i even 6 1
5040.2.a.bt 2 84.j odd 6 1
5915.2.a.l 2 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 \(S_{2}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{4} - T_{2}^{3} + 5T_{2}^{2} + 4T_{2} + 16 \) Copy content Toggle raw display
\( T_{3}^{4} + T_{3}^{3} + 5T_{3}^{2} - 4T_{3} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + 5 T^{2} + 4 T + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + 5 T^{2} - 4 T + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + 5 T^{2} - 4 T + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 44 T^{2} - 48 T + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + 20 T^{2} + 32 T + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - T - 38)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 5 T^{3} + 57 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + 20 T^{2} + 32 T + 256 \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + 180 T^{2} + \cdots + 20736 \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} + 80 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T - 8)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} + 116 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$79$ \( T^{4} - 9 T^{3} + 65 T^{2} - 144 T + 256 \) Copy content Toggle raw display
$83$ \( (T + 4)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 6 T^{3} + 44 T^{2} + 48 T + 64 \) Copy content Toggle raw display
$97$ \( (T^{2} - 9 T - 86)^{2} \) Copy content Toggle raw display
show more
show less