Properties

Label 525.4.a.v
Level $525$
Weight $4$
Character orbit 525.a
Self dual yes
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.26729725.1
Defining polynomial: \(x^{4} - 2 x^{3} - 18 x^{2} + 19 x + 44\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( 3 + 3 \beta_{1} ) q^{6} -7 q^{7} + ( 23 + 5 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( 3 + 3 \beta_{1} ) q^{6} -7 q^{7} + ( 23 + 5 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{8} + 9 q^{9} + ( 18 - 3 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{11} + ( 9 + 9 \beta_{1} + 3 \beta_{2} ) q^{12} + ( 11 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{13} + ( -7 - 7 \beta_{1} ) q^{14} + ( 45 + 27 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{16} + ( 17 + 17 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{17} + ( 9 + 9 \beta_{1} ) q^{18} + ( -16 + 6 \beta_{1} - 20 \beta_{2} ) q^{19} -21 q^{21} + ( -26 + 30 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{22} + ( 39 - 4 \beta_{2} + 8 \beta_{3} ) q^{23} + ( 69 + 15 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} ) q^{24} + ( -29 - 5 \beta_{1} - 19 \beta_{2} - 7 \beta_{3} ) q^{26} + 27 q^{27} + ( -21 - 21 \beta_{1} - 7 \beta_{2} ) q^{28} + ( 85 + 16 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -27 + 3 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} ) q^{31} + ( 143 + 95 \beta_{1} + 31 \beta_{2} + 10 \beta_{3} ) q^{32} + ( 54 - 9 \beta_{1} + 15 \beta_{2} - 3 \beta_{3} ) q^{33} + ( 169 + 57 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{34} + ( 27 + 27 \beta_{1} + 9 \beta_{2} ) q^{36} + ( 42 - 51 \beta_{1} - 3 \beta_{2} - 9 \beta_{3} ) q^{37} + ( 84 - 84 \beta_{1} - 14 \beta_{2} - 20 \beta_{3} ) q^{38} + ( 33 - 9 \beta_{1} - 3 \beta_{2} - 9 \beta_{3} ) q^{39} + ( -125 - 5 \beta_{1} - 33 \beta_{2} + 21 \beta_{3} ) q^{41} + ( -21 - 21 \beta_{1} ) q^{42} + ( 43 - 114 \beta_{1} + 4 \beta_{2} ) q^{43} + ( 148 + 52 \beta_{1} + 2 \beta_{2} + 11 \beta_{3} ) q^{44} + ( 79 + 39 \beta_{1} + 36 \beta_{2} + 12 \beta_{3} ) q^{46} + ( 172 + 34 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{47} + ( 135 + 81 \beta_{1} + 18 \beta_{2} + 18 \beta_{3} ) q^{48} + 49 q^{49} + ( 51 + 51 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{51} + ( -157 - 105 \beta_{1} - 51 \beta_{2} - 9 \beta_{3} ) q^{52} + ( 251 - 151 \beta_{1} - 15 \beta_{2} + 3 \beta_{3} ) q^{53} + ( 27 + 27 \beta_{1} ) q^{54} + ( -161 - 35 \beta_{1} - 28 \beta_{2} - 7 \beta_{3} ) q^{56} + ( -48 + 18 \beta_{1} - 60 \beta_{2} ) q^{57} + ( 241 + 105 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{58} + ( 31 + 73 \beta_{1} + 19 \beta_{2} + \beta_{3} ) q^{59} + ( -321 - 33 \beta_{1} - 9 \beta_{2} + 21 \beta_{3} ) q^{61} + ( -63 - 39 \beta_{1} - 69 \beta_{2} - 27 \beta_{3} ) q^{62} -63 q^{63} + ( 711 + 261 \beta_{1} + 128 \beta_{2} + 3 \beta_{3} ) q^{64} + ( -78 + 90 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{66} + ( 424 - 63 \beta_{1} + 31 \beta_{2} + 21 \beta_{3} ) q^{67} + ( 581 + 161 \beta_{1} + 23 \beta_{2} + 23 \beta_{3} ) q^{68} + ( 117 - 12 \beta_{2} + 24 \beta_{3} ) q^{69} + ( -320 - 137 \beta_{1} + 91 \beta_{2} - 11 \beta_{3} ) q^{71} + ( 207 + 45 \beta_{1} + 36 \beta_{2} + 9 \beta_{3} ) q^{72} + ( 114 + 18 \beta_{1} - 28 \beta_{2} + 36 \beta_{3} ) q^{73} + ( -498 - 90 \beta_{1} - 99 \beta_{2} - 21 \beta_{3} ) q^{74} + ( -680 - 228 \beta_{1} - 38 \beta_{2} - 54 \beta_{3} ) q^{76} + ( -126 + 21 \beta_{1} - 35 \beta_{2} + 7 \beta_{3} ) q^{77} + ( -87 - 15 \beta_{1} - 57 \beta_{2} - 21 \beta_{3} ) q^{78} + ( 374 - 33 \beta_{1} - 95 \beta_{2} - 9 \beta_{3} ) q^{79} + 81 q^{81} + ( -25 - 225 \beta_{1} + 67 \beta_{2} + 9 \beta_{3} ) q^{82} + ( 71 - 79 \beta_{1} - 79 \beta_{2} - 37 \beta_{3} ) q^{83} + ( -63 - 63 \beta_{1} - 21 \beta_{2} ) q^{84} + ( -1105 - 169 \beta_{1} - 110 \beta_{2} + 4 \beta_{3} ) q^{86} + ( 255 + 48 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{87} + ( 916 + 42 \beta_{1} + 133 \beta_{2} ) q^{88} + ( -50 - 356 \beta_{1} + 38 \beta_{2} - 10 \beta_{3} ) q^{89} + ( -77 + 21 \beta_{1} + 7 \beta_{2} + 21 \beta_{3} ) q^{91} + ( 133 + 325 \beta_{1} + 167 \beta_{2} - 4 \beta_{3} ) q^{92} + ( -81 + 9 \beta_{1} + 9 \beta_{2} - 45 \beta_{3} ) q^{93} + ( 520 + 264 \beta_{1} + 58 \beta_{2} + 12 \beta_{3} ) q^{94} + ( 429 + 285 \beta_{1} + 93 \beta_{2} + 30 \beta_{3} ) q^{96} + ( 754 - 204 \beta_{1} - 16 \beta_{2} + 24 \beta_{3} ) q^{97} + ( 49 + 49 \beta_{1} ) q^{98} + ( 162 - 27 \beta_{1} + 45 \beta_{2} - 9 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{2} + 12q^{3} + 16q^{4} + 18q^{6} - 28q^{7} + 93q^{8} + 36q^{9} + O(q^{10}) \) \( 4q + 6q^{2} + 12q^{3} + 16q^{4} + 18q^{6} - 28q^{7} + 93q^{8} + 36q^{9} + 57q^{11} + 48q^{12} + 43q^{13} - 42q^{14} + 216q^{16} + 99q^{17} + 54q^{18} - 12q^{19} - 84q^{21} - 41q^{22} + 156q^{23} + 279q^{24} - 81q^{26} + 108q^{27} - 112q^{28} + 378q^{29} - 93q^{31} + 690q^{32} + 171q^{33} + 783q^{34} + 144q^{36} + 81q^{37} + 216q^{38} + 129q^{39} - 465q^{41} - 126q^{42} - 64q^{43} + 681q^{44} + 310q^{46} + 744q^{47} + 648q^{48} + 196q^{49} + 297q^{51} - 727q^{52} + 729q^{53} + 162q^{54} - 651q^{56} - 36q^{57} + 1172q^{58} + 231q^{59} - 1353q^{61} - 165q^{62} - 252q^{63} + 3107q^{64} - 123q^{66} + 1487q^{67} + 2577q^{68} + 468q^{69} - 1725q^{71} + 837q^{72} + 512q^{73} - 1953q^{74} - 3046q^{76} - 399q^{77} - 243q^{78} + 1629q^{79} + 324q^{81} - 693q^{82} + 321q^{83} - 336q^{84} - 4542q^{86} + 1134q^{87} + 3482q^{88} - 978q^{89} - 301q^{91} + 852q^{92} - 279q^{93} + 2480q^{94} + 2070q^{96} + 2616q^{97} + 294q^{98} + 513q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 18 x^{2} + 19 x + 44\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 10 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 14 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 10\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 15 \beta_{1} + 8\)

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
−3.56826
−1.21734
2.21734
4.56826
−2.56826 3.00000 −1.40404 0 −7.70478 −7.00000 24.1520 9.00000 0
1.2 −0.217342 3.00000 −7.95276 0 −0.652027 −7.00000 3.46721 9.00000 0
1.3 3.21734 3.00000 2.35129 0 9.65203 −7.00000 −18.1738 9.00000 0
1.4 5.56826 3.00000 23.0055 0 16.7048 −7.00000 83.5546 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

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 525.4.a.v yes 4
3.b odd 2 1 1575.4.a.bf 4
5.b even 2 1 525.4.a.s 4
5.c odd 4 2 525.4.d.o 8
15.d odd 2 1 1575.4.a.bm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.s 4 5.b even 2 1
525.4.a.v yes 4 1.a even 1 1 trivial
525.4.d.o 8 5.c odd 4 2
1575.4.a.bf 4 3.b odd 2 1
1575.4.a.bm 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2}^{4} - 6 T_{2}^{3} - 6 T_{2}^{2} + 45 T_{2} + 10 \)
\( T_{11}^{4} - 57 T_{11}^{3} - 1323 T_{11}^{2} + 44829 T_{11} + 99334 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T + 26 T^{2} - 99 T^{3} + 298 T^{4} - 792 T^{5} + 1664 T^{6} - 3072 T^{7} + 4096 T^{8} \)
$3$ \( ( 1 - 3 T )^{4} \)
$5$ 1
$7$ \( ( 1 + 7 T )^{4} \)
$11$ \( 1 - 57 T + 4001 T^{2} - 182772 T^{3} + 7206874 T^{4} - 243269532 T^{5} + 7088015561 T^{6} - 134403018387 T^{7} + 3138428376721 T^{8} \)
$13$ \( 1 - 43 T + 4464 T^{2} - 221093 T^{3} + 13201958 T^{4} - 485741321 T^{5} + 21546875376 T^{6} - 455993473039 T^{7} + 23298085122481 T^{8} \)
$17$ \( 1 - 99 T + 13790 T^{2} - 926229 T^{3} + 90687946 T^{4} - 4550563077 T^{5} + 332857076510 T^{6} - 11740199773203 T^{7} + 582622237229761 T^{8} \)
$19$ \( 1 + 12 T - 10212 T^{2} + 154020 T^{3} + 95206646 T^{4} + 1056423180 T^{5} - 480432536772 T^{6} + 3872252373348 T^{7} + 2213314919066161 T^{8} \)
$23$ \( 1 - 156 T + 28082 T^{2} - 3613896 T^{3} + 479553835 T^{4} - 43970272632 T^{5} + 4157143834898 T^{6} - 280979815188228 T^{7} + 21914624432020321 T^{8} \)
$29$ \( 1 - 378 T + 143760 T^{2} - 29729988 T^{3} + 5852097929 T^{4} - 725084677332 T^{5} + 85511800626960 T^{6} - 5483701178878482 T^{7} + 353814783205469041 T^{8} \)
$31$ \( 1 + 93 T + 16330 T^{2} - 936891 T^{3} + 385099458 T^{4} - 27910919781 T^{5} + 14492935110730 T^{6} + 2458884860942403 T^{7} + 787662783788549761 T^{8} \)
$37$ \( 1 - 81 T + 107185 T^{2} - 469746 T^{3} + 5937897606 T^{4} - 23794044138 T^{5} + 275007385148665 T^{6} - 10526900923401237 T^{7} + 6582952005840035281 T^{8} \)
$41$ \( 1 + 465 T + 105716 T^{2} + 566235 T^{3} - 2050285754 T^{4} + 39025482435 T^{5} + 502162019941556 T^{6} + 152232599493191865 T^{7} + 22563490300366186081 T^{8} \)
$43$ \( 1 + 64 T + 64662 T^{2} - 2492176 T^{3} + 5477627255 T^{4} - 198145437232 T^{5} + 408751977474438 T^{6} + 32165927163957952 T^{7} + 39959630797262576401 T^{8} \)
$47$ \( 1 - 744 T + 588276 T^{2} - 247094976 T^{3} + 101020317878 T^{4} - 25654141693248 T^{5} + 6341153676882804 T^{6} - 832633071988458648 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 - 729 T + 333446 T^{2} - 137056347 T^{3} + 63585804370 T^{4} - 20404537772319 T^{5} + 7390617561020534 T^{6} - 2405527658423754957 T^{7} + \)\(49\!\cdots\!41\)\( T^{8} \)
$59$ \( 1 - 231 T + 701280 T^{2} - 148812447 T^{3} + 204019205798 T^{4} - 30562951552413 T^{5} + 29580364631760480 T^{6} - 2001152034109290909 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \)
$61$ \( 1 + 1353 T + 1374808 T^{2} + 931878711 T^{3} + 511309944510 T^{4} + 211518761701491 T^{5} + 70830622834497688 T^{6} + 15822179663604592773 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \)
$67$ \( 1 - 1487 T + 1603371 T^{2} - 1080212116 T^{3} + 674138302124 T^{4} - 324887836644508 T^{5} + 145038346676691699 T^{6} - 40456116647290586189 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \)
$71$ \( 1 + 1725 T + 1434819 T^{2} + 706812600 T^{3} + 347044577276 T^{4} + 252976004478600 T^{5} + 183800721275245299 T^{6} + 79088663739324578475 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} \)
$73$ \( 1 - 512 T + 1030980 T^{2} - 484092568 T^{3} + 566560915526 T^{4} - 188320238525656 T^{5} + 156022560619433220 T^{6} - 30142252394633171456 T^{7} + \)\(22\!\cdots\!21\)\( T^{8} \)
$79$ \( 1 - 1629 T + 1989837 T^{2} - 1837769472 T^{3} + 1504809477470 T^{4} - 906092022705408 T^{5} + 483704413231540077 T^{6} - \)\(19\!\cdots\!51\)\( T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \)
$83$ \( 1 - 321 T + 661434 T^{2} - 144132717 T^{3} + 79870144034 T^{4} - 82413213855279 T^{5} + 216249478918951146 T^{6} - 60007821940880469363 T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \)
$89$ \( 1 + 978 T + 526228 T^{2} - 159100050 T^{3} - 360762420714 T^{4} - 112160603148450 T^{5} + 261525470779825108 T^{6} + \)\(34\!\cdots\!02\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} \)
$97$ \( 1 - 2616 T + 5183100 T^{2} - 6945540936 T^{3} + 7789652954246 T^{4} - 6339007682681928 T^{5} + 4317377198747499900 T^{6} - \)\(19\!\cdots\!72\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \)
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