Properties

Label 245.2.f.b
Level 245
Weight 2
Character orbit 245.f
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.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} ) q^{4} + ( -2 + \zeta_{12}^{3} ) q^{5} + ( 1 - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( -1 - \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{8} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} ) q^{4} + ( -2 + \zeta_{12}^{3} ) q^{5} + ( 1 - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( -1 - \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{8} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{9} + ( -2 + 3 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{10} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{11} + ( -1 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{12} + ( -2 - 2 \zeta_{12}^{3} ) q^{13} + ( -2 + 3 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{15} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{16} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{17} + ( -1 + \zeta_{12}^{3} ) q^{18} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{19} + ( -2 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{20} + ( 1 + \zeta_{12}^{3} ) q^{22} + ( -1 + 3 \zeta_{12} - 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{23} + q^{24} + ( 3 - 4 \zeta_{12}^{3} ) q^{25} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{26} + ( -1 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{27} -3 \zeta_{12}^{3} q^{29} + ( -4 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{30} + ( 2 - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{31} + ( 5 - \zeta_{12} - \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{32} + ( 1 + \zeta_{12}^{3} ) q^{33} + 2 q^{34} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{36} + ( -2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{37} + ( 3 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{38} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{39} + ( 2 + \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{40} + ( 2 - 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{41} + ( 1 + 5 \zeta_{12} - 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{43} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{44} + ( -1 + 2 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{45} + ( -7 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{46} + ( 5 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{47} + ( 5 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{48} + ( 3 - 7 \zeta_{12} + \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{50} + 2 q^{51} + ( -2 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{52} + ( -5 + 5 \zeta_{12}^{3} ) q^{53} + ( 5 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{54} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{55} + ( 3 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{57} + ( -3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{58} + ( 3 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{59} + ( -\zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{60} + ( -2 + 4 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{61} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{62} + ( 1 - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{64} + ( 6 + 2 \zeta_{12}^{3} ) q^{65} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{66} + ( 1 + 5 \zeta_{12} + 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{67} + ( 2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{68} + ( -7 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{69} + ( 3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{71} + ( 1 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{72} + ( -4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{73} + ( -2 + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{74} + ( 3 - 7 \zeta_{12} + \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{75} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{76} + ( 2 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{78} + ( -1 + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{79} + ( 8 \zeta_{12} - 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{80} + ( 2 - 10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{81} + ( -2 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{82} + ( 2 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{83} + ( 4 + 2 \zeta_{12} - 6 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{85} + ( -9 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{86} + ( -3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{87} + ( 3 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{88} + ( -8 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{89} + ( 1 - 3 \zeta_{12}^{3} ) q^{90} + ( -7 + 5 \zeta_{12} + 5 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{92} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{93} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{94} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{95} + ( 5 - 10 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{96} + ( -3 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{97} + ( -2 + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 2q^{3} - 8q^{5} - 2q^{8} + O(q^{10}) \) \( 4q + 2q^{2} + 2q^{3} - 8q^{5} - 2q^{8} - 6q^{10} - 4q^{11} - 6q^{12} - 8q^{13} - 6q^{15} + 4q^{16} - 4q^{17} - 4q^{18} + 4q^{19} + 4q^{22} - 10q^{23} + 4q^{24} + 12q^{25} + 2q^{27} - 8q^{30} + 18q^{32} + 4q^{33} + 8q^{34} - 12q^{36} + 8q^{38} + 2q^{40} - 6q^{43} + 4q^{45} - 28q^{46} + 12q^{47} + 14q^{48} + 14q^{50} + 8q^{51} - 20q^{53} + 20q^{54} + 8q^{55} + 8q^{57} + 6q^{58} + 12q^{59} + 6q^{60} - 4q^{62} + 24q^{65} + 14q^{67} + 12q^{68} - 28q^{69} + 12q^{71} + 8q^{72} - 24q^{73} + 14q^{75} + 16q^{78} - 8q^{80} + 8q^{81} - 6q^{82} + 2q^{83} + 4q^{85} - 36q^{86} + 6q^{87} + 8q^{88} - 32q^{89} + 4q^{90} - 18q^{92} - 4q^{93} - 12q^{94} - 8q^{95} + 4q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
48.1
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.366025 + 0.366025i −0.366025 + 0.366025i 1.73205i −2.00000 1.00000i 0.267949i 0 −1.36603 1.36603i 2.73205i 1.09808 0.366025i
48.2 1.36603 1.36603i 1.36603 1.36603i 1.73205i −2.00000 1.00000i 3.73205i 0 0.366025 + 0.366025i 0.732051i −4.09808 + 1.36603i
97.1 −0.366025 0.366025i −0.366025 0.366025i 1.73205i −2.00000 + 1.00000i 0.267949i 0 −1.36603 + 1.36603i 2.73205i 1.09808 + 0.366025i
97.2 1.36603 + 1.36603i 1.36603 + 1.36603i 1.73205i −2.00000 + 1.00000i 3.73205i 0 0.366025 0.366025i 0.732051i −4.09808 1.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.f even 4 1 inner

Twists

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

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 2 T^{2} - 2 T^{3} + T^{4} - 4 T^{5} + 8 T^{6} - 16 T^{7} + 16 T^{8} \)
$3$ \( 1 - 2 T + 2 T^{2} - 4 T^{3} + 7 T^{4} - 12 T^{5} + 18 T^{6} - 54 T^{7} + 81 T^{8} \)
$5$ \( ( 1 + 4 T + 5 T^{2} )^{2} \)
$7$ 1
$11$ \( ( 1 + 2 T + 20 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 4 T + 8 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \)
$17$ \( 1 + 4 T + 8 T^{2} + 52 T^{3} + 322 T^{4} + 884 T^{5} + 2312 T^{6} + 19652 T^{7} + 83521 T^{8} \)
$19$ \( ( 1 - 2 T + 36 T^{2} - 38 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 10 T + 50 T^{2} + 220 T^{3} + 967 T^{4} + 5060 T^{5} + 26450 T^{6} + 121670 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \)
$31$ \( 1 - 68 T^{2} + 2310 T^{4} - 65348 T^{6} + 923521 T^{8} \)
$37$ \( 1 - 238 T^{4} + 1874161 T^{8} \)
$41$ \( 1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8} \)
$43$ \( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 2580 T^{5} + 33282 T^{6} + 477042 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 12 T + 72 T^{2} - 492 T^{3} + 3326 T^{4} - 23124 T^{5} + 159048 T^{6} - 1245876 T^{7} + 4879681 T^{8} \)
$53$ \( ( 1 - 4 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 6 T + 100 T^{2} - 354 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( 1 - 170 T^{2} + 13467 T^{4} - 632570 T^{6} + 13845841 T^{8} \)
$67$ \( 1 - 14 T + 98 T^{2} - 756 T^{3} + 5663 T^{4} - 50652 T^{5} + 439922 T^{6} - 4210682 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 + 24 T + 288 T^{2} + 2904 T^{3} + 26978 T^{4} + 211992 T^{5} + 1534752 T^{6} + 9336408 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 260 T^{2} + 29082 T^{4} - 1622660 T^{6} + 38950081 T^{8} \)
$83$ \( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 11620 T^{5} + 13778 T^{6} - 1143574 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 1164 T^{5} + 75272 T^{6} - 3650692 T^{7} + 88529281 T^{8} \)
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