Properties

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

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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.

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

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.g 2
7.b odd 2 1 245.4.e.f 2
7.c even 3 1 245.4.a.a 1
7.c even 3 1 inner 245.4.e.g 2
7.d odd 6 1 5.4.a.a 1
7.d odd 6 1 245.4.e.f 2
21.g even 6 1 45.4.a.d 1
21.h odd 6 1 2205.4.a.q 1
28.f even 6 1 80.4.a.d 1
35.i odd 6 1 25.4.a.c 1
35.j even 6 1 1225.4.a.k 1
35.k even 12 2 25.4.b.a 2
56.j odd 6 1 320.4.a.g 1
56.m even 6 1 320.4.a.h 1
63.i even 6 1 405.4.e.c 2
63.k odd 6 1 405.4.e.l 2
63.s even 6 1 405.4.e.c 2
63.t odd 6 1 405.4.e.l 2
77.i even 6 1 605.4.a.d 1
84.j odd 6 1 720.4.a.u 1
91.s odd 6 1 845.4.a.b 1
105.p even 6 1 225.4.a.b 1
105.w odd 12 2 225.4.b.c 2
112.v even 12 2 1280.4.d.l 2
112.x odd 12 2 1280.4.d.e 2
119.h odd 6 1 1445.4.a.a 1
133.o even 6 1 1805.4.a.h 1
140.s even 6 1 400.4.a.m 1
140.x odd 12 2 400.4.c.k 2
280.ba even 6 1 1600.4.a.s 1
280.bk odd 6 1 1600.4.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 7.d odd 6 1
25.4.a.c 1 35.i odd 6 1
25.4.b.a 2 35.k even 12 2
45.4.a.d 1 21.g even 6 1
80.4.a.d 1 28.f even 6 1
225.4.a.b 1 105.p even 6 1
225.4.b.c 2 105.w odd 12 2
245.4.a.a 1 7.c even 3 1
245.4.e.f 2 7.b odd 2 1
245.4.e.f 2 7.d odd 6 1
245.4.e.g 2 1.a even 1 1 trivial
245.4.e.g 2 7.c even 3 1 inner
320.4.a.g 1 56.j odd 6 1
320.4.a.h 1 56.m even 6 1
400.4.a.m 1 140.s even 6 1
400.4.c.k 2 140.x odd 12 2
405.4.e.c 2 63.i even 6 1
405.4.e.c 2 63.s even 6 1
405.4.e.l 2 63.k odd 6 1
405.4.e.l 2 63.t odd 6 1
605.4.a.d 1 77.i even 6 1
720.4.a.u 1 84.j odd 6 1
845.4.a.b 1 91.s odd 6 1
1225.4.a.k 1 35.j even 6 1
1280.4.d.e 2 112.x odd 12 2
1280.4.d.l 2 112.v even 12 2
1445.4.a.a 1 119.h odd 6 1
1600.4.a.s 1 280.ba even 6 1
1600.4.a.bi 1 280.bk odd 6 1
1805.4.a.h 1 133.o even 6 1
2205.4.a.q 1 21.h odd 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 8 T^{2} - 32 T^{3} + 64 T^{4} \)
$3$ \( 1 - 2 T - 23 T^{2} - 54 T^{3} + 729 T^{4} \)
$5$ \( 1 + 5 T + 25 T^{2} \)
$7$ 1
$11$ \( 1 + 32 T - 307 T^{2} + 42592 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 - 38 T + 2197 T^{2} )^{2} \)
$17$ \( 1 - 26 T - 4237 T^{2} - 127738 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 100 T + 3141 T^{2} - 685900 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 78 T - 6083 T^{2} - 949026 T^{3} + 148035889 T^{4} \)
$29$ \( ( 1 + 50 T + 24389 T^{2} )^{2} \)
$31$ \( 1 + 108 T - 18127 T^{2} + 3217428 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 266 T + 20103 T^{2} + 13473698 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 + 22 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 - 442 T + 79507 T^{2} )^{2} \)
$47$ \( 1 + 514 T + 160373 T^{2} + 53365022 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 2 T - 148873 T^{2} + 297754 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 500 T + 44621 T^{2} - 102689500 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 518 T + 41343 T^{2} + 117576158 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 126 T - 284887 T^{2} + 37896138 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 - 412 T + 357911 T^{2} )^{2} \)
$73$ \( 1 + 878 T + 381867 T^{2} + 341556926 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 600 T - 133039 T^{2} + 295823400 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 + 282 T + 571787 T^{2} )^{2} \)
$89$ \( 1 + 150 T - 682469 T^{2} + 105745350 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 + 386 T + 912673 T^{2} )^{2} \)
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