Properties

Label 245.3.h.b
Level $245$
Weight $3$
Character orbit 245.h
Analytic conductor $6.676$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 245.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.67576647683\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - 2 \beta_1 q^{3} + ( - 3 \beta_{2} + 3) q^{4} + ( - \beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{6} + 7 q^{8} + 11 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - 2 \beta_1 q^{3} + ( - 3 \beta_{2} + 3) q^{4} + ( - \beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{6} + 7 q^{8} + 11 \beta_{2} q^{9} + \beta_1 q^{10} + (2 \beta_{2} - 2) q^{11} + (6 \beta_{3} - 6 \beta_1) q^{12} - 6 \beta_{3} q^{13} - 10 q^{15} - 5 \beta_{2} q^{16} - 12 \beta_1 q^{17} + (11 \beta_{2} - 11) q^{18} + (6 \beta_{3} - 6 \beta_1) q^{19} - 3 \beta_{3} q^{20} - 2 q^{22} - 26 \beta_{2} q^{23} - 14 \beta_1 q^{24} + ( - 5 \beta_{2} + 5) q^{25} + ( - 6 \beta_{3} + 6 \beta_1) q^{26} - 4 \beta_{3} q^{27} - 22 q^{29} - 10 \beta_{2} q^{30} + 24 \beta_1 q^{31} + ( - 33 \beta_{2} + 33) q^{32} + ( - 4 \beta_{3} + 4 \beta_1) q^{33} - 12 \beta_{3} q^{34} + 33 q^{36} - 14 \beta_{2} q^{37} - 6 \beta_1 q^{38} + (60 \beta_{2} - 60) q^{39} + ( - 7 \beta_{3} + 7 \beta_1) q^{40} - 12 \beta_{3} q^{41} - 34 q^{43} + 6 \beta_{2} q^{44} + 11 \beta_1 q^{45} + ( - 26 \beta_{2} + 26) q^{46} + (12 \beta_{3} - 12 \beta_1) q^{47} + 10 \beta_{3} q^{48} + 5 q^{50} + 120 \beta_{2} q^{51} - 18 \beta_1 q^{52} + ( - 34 \beta_{2} + 34) q^{53} + ( - 4 \beta_{3} + 4 \beta_1) q^{54} + 2 \beta_{3} q^{55} + 60 q^{57} - 22 \beta_{2} q^{58} + 18 \beta_1 q^{59} + (30 \beta_{2} - 30) q^{60} + ( - 42 \beta_{3} + 42 \beta_1) q^{61} + 24 \beta_{3} q^{62} + 13 q^{64} - 30 \beta_{2} q^{65} + 4 \beta_1 q^{66} + (14 \beta_{2} - 14) q^{67} + (36 \beta_{3} - 36 \beta_1) q^{68} + 52 \beta_{3} q^{69} + 62 q^{71} + 77 \beta_{2} q^{72} + 24 \beta_1 q^{73} + ( - 14 \beta_{2} + 14) q^{74} + (10 \beta_{3} - 10 \beta_1) q^{75} + 18 \beta_{3} q^{76} - 60 q^{78} - 38 \beta_{2} q^{79} - 5 \beta_1 q^{80} + ( - 59 \beta_{2} + 59) q^{81} + ( - 12 \beta_{3} + 12 \beta_1) q^{82} + 18 \beta_{3} q^{83} - 60 q^{85} - 34 \beta_{2} q^{86} + 44 \beta_1 q^{87} + (14 \beta_{2} - 14) q^{88} + (12 \beta_{3} - 12 \beta_1) q^{89} + 11 \beta_{3} q^{90} - 78 q^{92} - 240 \beta_{2} q^{93} - 12 \beta_1 q^{94} + (30 \beta_{2} - 30) q^{95} + (66 \beta_{3} - 66 \beta_1) q^{96} - 12 \beta_{3} q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 6 q^{4} + 28 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 6 q^{4} + 28 q^{8} + 22 q^{9} - 4 q^{11} - 40 q^{15} - 10 q^{16} - 22 q^{18} - 8 q^{22} - 52 q^{23} + 10 q^{25} - 88 q^{29} - 20 q^{30} + 66 q^{32} + 132 q^{36} - 28 q^{37} - 120 q^{39} - 136 q^{43} + 12 q^{44} + 52 q^{46} + 20 q^{50} + 240 q^{51} + 68 q^{53} + 240 q^{57} - 44 q^{58} - 60 q^{60} + 52 q^{64} - 60 q^{65} - 28 q^{67} + 248 q^{71} + 154 q^{72} + 28 q^{74} - 240 q^{78} - 76 q^{79} + 118 q^{81} - 240 q^{85} - 68 q^{86} - 28 q^{88} - 312 q^{92} - 480 q^{93} - 60 q^{95} - 88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.93649 + 1.11803i
−1.93649 1.11803i
1.93649 1.11803i
−1.93649 + 1.11803i
0.500000 + 0.866025i −3.87298 2.23607i 1.50000 2.59808i 1.93649 1.11803i 4.47214i 0 7.00000 5.50000 + 9.52628i 1.93649 + 1.11803i
31.2 0.500000 + 0.866025i 3.87298 + 2.23607i 1.50000 2.59808i −1.93649 + 1.11803i 4.47214i 0 7.00000 5.50000 + 9.52628i −1.93649 1.11803i
166.1 0.500000 0.866025i −3.87298 + 2.23607i 1.50000 + 2.59808i 1.93649 + 1.11803i 4.47214i 0 7.00000 5.50000 9.52628i 1.93649 1.11803i
166.2 0.500000 0.866025i 3.87298 2.23607i 1.50000 + 2.59808i −1.93649 1.11803i 4.47214i 0 7.00000 5.50000 9.52628i −1.93649 + 1.11803i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.3.h.b 4
7.b odd 2 1 inner 245.3.h.b 4
7.c even 3 1 35.3.d.a 2
7.c even 3 1 inner 245.3.h.b 4
7.d odd 6 1 35.3.d.a 2
7.d odd 6 1 inner 245.3.h.b 4
21.g even 6 1 315.3.h.b 2
21.h odd 6 1 315.3.h.b 2
28.f even 6 1 560.3.f.a 2
28.g odd 6 1 560.3.f.a 2
35.i odd 6 1 175.3.d.f 2
35.j even 6 1 175.3.d.f 2
35.k even 12 2 175.3.c.d 4
35.l odd 12 2 175.3.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.a 2 7.c even 3 1
35.3.d.a 2 7.d odd 6 1
175.3.c.d 4 35.k even 12 2
175.3.c.d 4 35.l odd 12 2
175.3.d.f 2 35.i odd 6 1
175.3.d.f 2 35.j even 6 1
245.3.h.b 4 1.a even 1 1 trivial
245.3.h.b 4 7.b odd 2 1 inner
245.3.h.b 4 7.c even 3 1 inner
245.3.h.b 4 7.d odd 6 1 inner
315.3.h.b 2 21.g even 6 1
315.3.h.b 2 21.h odd 6 1
560.3.f.a 2 28.f even 6 1
560.3.f.a 2 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(245, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 20T^{2} + 400 \) Copy content Toggle raw display
$5$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 720 T^{2} + 518400 \) Copy content Toggle raw display
$19$ \( T^{4} - 180 T^{2} + 32400 \) Copy content Toggle raw display
$23$ \( (T^{2} + 26 T + 676)^{2} \) Copy content Toggle raw display
$29$ \( (T + 22)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 2880 T^{2} + \cdots + 8294400 \) Copy content Toggle raw display
$37$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 720)^{2} \) Copy content Toggle raw display
$43$ \( (T + 34)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 720 T^{2} + 518400 \) Copy content Toggle raw display
$53$ \( (T^{2} - 34 T + 1156)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 1620 T^{2} + \cdots + 2624400 \) Copy content Toggle raw display
$61$ \( T^{4} - 8820 T^{2} + \cdots + 77792400 \) Copy content Toggle raw display
$67$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$71$ \( (T - 62)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 2880 T^{2} + \cdots + 8294400 \) Copy content Toggle raw display
$79$ \( (T^{2} + 38 T + 1444)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1620)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 720 T^{2} + 518400 \) Copy content Toggle raw display
$97$ \( (T^{2} + 720)^{2} \) Copy content Toggle raw display
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