Properties

Label 245.4.a.k
Level $245$
Weight $4$
Character orbit 245.a
Self dual yes
Analytic conductor $14.455$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 + \beta ) q^{2} + ( -1 + 4 \beta ) q^{3} + ( 10 + 8 \beta ) q^{4} + 5 q^{5} + ( 4 + 15 \beta ) q^{6} + ( 24 + 34 \beta ) q^{8} + ( 6 - 8 \beta ) q^{9} +O(q^{10})\) \( q + ( 4 + \beta ) q^{2} + ( -1 + 4 \beta ) q^{3} + ( 10 + 8 \beta ) q^{4} + 5 q^{5} + ( 4 + 15 \beta ) q^{6} + ( 24 + 34 \beta ) q^{8} + ( 6 - 8 \beta ) q^{9} + ( 20 + 5 \beta ) q^{10} + ( -7 - 32 \beta ) q^{11} + ( 54 + 32 \beta ) q^{12} + ( -25 - 4 \beta ) q^{13} + ( -5 + 20 \beta ) q^{15} + ( 84 + 96 \beta ) q^{16} + ( 25 - 44 \beta ) q^{17} + ( 8 - 26 \beta ) q^{18} + ( -18 - 44 \beta ) q^{19} + ( 50 + 40 \beta ) q^{20} + ( -92 - 135 \beta ) q^{22} + ( 122 - 68 \beta ) q^{23} + ( 248 + 62 \beta ) q^{24} + 25 q^{25} + ( -108 - 41 \beta ) q^{26} + ( -43 - 76 \beta ) q^{27} + ( -13 + 24 \beta ) q^{29} + ( 20 + 75 \beta ) q^{30} + ( 60 + 180 \beta ) q^{31} + ( 336 + 196 \beta ) q^{32} + ( -249 + 4 \beta ) q^{33} + ( 12 - 151 \beta ) q^{34} + ( -68 - 32 \beta ) q^{36} + ( 282 - 60 \beta ) q^{37} + ( -160 - 194 \beta ) q^{38} + ( -7 - 96 \beta ) q^{39} + ( 120 + 170 \beta ) q^{40} + ( 164 - 124 \beta ) q^{41} + ( -130 + 68 \beta ) q^{43} + ( -582 - 376 \beta ) q^{44} + ( 30 - 40 \beta ) q^{45} + ( 352 - 150 \beta ) q^{46} + ( 175 + 132 \beta ) q^{47} + ( 684 + 240 \beta ) q^{48} + ( 100 + 25 \beta ) q^{50} + ( -377 + 144 \beta ) q^{51} + ( -314 - 240 \beta ) q^{52} + ( -28 + 128 \beta ) q^{53} + ( -324 - 347 \beta ) q^{54} + ( -35 - 160 \beta ) q^{55} + ( -334 - 28 \beta ) q^{57} + ( -4 + 83 \beta ) q^{58} + 616 q^{59} + ( 270 + 160 \beta ) q^{60} + ( -168 + 108 \beta ) q^{61} + ( 600 + 780 \beta ) q^{62} + ( 1064 + 352 \beta ) q^{64} + ( -125 - 20 \beta ) q^{65} + ( -988 - 233 \beta ) q^{66} + ( -76 - 64 \beta ) q^{67} + ( -454 - 240 \beta ) q^{68} + ( -666 + 556 \beta ) q^{69} -952 q^{71} + ( -400 + 12 \beta ) q^{72} + ( -338 + 344 \beta ) q^{73} + ( 1008 + 42 \beta ) q^{74} + ( -25 + 100 \beta ) q^{75} + ( -884 - 584 \beta ) q^{76} + ( -220 - 391 \beta ) q^{78} + ( 507 + 248 \beta ) q^{79} + ( 420 + 480 \beta ) q^{80} + ( -727 + 120 \beta ) q^{81} + ( 408 - 332 \beta ) q^{82} + ( 188 - 600 \beta ) q^{83} + ( 125 - 220 \beta ) q^{85} + ( -384 + 142 \beta ) q^{86} + ( 205 - 76 \beta ) q^{87} + ( -2344 - 1006 \beta ) q^{88} + ( 108 - 44 \beta ) q^{89} + ( 40 - 130 \beta ) q^{90} + ( 132 + 296 \beta ) q^{92} + ( 1380 + 60 \beta ) q^{93} + ( 964 + 703 \beta ) q^{94} + ( -90 - 220 \beta ) q^{95} + ( 1232 + 1148 \beta ) q^{96} + ( -1371 - 220 \beta ) q^{97} + ( 470 - 136 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 2 q^{3} + 20 q^{4} + 10 q^{5} + 8 q^{6} + 48 q^{8} + 12 q^{9} + O(q^{10}) \) \( 2 q + 8 q^{2} - 2 q^{3} + 20 q^{4} + 10 q^{5} + 8 q^{6} + 48 q^{8} + 12 q^{9} + 40 q^{10} - 14 q^{11} + 108 q^{12} - 50 q^{13} - 10 q^{15} + 168 q^{16} + 50 q^{17} + 16 q^{18} - 36 q^{19} + 100 q^{20} - 184 q^{22} + 244 q^{23} + 496 q^{24} + 50 q^{25} - 216 q^{26} - 86 q^{27} - 26 q^{29} + 40 q^{30} + 120 q^{31} + 672 q^{32} - 498 q^{33} + 24 q^{34} - 136 q^{36} + 564 q^{37} - 320 q^{38} - 14 q^{39} + 240 q^{40} + 328 q^{41} - 260 q^{43} - 1164 q^{44} + 60 q^{45} + 704 q^{46} + 350 q^{47} + 1368 q^{48} + 200 q^{50} - 754 q^{51} - 628 q^{52} - 56 q^{53} - 648 q^{54} - 70 q^{55} - 668 q^{57} - 8 q^{58} + 1232 q^{59} + 540 q^{60} - 336 q^{61} + 1200 q^{62} + 2128 q^{64} - 250 q^{65} - 1976 q^{66} - 152 q^{67} - 908 q^{68} - 1332 q^{69} - 1904 q^{71} - 800 q^{72} - 676 q^{73} + 2016 q^{74} - 50 q^{75} - 1768 q^{76} - 440 q^{78} + 1014 q^{79} + 840 q^{80} - 1454 q^{81} + 816 q^{82} + 376 q^{83} + 250 q^{85} - 768 q^{86} + 410 q^{87} - 4688 q^{88} + 216 q^{89} + 80 q^{90} + 264 q^{92} + 2760 q^{93} + 1928 q^{94} - 180 q^{95} + 2464 q^{96} - 2742 q^{97} + 940 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
2.58579 −6.65685 −1.31371 5.00000 −17.2132 0 −24.0833 17.3137 12.9289
1.2 5.41421 4.65685 21.3137 5.00000 25.2132 0 72.0833 −5.31371 27.0711
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.a.k 2
3.b odd 2 1 2205.4.a.u 2
5.b even 2 1 1225.4.a.m 2
7.b odd 2 1 35.4.a.b 2
7.c even 3 2 245.4.e.i 4
7.d odd 6 2 245.4.e.h 4
21.c even 2 1 315.4.a.f 2
28.d even 2 1 560.4.a.r 2
35.c odd 2 1 175.4.a.c 2
35.f even 4 2 175.4.b.c 4
56.e even 2 1 2240.4.a.bo 2
56.h odd 2 1 2240.4.a.bn 2
105.g even 2 1 1575.4.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 7.b odd 2 1
175.4.a.c 2 35.c odd 2 1
175.4.b.c 4 35.f even 4 2
245.4.a.k 2 1.a even 1 1 trivial
245.4.e.h 4 7.d odd 6 2
245.4.e.i 4 7.c even 3 2
315.4.a.f 2 21.c even 2 1
560.4.a.r 2 28.d even 2 1
1225.4.a.m 2 5.b even 2 1
1575.4.a.z 2 105.g even 2 1
2205.4.a.u 2 3.b odd 2 1
2240.4.a.bn 2 56.h odd 2 1
2240.4.a.bo 2 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{2} - 8 T_{2} + 14 \)
\( T_{3}^{2} + 2 T_{3} - 31 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 14 - 8 T + T^{2} \)
$3$ \( -31 + 2 T + T^{2} \)
$5$ \( ( -5 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -1999 + 14 T + T^{2} \)
$13$ \( 593 + 50 T + T^{2} \)
$17$ \( -3247 - 50 T + T^{2} \)
$19$ \( -3548 + 36 T + T^{2} \)
$23$ \( 5636 - 244 T + T^{2} \)
$29$ \( -983 + 26 T + T^{2} \)
$31$ \( -61200 - 120 T + T^{2} \)
$37$ \( 72324 - 564 T + T^{2} \)
$41$ \( -3856 - 328 T + T^{2} \)
$43$ \( 7652 + 260 T + T^{2} \)
$47$ \( -4223 - 350 T + T^{2} \)
$53$ \( -31984 + 56 T + T^{2} \)
$59$ \( ( -616 + T )^{2} \)
$61$ \( 4896 + 336 T + T^{2} \)
$67$ \( -2416 + 152 T + T^{2} \)
$71$ \( ( 952 + T )^{2} \)
$73$ \( -122428 + 676 T + T^{2} \)
$79$ \( 134041 - 1014 T + T^{2} \)
$83$ \( -684656 - 376 T + T^{2} \)
$89$ \( 7792 - 216 T + T^{2} \)
$97$ \( 1782841 + 2742 T + T^{2} \)
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