Properties

Label 245.2.l.c
Level 245
Weight 2
Character orbit 245.l
Analytic conductor 1.956
Analytic rank 0
Dimension 8
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.l (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.3317760000.2
Defining polynomial: \(x^{8} - 25 x^{4} + 625\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{2} - \beta_{4} ) q^{2} + \beta_{1} q^{3} + \beta_{5} q^{5} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{6} + ( -2 - 2 \beta_{6} ) q^{8} + 2 \beta_{2} q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{2} - \beta_{4} ) q^{2} + \beta_{1} q^{3} + \beta_{5} q^{5} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{6} + ( -2 - 2 \beta_{6} ) q^{8} + 2 \beta_{2} q^{9} + ( \beta_{1} - \beta_{7} ) q^{10} + \beta_{4} q^{11} + ( \beta_{1} - \beta_{5} ) q^{13} + 5 \beta_{6} q^{15} + ( -4 + 4 \beta_{4} ) q^{16} + \beta_{7} q^{17} + ( 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{18} + ( \beta_{3} - \beta_{5} - \beta_{7} ) q^{19} + ( 1 - \beta_{6} ) q^{22} + ( -2 - 2 \beta_{2} + 2 \beta_{4} ) q^{23} + ( -2 \beta_{1} - 2 \beta_{7} ) q^{24} + ( -5 \beta_{2} + 5 \beta_{6} ) q^{25} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{26} -\beta_{3} q^{27} -3 \beta_{6} q^{29} + ( 5 + 5 \beta_{2} - 5 \beta_{4} ) q^{30} + ( -\beta_{1} + \beta_{7} ) q^{31} + \beta_{5} q^{33} + ( \beta_{1} + \beta_{3} - \beta_{5} ) q^{34} + ( 6 - 6 \beta_{2} - 6 \beta_{4} ) q^{37} -2 \beta_{1} q^{38} + ( 5 \beta_{2} - 5 \beta_{6} ) q^{39} + ( 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{40} + ( -3 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} ) q^{41} + ( -3 - 3 \beta_{6} ) q^{43} + 2 \beta_{7} q^{45} + 4 \beta_{4} q^{46} + ( -3 \beta_{3} + 3 \beta_{7} ) q^{47} + ( -4 \beta_{1} + 4 \beta_{5} ) q^{48} + ( 5 + 5 \beta_{6} ) q^{50} + ( -5 + 5 \beta_{4} ) q^{51} + ( \beta_{2} - \beta_{4} - \beta_{6} ) q^{53} + ( -\beta_{3} + \beta_{5} + \beta_{7} ) q^{54} + ( -\beta_{1} + \beta_{5} ) q^{55} + ( 5 - 5 \beta_{6} ) q^{57} + ( -3 - 3 \beta_{2} + 3 \beta_{4} ) q^{58} + ( 3 \beta_{1} + 3 \beta_{7} ) q^{59} + ( 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{61} + 2 \beta_{3} q^{62} + 8 \beta_{6} q^{64} + 5 \beta_{2} q^{65} + ( \beta_{1} - \beta_{7} ) q^{66} + ( \beta_{2} + \beta_{4} - \beta_{6} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{69} -6 q^{71} + ( 4 - 4 \beta_{2} - 4 \beta_{4} ) q^{72} + ( -12 \beta_{2} + 12 \beta_{6} ) q^{74} + ( -5 \beta_{3} + 5 \beta_{7} ) q^{75} + ( -5 - 5 \beta_{6} ) q^{78} + 13 \beta_{2} q^{79} -4 \beta_{1} q^{80} -11 \beta_{4} q^{81} + ( 6 \beta_{3} - 6 \beta_{7} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{83} -5 q^{85} + ( -6 + 6 \beta_{4} ) q^{86} -3 \beta_{7} q^{87} + ( 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{88} + ( -2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{89} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{90} + ( -5 - 5 \beta_{2} + 5 \beta_{4} ) q^{93} + ( 3 \beta_{1} + 3 \beta_{7} ) q^{94} + ( 5 \beta_{2} + 5 \beta_{4} - 5 \beta_{6} ) q^{95} + \beta_{3} q^{97} + 2 \beta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{2} - 16q^{8} + O(q^{10}) \) \( 8q + 4q^{2} - 16q^{8} + 4q^{11} - 16q^{16} - 8q^{18} + 8q^{22} - 8q^{23} + 20q^{30} + 24q^{37} - 24q^{43} + 16q^{46} + 40q^{50} - 20q^{51} - 4q^{53} + 40q^{57} - 12q^{58} + 4q^{67} - 48q^{71} + 16q^{72} - 40q^{78} - 44q^{81} - 40q^{85} - 24q^{86} - 8q^{88} - 20q^{93} + 20q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 25 x^{4} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/25\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/25\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/125\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/125\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)
\(\nu^{4}\)\(=\)\(25 \beta_{4}\)
\(\nu^{5}\)\(=\)\(25 \beta_{5}\)
\(\nu^{6}\)\(=\)\(125 \beta_{6}\)
\(\nu^{7}\)\(=\)\(125 \beta_{7}\)

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1 - \beta_{4}\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
−2.15988 0.578737i
2.15988 + 0.578737i
−0.578737 + 2.15988i
0.578737 2.15988i
−0.578737 2.15988i
0.578737 + 2.15988i
−2.15988 + 0.578737i
2.15988 0.578737i
−0.366025 1.36603i −2.15988 0.578737i 0 −0.578737 2.15988i 3.16228i 0 −2.00000 2.00000i 1.73205 + 1.00000i −2.73861 + 1.58114i
68.2 −0.366025 1.36603i 2.15988 + 0.578737i 0 0.578737 + 2.15988i 3.16228i 0 −2.00000 2.00000i 1.73205 + 1.00000i 2.73861 1.58114i
117.1 1.36603 0.366025i −0.578737 + 2.15988i 0 −2.15988 + 0.578737i 3.16228i 0 −2.00000 + 2.00000i −1.73205 1.00000i −2.73861 + 1.58114i
117.2 1.36603 0.366025i 0.578737 2.15988i 0 2.15988 0.578737i 3.16228i 0 −2.00000 + 2.00000i −1.73205 1.00000i 2.73861 1.58114i
178.1 1.36603 + 0.366025i −0.578737 2.15988i 0 −2.15988 0.578737i 3.16228i 0 −2.00000 2.00000i −1.73205 + 1.00000i −2.73861 1.58114i
178.2 1.36603 + 0.366025i 0.578737 + 2.15988i 0 2.15988 + 0.578737i 3.16228i 0 −2.00000 2.00000i −1.73205 + 1.00000i 2.73861 + 1.58114i
227.1 −0.366025 + 1.36603i −2.15988 + 0.578737i 0 −0.578737 + 2.15988i 3.16228i 0 −2.00000 + 2.00000i 1.73205 1.00000i −2.73861 1.58114i
227.2 −0.366025 + 1.36603i 2.15988 0.578737i 0 0.578737 2.15988i 3.16228i 0 −2.00000 + 2.00000i 1.73205 1.00000i 2.73861 + 1.58114i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 227.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.l.c 8
5.c odd 4 1 inner 245.2.l.c 8
7.b odd 2 1 inner 245.2.l.c 8
7.c even 3 1 35.2.f.a 4
7.c even 3 1 inner 245.2.l.c 8
7.d odd 6 1 35.2.f.a 4
7.d odd 6 1 inner 245.2.l.c 8
21.g even 6 1 315.2.p.c 4
21.h odd 6 1 315.2.p.c 4
28.f even 6 1 560.2.bj.a 4
28.g odd 6 1 560.2.bj.a 4
35.f even 4 1 inner 245.2.l.c 8
35.i odd 6 1 175.2.f.c 4
35.j even 6 1 175.2.f.c 4
35.k even 12 1 35.2.f.a 4
35.k even 12 1 175.2.f.c 4
35.k even 12 1 inner 245.2.l.c 8
35.l odd 12 1 35.2.f.a 4
35.l odd 12 1 175.2.f.c 4
35.l odd 12 1 inner 245.2.l.c 8
105.w odd 12 1 315.2.p.c 4
105.x even 12 1 315.2.p.c 4
140.w even 12 1 560.2.bj.a 4
140.x odd 12 1 560.2.bj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.f.a 4 7.c even 3 1
35.2.f.a 4 7.d odd 6 1
35.2.f.a 4 35.k even 12 1
35.2.f.a 4 35.l odd 12 1
175.2.f.c 4 35.i odd 6 1
175.2.f.c 4 35.j even 6 1
175.2.f.c 4 35.k even 12 1
175.2.f.c 4 35.l odd 12 1
245.2.l.c 8 1.a even 1 1 trivial
245.2.l.c 8 5.c odd 4 1 inner
245.2.l.c 8 7.b odd 2 1 inner
245.2.l.c 8 7.c even 3 1 inner
245.2.l.c 8 7.d odd 6 1 inner
245.2.l.c 8 35.f even 4 1 inner
245.2.l.c 8 35.k even 12 1 inner
245.2.l.c 8 35.l odd 12 1 inner
315.2.p.c 4 21.g even 6 1
315.2.p.c 4 21.h odd 6 1
315.2.p.c 4 105.w odd 12 1
315.2.p.c 4 105.x even 12 1
560.2.bj.a 4 28.f even 6 1
560.2.bj.a 4 28.g odd 6 1
560.2.bj.a 4 140.w even 12 1
560.2.bj.a 4 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2 T_{2}^{3} + 2 T_{2}^{2} - 4 T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 2 T^{2} )^{4}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2} \)
$3$ \( 1 + 17 T^{4} + 208 T^{8} + 1377 T^{12} + 6561 T^{16} \)
$5$ \( 1 - 25 T^{4} + 625 T^{8} \)
$7$ 1
$11$ \( ( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} )^{4} \)
$13$ \( ( 1 + 103 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( 1 - 263 T^{4} - 14352 T^{8} - 21966023 T^{12} + 6975757441 T^{16} \)
$19$ \( ( 1 - 28 T^{2} + 423 T^{4} - 10108 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 4 T + 8 T^{2} - 152 T^{3} - 833 T^{4} - 3496 T^{5} + 4232 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 49 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 + 52 T^{2} + 1743 T^{4} + 49972 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 12 T + 72 T^{2} + 24 T^{3} - 1513 T^{4} + 888 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 + 8 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 + 6 T + 18 T^{2} + 258 T^{3} + 1849 T^{4} )^{4} \)
$47$ \( 1 + 2017 T^{4} - 811392 T^{8} + 9842316577 T^{12} + 23811286661761 T^{16} \)
$53$ \( ( 1 + 2 T + 2 T^{2} - 208 T^{3} - 3017 T^{4} - 11024 T^{5} + 5618 T^{6} + 297754 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 28 T^{2} - 2697 T^{4} - 97468 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 82 T^{2} + 3003 T^{4} + 305122 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 2 T + 2 T^{2} + 264 T^{3} - 4753 T^{4} + 17688 T^{5} + 8978 T^{6} - 601526 T^{7} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{8} \)
$73$ \( ( 1 - 5329 T^{4} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 142 T^{2} + 6241 T^{4} )^{2}( 1 + 131 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 7538 T^{4} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 138 T^{2} + 11123 T^{4} - 1093098 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 16903 T^{4} + 88529281 T^{8} )^{2} \)
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