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: \( x^{2} - 2 \) Copy content Toggle raw display
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.

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

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{2} - 8T_{2} + 14 \) Copy content Toggle raw display
\( T_{3}^{2} + 2T_{3} - 31 \) Copy content Toggle raw display

Hecke characteristic polynomials

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