Properties

Label 245.4.e.f
Level $245$
Weight $4$
Character orbit 245.e
Analytic conductor $14.455$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} + ( -8 + 8 \zeta_{6} ) q^{4} + 5 \zeta_{6} q^{5} -8 q^{6} + 23 \zeta_{6} q^{9} +O(q^{10})\) \( q + 4 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} + ( -8 + 8 \zeta_{6} ) q^{4} + 5 \zeta_{6} q^{5} -8 q^{6} + 23 \zeta_{6} q^{9} + ( -20 + 20 \zeta_{6} ) q^{10} + ( -32 + 32 \zeta_{6} ) q^{11} -16 \zeta_{6} q^{12} -38 q^{13} -10 q^{15} + 64 \zeta_{6} q^{16} + ( -26 + 26 \zeta_{6} ) q^{17} + ( -92 + 92 \zeta_{6} ) q^{18} -100 \zeta_{6} q^{19} -40 q^{20} -128 q^{22} + 78 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} -152 \zeta_{6} q^{26} -100 q^{27} -50 q^{29} -40 \zeta_{6} q^{30} + ( 108 - 108 \zeta_{6} ) q^{31} + ( -256 + 256 \zeta_{6} ) q^{32} -64 \zeta_{6} q^{33} -104 q^{34} -184 q^{36} -266 \zeta_{6} q^{37} + ( 400 - 400 \zeta_{6} ) q^{38} + ( 76 - 76 \zeta_{6} ) q^{39} + 22 q^{41} + 442 q^{43} -256 \zeta_{6} q^{44} + ( -115 + 115 \zeta_{6} ) q^{45} + ( -312 + 312 \zeta_{6} ) q^{46} + 514 \zeta_{6} q^{47} -128 q^{48} -100 q^{50} -52 \zeta_{6} q^{51} + ( 304 - 304 \zeta_{6} ) q^{52} + ( -2 + 2 \zeta_{6} ) q^{53} -400 \zeta_{6} q^{54} -160 q^{55} + 200 q^{57} -200 \zeta_{6} q^{58} + ( -500 + 500 \zeta_{6} ) q^{59} + ( 80 - 80 \zeta_{6} ) q^{60} + 518 \zeta_{6} q^{61} + 432 q^{62} -512 q^{64} -190 \zeta_{6} q^{65} + ( 256 - 256 \zeta_{6} ) q^{66} + ( -126 + 126 \zeta_{6} ) q^{67} -208 \zeta_{6} q^{68} -156 q^{69} + 412 q^{71} + ( 878 - 878 \zeta_{6} ) q^{73} + ( 1064 - 1064 \zeta_{6} ) q^{74} -50 \zeta_{6} q^{75} + 800 q^{76} + 304 q^{78} -600 \zeta_{6} q^{79} + ( -320 + 320 \zeta_{6} ) q^{80} + ( -421 + 421 \zeta_{6} ) q^{81} + 88 \zeta_{6} q^{82} + 282 q^{83} -130 q^{85} + 1768 \zeta_{6} q^{86} + ( 100 - 100 \zeta_{6} ) q^{87} + 150 \zeta_{6} q^{89} -460 q^{90} -624 q^{92} + 216 \zeta_{6} q^{93} + ( -2056 + 2056 \zeta_{6} ) q^{94} + ( 500 - 500 \zeta_{6} ) q^{95} -512 \zeta_{6} q^{96} + 386 q^{97} -736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 2 q^{3} - 8 q^{4} + 5 q^{5} - 16 q^{6} + 23 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{2} - 2 q^{3} - 8 q^{4} + 5 q^{5} - 16 q^{6} + 23 q^{9} - 20 q^{10} - 32 q^{11} - 16 q^{12} - 76 q^{13} - 20 q^{15} + 64 q^{16} - 26 q^{17} - 92 q^{18} - 100 q^{19} - 80 q^{20} - 256 q^{22} + 78 q^{23} - 25 q^{25} - 152 q^{26} - 200 q^{27} - 100 q^{29} - 40 q^{30} + 108 q^{31} - 256 q^{32} - 64 q^{33} - 208 q^{34} - 368 q^{36} - 266 q^{37} + 400 q^{38} + 76 q^{39} + 44 q^{41} + 884 q^{43} - 256 q^{44} - 115 q^{45} - 312 q^{46} + 514 q^{47} - 256 q^{48} - 200 q^{50} - 52 q^{51} + 304 q^{52} - 2 q^{53} - 400 q^{54} - 320 q^{55} + 400 q^{57} - 200 q^{58} - 500 q^{59} + 80 q^{60} + 518 q^{61} + 864 q^{62} - 1024 q^{64} - 190 q^{65} + 256 q^{66} - 126 q^{67} - 208 q^{68} - 312 q^{69} + 824 q^{71} + 878 q^{73} + 1064 q^{74} - 50 q^{75} + 1600 q^{76} + 608 q^{78} - 600 q^{79} - 320 q^{80} - 421 q^{81} + 88 q^{82} + 564 q^{83} - 260 q^{85} + 1768 q^{86} + 100 q^{87} + 150 q^{89} - 920 q^{90} - 1248 q^{92} + 216 q^{93} - 2056 q^{94} + 500 q^{95} - 512 q^{96} + 772 q^{97} - 1472 q^{99} + O(q^{100}) \)

Character values

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

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

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
0.500000 + 0.866025i
0.500000 0.866025i
2.00000 + 3.46410i −1.00000 + 1.73205i −4.00000 + 6.92820i 2.50000 + 4.33013i −8.00000 0 0 11.5000 + 19.9186i −10.0000 + 17.3205i
226.1 2.00000 3.46410i −1.00000 1.73205i −4.00000 6.92820i 2.50000 4.33013i −8.00000 0 0 11.5000 19.9186i −10.0000 17.3205i
\(n\): 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 245.4.e.f 2
7.b odd 2 1 245.4.e.g 2
7.c even 3 1 5.4.a.a 1
7.c even 3 1 inner 245.4.e.f 2
7.d odd 6 1 245.4.a.a 1
7.d odd 6 1 245.4.e.g 2
21.g even 6 1 2205.4.a.q 1
21.h odd 6 1 45.4.a.d 1
28.g odd 6 1 80.4.a.d 1
35.i odd 6 1 1225.4.a.k 1
35.j even 6 1 25.4.a.c 1
35.l odd 12 2 25.4.b.a 2
56.k odd 6 1 320.4.a.h 1
56.p even 6 1 320.4.a.g 1
63.g even 3 1 405.4.e.l 2
63.h even 3 1 405.4.e.l 2
63.j odd 6 1 405.4.e.c 2
63.n odd 6 1 405.4.e.c 2
77.h odd 6 1 605.4.a.d 1
84.n even 6 1 720.4.a.u 1
91.r even 6 1 845.4.a.b 1
105.o odd 6 1 225.4.a.b 1
105.x even 12 2 225.4.b.c 2
112.u odd 12 2 1280.4.d.l 2
112.w even 12 2 1280.4.d.e 2
119.j even 6 1 1445.4.a.a 1
133.r odd 6 1 1805.4.a.h 1
140.p odd 6 1 400.4.a.m 1
140.w even 12 2 400.4.c.k 2
280.bf even 6 1 1600.4.a.bi 1
280.bi odd 6 1 1600.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 7.c even 3 1
25.4.a.c 1 35.j even 6 1
25.4.b.a 2 35.l odd 12 2
45.4.a.d 1 21.h odd 6 1
80.4.a.d 1 28.g odd 6 1
225.4.a.b 1 105.o odd 6 1
225.4.b.c 2 105.x even 12 2
245.4.a.a 1 7.d odd 6 1
245.4.e.f 2 1.a even 1 1 trivial
245.4.e.f 2 7.c even 3 1 inner
245.4.e.g 2 7.b odd 2 1
245.4.e.g 2 7.d odd 6 1
320.4.a.g 1 56.p even 6 1
320.4.a.h 1 56.k odd 6 1
400.4.a.m 1 140.p odd 6 1
400.4.c.k 2 140.w even 12 2
405.4.e.c 2 63.j odd 6 1
405.4.e.c 2 63.n odd 6 1
405.4.e.l 2 63.g even 3 1
405.4.e.l 2 63.h even 3 1
605.4.a.d 1 77.h odd 6 1
720.4.a.u 1 84.n even 6 1
845.4.a.b 1 91.r even 6 1
1225.4.a.k 1 35.i odd 6 1
1280.4.d.e 2 112.w even 12 2
1280.4.d.l 2 112.u odd 12 2
1445.4.a.a 1 119.j even 6 1
1600.4.a.s 1 280.bi odd 6 1
1600.4.a.bi 1 280.bf even 6 1
1805.4.a.h 1 133.r odd 6 1
2205.4.a.q 1 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{2} - 4 T_{2} + 16 \)
\( T_{3}^{2} + 2 T_{3} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 4 T + T^{2} \)
$3$ \( 4 + 2 T + T^{2} \)
$5$ \( 25 - 5 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 1024 + 32 T + T^{2} \)
$13$ \( ( 38 + T )^{2} \)
$17$ \( 676 + 26 T + T^{2} \)
$19$ \( 10000 + 100 T + T^{2} \)
$23$ \( 6084 - 78 T + T^{2} \)
$29$ \( ( 50 + T )^{2} \)
$31$ \( 11664 - 108 T + T^{2} \)
$37$ \( 70756 + 266 T + T^{2} \)
$41$ \( ( -22 + T )^{2} \)
$43$ \( ( -442 + T )^{2} \)
$47$ \( 264196 - 514 T + T^{2} \)
$53$ \( 4 + 2 T + T^{2} \)
$59$ \( 250000 + 500 T + T^{2} \)
$61$ \( 268324 - 518 T + T^{2} \)
$67$ \( 15876 + 126 T + T^{2} \)
$71$ \( ( -412 + T )^{2} \)
$73$ \( 770884 - 878 T + T^{2} \)
$79$ \( 360000 + 600 T + T^{2} \)
$83$ \( ( -282 + T )^{2} \)
$89$ \( 22500 - 150 T + T^{2} \)
$97$ \( ( -386 + T )^{2} \)
show more
show less