Properties

Label 245.4.a.l
Level $245$
Weight $4$
Character orbit 245.a
Self dual yes
Analytic conductor $14.455$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.14360.1
Defining polynomial: \( x^{3} - 17x - 14 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} - 5 q^{5} + ( - 3 \beta_{2} + 4 \beta_1 - 7) q^{6} + ( - 3 \beta_{2} + \beta_1 - 4) q^{8} + ( - 3 \beta_{2} - 9 \beta_1 + 28) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} - 5 q^{5} + ( - 3 \beta_{2} + 4 \beta_1 - 7) q^{6} + ( - 3 \beta_{2} + \beta_1 - 4) q^{8} + ( - 3 \beta_{2} - 9 \beta_1 + 28) q^{9} + ( - 5 \beta_1 + 5) q^{10} + (\beta_{2} + 3 \beta_1 - 25) q^{11} + (2 \beta_{2} - 14 \beta_1 + 50) q^{12} + ( - 5 \beta_{2} + 13 \beta_1 - 13) q^{13} + ( - 5 \beta_{2} + 5 \beta_1 + 5) q^{15} + ( - \beta_{2} - 11 \beta_1 - 26) q^{16} + ( - 11 \beta_{2} - 13 \beta_1 + 21) q^{17} + ( - 3 \beta_{2} + 13 \beta_1 - 136) q^{18} + ( - 6 \beta_{2} + 10 \beta_1 - 54) q^{19} + ( - 5 \beta_{2} + 5 \beta_1 - 20) q^{20} + (\beta_{2} - 20 \beta_1 + 61) q^{22} + (2 \beta_{2} - 14 \beta_1 - 42) q^{23} + (6 \beta_{2} + 28 \beta_1 - 142) q^{24} + 25 q^{25} + (23 \beta_{2} - 38 \beta_1 + 141) q^{26} + (31 \beta_{2} - 7 \beta_1 - 67) q^{27} + (17 \beta_{2} + 19 \beta_1 + 105) q^{29} + (15 \beta_{2} - 20 \beta_1 + 35) q^{30} + (4 \beta_{2} - 24 \beta_1 - 108) q^{31} + (15 \beta_{2} - 39 \beta_1 - 66) q^{32} + ( - 35 \beta_{2} + 27 \beta_1 + 47) q^{33} + (9 \beta_{2} - 34 \beta_1 - 197) q^{34} + (43 \beta_{2} - 79 \beta_1 + 46) q^{36} + ( - 12 \beta_{2} + 16 \beta_1 - 14) q^{37} + (22 \beta_{2} - 84 \beta_1 + 146) q^{38} + ( - 19 \beta_{2} + 87 \beta_1 - 321) q^{39} + (15 \beta_{2} - 5 \beta_1 + 20) q^{40} + ( - 2 \beta_{2} - 10 \beta_1 - 120) q^{41} + ( - 34 \beta_{2} + 30 \beta_1 + 6) q^{43} + ( - 30 \beta_{2} + 42 \beta_1 - 78) q^{44} + (15 \beta_{2} + 45 \beta_1 - 140) q^{45} + ( - 18 \beta_{2} - 32 \beta_1 - 106) q^{46} + (13 \beta_{2} + 51 \beta_1 + 239) q^{47} + 68 q^{48} + (25 \beta_1 - 25) q^{50} + (91 \beta_{2} + 17 \beta_1 - 423) q^{51} + ( - 44 \beta_{2} + 152 \beta_1 - 386) q^{52} + ( - 22 \beta_{2} - 130 \beta_1 + 44) q^{53} + ( - 69 \beta_{2} + 88 \beta_1 + 83) q^{54} + ( - 5 \beta_{2} - 15 \beta_1 + 125) q^{55} + ( - 50 \beta_{2} + 126 \beta_1 - 302) q^{57} + ( - 15 \beta_{2} + 190 \beta_1 + 155) q^{58} + ( - 48 \beta_{2} + 176 \beta_1 + 76) q^{59} + ( - 10 \beta_{2} + 70 \beta_1 - 250) q^{60} + (26 \beta_{2} + 34 \beta_1 - 416) q^{61} + ( - 32 \beta_{2} - 88 \beta_1 - 144) q^{62} + ( - 61 \beta_{2} + 97 \beta_1 - 110) q^{64} + (25 \beta_{2} - 65 \beta_1 + 65) q^{65} + (97 \beta_{2} - 128 \beta_1 + 145) q^{66} + (108 \beta_{2} - 12 \beta_1 + 32) q^{67} + (36 \beta_{2} - 48 \beta_1 - 318) q^{68} + ( - 22 \beta_{2} - 14 \beta_1 + 246) q^{69} + ( - 40 \beta_{2} + 72 \beta_1 - 32) q^{71} + ( - 141 \beta_{2} + 157 \beta_1 + 302) q^{72} + ( - 76 \beta_{2} - 124 \beta_1 - 78) q^{73} + (40 \beta_{2} - 74 \beta_1 + 154) q^{74} + (25 \beta_{2} - 25 \beta_1 - 25) q^{75} + ( - 80 \beta_{2} + 176 \beta_1 - 572) q^{76} + (125 \beta_{2} - 416 \beta_1 + 1221) q^{78} + ( - 89 \beta_{2} - 83 \beta_1 - 315) q^{79} + (5 \beta_{2} + 55 \beta_1 + 130) q^{80} + ( - 96 \beta_{2} + 72 \beta_1 + 793) q^{81} + ( - 6 \beta_{2} - 130 \beta_1 + 4) q^{82} + ( - 8 \beta_{2} + 160 \beta_1 + 556) q^{83} + (55 \beta_{2} + 65 \beta_1 - 105) q^{85} + (98 \beta_{2} - 164 \beta_1 + 222) q^{86} + ( - \beta_{2} - 167 \beta_1 + 525) q^{87} + (94 \beta_{2} - 68 \beta_1 - 38) q^{88} + (82 \beta_{2} + 98 \beta_1 - 108) q^{89} + (15 \beta_{2} - 65 \beta_1 + 680) q^{90} + ( - 12 \beta_{2} - 84 \beta_1 + 36) q^{92} + ( - 76 \beta_{2} + 8 \beta_1 + 484) q^{93} + (25 \beta_{2} + 304 \beta_1 + 361) q^{94} + (30 \beta_{2} - 50 \beta_1 + 270) q^{95} + ( - 48 \beta_{2} - 156 \beta_1 + 1068) q^{96} + (65 \beta_{2} + 87 \beta_1 - 55) q^{97} + (106 \beta_{2} + 198 \beta_1 - 1198) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 2 q^{3} + 13 q^{4} - 15 q^{5} - 24 q^{6} - 15 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 2 q^{3} + 13 q^{4} - 15 q^{5} - 24 q^{6} - 15 q^{8} + 81 q^{9} + 15 q^{10} - 74 q^{11} + 152 q^{12} - 44 q^{13} + 10 q^{15} - 79 q^{16} + 52 q^{17} - 411 q^{18} - 168 q^{19} - 65 q^{20} + 184 q^{22} - 124 q^{23} - 420 q^{24} + 75 q^{25} + 446 q^{26} - 170 q^{27} + 332 q^{29} + 120 q^{30} - 320 q^{31} - 183 q^{32} + 106 q^{33} - 582 q^{34} + 181 q^{36} - 54 q^{37} + 460 q^{38} - 982 q^{39} + 75 q^{40} - 362 q^{41} - 16 q^{43} - 264 q^{44} - 405 q^{45} - 336 q^{46} + 730 q^{47} + 204 q^{48} - 75 q^{50} - 1178 q^{51} - 1202 q^{52} + 110 q^{53} + 180 q^{54} + 370 q^{55} - 956 q^{57} + 450 q^{58} + 180 q^{59} - 760 q^{60} - 1222 q^{61} - 464 q^{62} - 391 q^{64} + 220 q^{65} + 532 q^{66} + 204 q^{67} - 918 q^{68} + 716 q^{69} - 136 q^{71} + 765 q^{72} - 310 q^{73} + 502 q^{74} - 50 q^{75} - 1796 q^{76} + 3788 q^{78} - 1034 q^{79} + 395 q^{80} + 2283 q^{81} + 6 q^{82} + 1660 q^{83} - 260 q^{85} + 764 q^{86} + 1574 q^{87} - 20 q^{88} - 242 q^{89} + 2055 q^{90} + 96 q^{92} + 1376 q^{93} + 1108 q^{94} + 840 q^{95} + 3156 q^{96} - 100 q^{97} - 3488 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 17x - 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) 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
−3.62456
−0.861086
4.48565
−4.62456 8.38660 13.3866 −5.00000 −38.7844 0 −24.9107 43.3350 23.1228
1.2 −1.86109 −9.53636 −4.53636 −5.00000 17.7480 0 23.3312 63.9421 9.30543
1.3 3.48565 −0.850238 4.14976 −5.00000 −2.96363 0 −13.4206 −26.2771 −17.4283
\(n\): 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.l 3
3.b odd 2 1 2205.4.a.bm 3
5.b even 2 1 1225.4.a.y 3
7.b odd 2 1 35.4.a.c 3
7.c even 3 2 245.4.e.n 6
7.d odd 6 2 245.4.e.m 6
21.c even 2 1 315.4.a.p 3
28.d even 2 1 560.4.a.u 3
35.c odd 2 1 175.4.a.f 3
35.f even 4 2 175.4.b.e 6
56.e even 2 1 2240.4.a.bv 3
56.h odd 2 1 2240.4.a.bt 3
105.g even 2 1 1575.4.a.ba 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.c 3 7.b odd 2 1
175.4.a.f 3 35.c odd 2 1
175.4.b.e 6 35.f even 4 2
245.4.a.l 3 1.a even 1 1 trivial
245.4.e.m 6 7.d odd 6 2
245.4.e.n 6 7.c even 3 2
315.4.a.p 3 21.c even 2 1
560.4.a.u 3 28.d even 2 1
1225.4.a.y 3 5.b even 2 1
1575.4.a.ba 3 105.g even 2 1
2205.4.a.bm 3 3.b odd 2 1
2240.4.a.bt 3 56.h odd 2 1
2240.4.a.bv 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{3} + 3T_{2}^{2} - 14T_{2} - 30 \) Copy content Toggle raw display
\( T_{3}^{3} + 2T_{3}^{2} - 79T_{3} - 68 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3 T^{2} - 14 T - 30 \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 79 T - 68 \) Copy content Toggle raw display
$5$ \( (T + 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 74 T^{2} + 1577 T + 7692 \) Copy content Toggle raw display
$13$ \( T^{3} + 44 T^{2} - 3491 T + 44870 \) Copy content Toggle raw display
$17$ \( T^{3} - 52 T^{2} - 11747 T + 56706 \) Copy content Toggle raw display
$19$ \( T^{3} + 168 T^{2} + 5620 T + 28720 \) Copy content Toggle raw display
$23$ \( T^{3} + 124 T^{2} + 1732 T - 94368 \) Copy content Toggle raw display
$29$ \( T^{3} - 332 T^{2} + 7405 T + 2565450 \) Copy content Toggle raw display
$31$ \( T^{3} + 320 T^{2} + 23968 T - 50176 \) Copy content Toggle raw display
$37$ \( T^{3} + 54 T^{2} - 12116 T + 25736 \) Copy content Toggle raw display
$41$ \( T^{3} + 362 T^{2} + 41536 T + 1536192 \) Copy content Toggle raw display
$43$ \( T^{3} + 16 T^{2} - 89516 T - 1524560 \) Copy content Toggle raw display
$47$ \( T^{3} - 730 T^{2} + 116057 T - 4968912 \) Copy content Toggle raw display
$53$ \( T^{3} - 110 T^{2} + \cdots + 90318336 \) Copy content Toggle raw display
$59$ \( T^{3} - 180 T^{2} + \cdots + 202459200 \) Copy content Toggle raw display
$61$ \( T^{3} + 1222 T^{2} + \cdots + 38393792 \) Copy content Toggle raw display
$67$ \( T^{3} - 204 T^{2} + \cdots + 324944128 \) Copy content Toggle raw display
$71$ \( T^{3} + 136 T^{2} + \cdots + 15575040 \) Copy content Toggle raw display
$73$ \( T^{3} + 310 T^{2} + \cdots - 48718616 \) Copy content Toggle raw display
$79$ \( T^{3} + 1034 T^{2} + \cdots - 343615600 \) Copy content Toggle raw display
$83$ \( T^{3} - 1660 T^{2} + \cdots + 42727104 \) Copy content Toggle raw display
$89$ \( T^{3} + 242 T^{2} - 687680 T + 6359520 \) Copy content Toggle raw display
$97$ \( T^{3} + 100 T^{2} - 471963 T + 1978018 \) Copy content Toggle raw display
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