Properties

Label 525.4.a.w
Level $525$
Weight $4$
Character orbit 525.a
Self dual yes
Analytic conductor $30.976$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.78066700.1
Defining polynomial: \(x^{5} - x^{4} - 18 x^{3} + 7 x^{2} + 30 x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 5 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 3 q^{3} + ( 5 - \beta_{1} - \beta_{3} ) q^{4} + 3 \beta_{1} q^{6} -7 q^{7} + ( -6 + 5 \beta_{1} + \beta_{2} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + 3 q^{3} + ( 5 - \beta_{1} - \beta_{3} ) q^{4} + 3 \beta_{1} q^{6} -7 q^{7} + ( -6 + 5 \beta_{1} + \beta_{2} ) q^{8} + 9 q^{9} + ( 15 + 8 \beta_{1} + \beta_{4} ) q^{11} + ( 15 - 3 \beta_{1} - 3 \beta_{3} ) q^{12} + ( 1 + 6 \beta_{1} + \beta_{2} - \beta_{4} ) q^{13} -7 \beta_{1} q^{14} + ( 29 - 5 \beta_{1} - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} ) q^{16} + ( -18 + 6 \beta_{1} - \beta_{2} + 8 \beta_{3} + 2 \beta_{4} ) q^{17} + 9 \beta_{1} q^{18} + ( 33 - 2 \beta_{1} + \beta_{2} - 6 \beta_{3} + \beta_{4} ) q^{19} -21 q^{21} + ( 102 + 10 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} ) q^{22} + ( 27 + 12 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{23} + ( -18 + 15 \beta_{1} + 3 \beta_{2} ) q^{24} + ( 84 - 10 \beta_{1} + \beta_{2} - 12 \beta_{3} - 2 \beta_{4} ) q^{26} + 27 q^{27} + ( -35 + 7 \beta_{1} + 7 \beta_{3} ) q^{28} + ( 72 - 8 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{29} + ( 65 - 26 \beta_{1} - \beta_{2} + 8 \beta_{3} - \beta_{4} ) q^{31} + ( 18 + 25 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} + 2 \beta_{4} ) q^{32} + ( 45 + 24 \beta_{1} + 3 \beta_{4} ) q^{33} + ( 14 - 72 \beta_{1} - 11 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} ) q^{34} + ( 45 - 9 \beta_{1} - 9 \beta_{3} ) q^{36} + ( 74 - 24 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} - 8 \beta_{4} ) q^{37} + ( 18 + 78 \beta_{1} + 3 \beta_{2} - 14 \beta_{3} - 2 \beta_{4} ) q^{38} + ( 3 + 18 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} ) q^{39} + ( 171 + 32 \beta_{1} + 3 \beta_{2} + 20 \beta_{3} + 7 \beta_{4} ) q^{41} -21 \beta_{1} q^{42} + ( 122 + 16 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 8 \beta_{4} ) q^{43} + ( 72 + 102 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{44} + ( 132 + 4 \beta_{1} + 3 \beta_{2} + 16 \beta_{3} + 6 \beta_{4} ) q^{46} + ( -138 + 36 \beta_{1} + 6 \beta_{2} + 10 \beta_{3} - 8 \beta_{4} ) q^{47} + ( 87 - 15 \beta_{1} - 3 \beta_{2} - 15 \beta_{3} - 6 \beta_{4} ) q^{48} + 49 q^{49} + ( -54 + 18 \beta_{1} - 3 \beta_{2} + 24 \beta_{3} + 6 \beta_{4} ) q^{51} + ( -46 + 122 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} ) q^{52} + ( -15 + 14 \beta_{1} - 11 \beta_{2} - 14 \beta_{3} + \beta_{4} ) q^{53} + 27 \beta_{1} q^{54} + ( 42 - 35 \beta_{1} - 7 \beta_{2} ) q^{56} + ( 99 - 6 \beta_{1} + 3 \beta_{2} - 18 \beta_{3} + 3 \beta_{4} ) q^{57} + ( -70 + 126 \beta_{1} + 8 \beta_{2} + 18 \beta_{3} + 4 \beta_{4} ) q^{58} + ( 144 - 28 \beta_{1} - 12 \beta_{2} + 28 \beta_{3} - 8 \beta_{4} ) q^{59} + ( 118 - 36 \beta_{1} + 16 \beta_{2} + 18 \beta_{3} - 10 \beta_{4} ) q^{61} + ( -396 + 34 \beta_{1} - 5 \beta_{2} + 44 \beta_{3} + 2 \beta_{4} ) q^{62} -63 q^{63} + ( 13 - 49 \beta_{1} - 10 \beta_{2} + 7 \beta_{3} + 12 \beta_{4} ) q^{64} + ( 306 + 30 \beta_{1} - 6 \beta_{2} - 30 \beta_{3} ) q^{66} + ( -52 - 64 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} ) q^{67} + ( -882 + 24 \beta_{1} + 9 \beta_{2} + 98 \beta_{3} + 6 \beta_{4} ) q^{68} + ( 81 + 36 \beta_{1} - 9 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} ) q^{69} + ( 291 - 16 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - 17 \beta_{4} ) q^{71} + ( -54 + 45 \beta_{1} + 9 \beta_{2} ) q^{72} + ( 87 - 102 \beta_{1} - 13 \beta_{2} - 24 \beta_{3} + \beta_{4} ) q^{73} + ( -234 + 148 \beta_{1} + 28 \beta_{2} + 46 \beta_{3} + 4 \beta_{4} ) q^{74} + ( 864 + 42 \beta_{1} + 7 \beta_{2} - 64 \beta_{3} - 14 \beta_{4} ) q^{76} + ( -105 - 56 \beta_{1} - 7 \beta_{4} ) q^{77} + ( 252 - 30 \beta_{1} + 3 \beta_{2} - 36 \beta_{3} - 6 \beta_{4} ) q^{78} + ( -182 - 96 \beta_{1} + 14 \beta_{2} + 36 \beta_{3} - 2 \beta_{4} ) q^{79} + 81 q^{81} + ( 274 + 14 \beta_{1} - 37 \beta_{2} - 50 \beta_{3} - 6 \beta_{4} ) q^{82} + ( -312 - 48 \beta_{1} + 4 \beta_{2} + 20 \beta_{3} - 4 \beta_{4} ) q^{83} + ( -105 + 21 \beta_{1} + 21 \beta_{3} ) q^{84} + ( 198 + 148 \beta_{1} - 12 \beta_{2} - 18 \beta_{3} + 4 \beta_{4} ) q^{86} + ( 216 - 24 \beta_{1} - 6 \beta_{2} - 18 \beta_{3} ) q^{87} + ( 594 - 118 \beta_{1} + 16 \beta_{2} - 114 \beta_{3} - 24 \beta_{4} ) q^{88} + ( 81 + 72 \beta_{1} - 15 \beta_{2} - 8 \beta_{3} + 25 \beta_{4} ) q^{89} + ( -7 - 42 \beta_{1} - 7 \beta_{2} + 7 \beta_{4} ) q^{91} + ( -276 - 68 \beta_{1} - 7 \beta_{2} - 40 \beta_{3} + 2 \beta_{4} ) q^{92} + ( 195 - 78 \beta_{1} - 3 \beta_{2} + 24 \beta_{3} - 3 \beta_{4} ) q^{93} + ( 438 - 280 \beta_{1} - 58 \beta_{3} - 12 \beta_{4} ) q^{94} + ( 54 + 75 \beta_{1} + 6 \beta_{2} + 36 \beta_{3} + 6 \beta_{4} ) q^{96} + ( -83 - 162 \beta_{1} + \beta_{2} + 20 \beta_{3} + 7 \beta_{4} ) q^{97} + 49 \beta_{1} q^{98} + ( 135 + 72 \beta_{1} + 9 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - q^{2} + 15q^{3} + 27q^{4} - 3q^{6} - 35q^{7} - 33q^{8} + 45q^{9} + O(q^{10}) \) \( 5q - q^{2} + 15q^{3} + 27q^{4} - 3q^{6} - 35q^{7} - 33q^{8} + 45q^{9} + 66q^{11} + 81q^{12} + 2q^{13} + 7q^{14} + 155q^{16} - 108q^{17} - 9q^{18} + 174q^{19} - 105q^{21} + 506q^{22} + 116q^{23} - 99q^{24} + 446q^{26} + 135q^{27} - 189q^{28} + 370q^{29} + 342q^{31} + 55q^{32} + 198q^{33} + 112q^{34} + 243q^{36} + 408q^{37} + 34q^{38} + 6q^{39} + 802q^{41} + 21q^{42} + 584q^{43} + 290q^{44} + 640q^{46} - 716q^{47} + 465q^{48} + 245q^{49} - 324q^{51} - 338q^{52} - 98q^{53} - 27q^{54} + 231q^{56} + 522q^{57} - 482q^{58} + 704q^{59} + 650q^{61} - 2070q^{62} - 315q^{63} + 75q^{64} + 1518q^{66} - 180q^{67} - 4520q^{68} + 348q^{69} + 1470q^{71} - 297q^{72} + 534q^{73} - 1312q^{74} + 4370q^{76} - 462q^{77} + 1338q^{78} - 820q^{79} + 405q^{81} + 1338q^{82} - 1520q^{83} - 567q^{84} + 832q^{86} + 1110q^{87} + 3258q^{88} + 286q^{89} - 14q^{91} - 1288q^{92} + 1026q^{93} + 2540q^{94} + 165q^{96} - 278q^{97} - 49q^{98} + 594q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 18 x^{3} + 7 x^{2} + 30 x - 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{4} + 3 \nu^{3} + 27 \nu^{2} - 20 \nu + 10 \)\()/15\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{4} + 24 \nu^{3} - 84 \nu^{2} - 320 \nu + 70 \)\()/15\)
\(\beta_{3}\)\(=\)\((\)\( 8 \nu^{4} - 12 \nu^{3} - 138 \nu^{2} + 140 \nu + 155 \)\()/15\)
\(\beta_{4}\)\(=\)\((\)\( -32 \nu^{4} + 18 \nu^{3} + 582 \nu^{2} + 100 \nu - 785 \)\()/15\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + 4 \beta_{3} + \beta_{2} + 2 \beta_{1} + 5\)\()/20\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} - \beta_{3} + \beta_{2} - 18 \beta_{1} + 70\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{4} + 23 \beta_{3} + 12 \beta_{2} + 4 \beta_{1} + 70\)\()/10\)
\(\nu^{4}\)\(=\)\((\)\(38 \beta_{4} + 2 \beta_{3} + 53 \beta_{2} - 644 \beta_{1} + 2150\)\()/20\)

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
−3.71490
4.40248
0.329739
1.35311
−1.37042
−5.18660 3.00000 18.9008 0 −15.5598 −7.00000 −56.5383 9.00000 0
1.2 −3.33774 3.00000 3.14050 0 −10.0132 −7.00000 16.2197 9.00000 0
1.3 0.428319 3.00000 −7.81654 0 1.28496 −7.00000 −6.77452 9.00000 0
1.4 2.20666 3.00000 −3.13065 0 6.61998 −7.00000 −24.5616 9.00000 0
1.5 4.88936 3.00000 15.9059 0 14.6681 −7.00000 38.6546 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.w 5
3.b odd 2 1 1575.4.a.bp 5
5.b even 2 1 525.4.a.x 5
5.c odd 4 2 105.4.d.b 10
15.d odd 2 1 1575.4.a.bo 5
15.e even 4 2 315.4.d.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.d.b 10 5.c odd 4 2
315.4.d.b 10 15.e even 4 2
525.4.a.w 5 1.a even 1 1 trivial
525.4.a.x 5 5.b even 2 1
1575.4.a.bo 5 15.d odd 2 1
1575.4.a.bp 5 3.b odd 2 1

Hecke kernels

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

\( T_{2}^{5} + T_{2}^{4} - 33 T_{2}^{3} - 17 T_{2}^{2} + 200 T_{2} - 80 \)
\( T_{11}^{5} - 66 T_{11}^{4} - 2100 T_{11}^{3} + 140456 T_{11}^{2} + 1472448 T_{11} - 55852416 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 7 T^{2} + 15 T^{3} + 48 T^{4} + 32 T^{5} + 384 T^{6} + 960 T^{7} + 3584 T^{8} + 4096 T^{9} + 32768 T^{10} \)
$3$ \( ( 1 - 3 T )^{5} \)
$5$ 1
$7$ \( ( 1 + 7 T )^{5} \)
$11$ \( 1 - 66 T + 4555 T^{2} - 210928 T^{3} + 10802758 T^{4} - 383496700 T^{5} + 14378470898 T^{6} - 373671818608 T^{7} + 10740451732505 T^{8} - 207136272863586 T^{9} + 4177248169415651 T^{10} \)
$13$ \( 1 - 2 T + 6833 T^{2} - 33080 T^{3} + 25238338 T^{4} - 84665324 T^{5} + 55448628586 T^{6} - 159670841720 T^{7} + 72460544215709 T^{8} - 46596170244962 T^{9} + 51185893014090757 T^{10} \)
$17$ \( 1 + 108 T + 7625 T^{2} + 580624 T^{3} + 28716998 T^{4} + 723515208 T^{5} + 141086611174 T^{6} + 14014851863056 T^{7} + 904232558289625 T^{8} + 62923201620814188 T^{9} + 2862423051509815793 T^{10} \)
$19$ \( 1 - 174 T + 30655 T^{2} - 3260120 T^{3} + 355132498 T^{4} - 29134734196 T^{5} + 2435853803782 T^{6} - 153375217565720 T^{7} + 9891991375415245 T^{8} - 385116795917512014 T^{9} + 15181127029874798299 T^{10} \)
$23$ \( 1 - 116 T + 41159 T^{2} - 2945168 T^{3} + 716339614 T^{4} - 37811551864 T^{5} + 8715704083538 T^{6} - 435990563134352 T^{7} + 74133642393155617 T^{8} - 2542096434114357236 T^{9} + \)\(26\!\cdots\!07\)\( T^{10} \)
$29$ \( 1 - 370 T + 160385 T^{2} - 37125560 T^{3} + 8745786530 T^{4} - 1370740158092 T^{5} + 213300987680170 T^{6} - 22083148893184760 T^{7} + 2326728607339749565 T^{8} - \)\(13\!\cdots\!70\)\( T^{9} + \)\(86\!\cdots\!49\)\( T^{10} \)
$31$ \( 1 - 342 T + 142051 T^{2} - 26927448 T^{3} + 6639779138 T^{4} - 944601593732 T^{5} + 197805660300158 T^{6} - 23898209219936088 T^{7} + 3755774767545476221 T^{8} - \)\(26\!\cdots\!62\)\( T^{9} + \)\(23\!\cdots\!51\)\( T^{10} \)
$37$ \( 1 - 408 T + 163665 T^{2} - 51926944 T^{3} + 15115947738 T^{4} - 3482080406928 T^{5} + 765668100772914 T^{6} - 133230331559464096 T^{7} + 21270188143561277205 T^{8} - \)\(26\!\cdots\!48\)\( T^{9} + \)\(33\!\cdots\!93\)\( T^{10} \)
$41$ \( 1 - 802 T + 392457 T^{2} - 146273472 T^{3} + 44792512462 T^{4} - 12012058617212 T^{5} + 3087144751393502 T^{6} - 694814239692994752 T^{7} + \)\(12\!\cdots\!77\)\( T^{8} - \)\(18\!\cdots\!62\)\( T^{9} + \)\(15\!\cdots\!01\)\( T^{10} \)
$43$ \( 1 - 584 T + 413407 T^{2} - 149329184 T^{3} + 61499850522 T^{4} - 16229990754224 T^{5} + 4889668615452654 T^{6} - 943963985874922016 T^{7} + \)\(20\!\cdots\!01\)\( T^{8} - \)\(23\!\cdots\!84\)\( T^{9} + \)\(31\!\cdots\!07\)\( T^{10} \)
$47$ \( 1 + 716 T + 507419 T^{2} + 209415632 T^{3} + 90503203546 T^{4} + 27982753228744 T^{5} + 9396314101756358 T^{6} + 2257336190586622928 T^{7} + \)\(56\!\cdots\!73\)\( T^{8} + \)\(83\!\cdots\!56\)\( T^{9} + \)\(12\!\cdots\!43\)\( T^{10} \)
$53$ \( 1 + 98 T + 503305 T^{2} + 71121272 T^{3} + 122661090658 T^{4} + 16369960190572 T^{5} + 18261415193891066 T^{6} + 1576357556561836088 T^{7} + \)\(16\!\cdots\!65\)\( T^{8} + \)\(48\!\cdots\!18\)\( T^{9} + \)\(73\!\cdots\!57\)\( T^{10} \)
$59$ \( 1 - 704 T + 427487 T^{2} - 197865856 T^{3} + 99740405354 T^{4} - 55609053370240 T^{5} + 20484584711199166 T^{6} - 8346087395413261696 T^{7} + \)\(37\!\cdots\!93\)\( T^{8} - \)\(12\!\cdots\!24\)\( T^{9} + \)\(36\!\cdots\!99\)\( T^{10} \)
$61$ \( 1 - 650 T + 689345 T^{2} - 375251480 T^{3} + 259020342850 T^{4} - 102268282520348 T^{5} + 58792696440435850 T^{6} - 19333096729119304280 T^{7} + \)\(80\!\cdots\!45\)\( T^{8} - \)\(17\!\cdots\!50\)\( T^{9} + \)\(60\!\cdots\!01\)\( T^{10} \)
$67$ \( 1 + 180 T + 1024695 T^{2} - 16666000 T^{3} + 437663187690 T^{4} - 44169474067848 T^{5} + 131632893319207470 T^{6} - 1507579397228554000 T^{7} + \)\(27\!\cdots\!65\)\( T^{8} + \)\(14\!\cdots\!80\)\( T^{9} + \)\(24\!\cdots\!43\)\( T^{10} \)
$71$ \( 1 - 1470 T + 1881615 T^{2} - 1727909680 T^{3} + 1338943951990 T^{4} - 860023408325220 T^{5} + 479222768800692890 T^{6} - \)\(22\!\cdots\!80\)\( T^{7} + \)\(86\!\cdots\!65\)\( T^{8} - \)\(24\!\cdots\!70\)\( T^{9} + \)\(58\!\cdots\!51\)\( T^{10} \)
$73$ \( 1 - 534 T + 1342021 T^{2} - 930732232 T^{3} + 817882087762 T^{4} - 560574711779204 T^{5} + 318170036134909954 T^{6} - \)\(14\!\cdots\!48\)\( T^{7} + \)\(79\!\cdots\!73\)\( T^{8} - \)\(12\!\cdots\!14\)\( T^{9} + \)\(89\!\cdots\!57\)\( T^{10} \)
$79$ \( 1 + 820 T + 1736795 T^{2} + 1027136240 T^{3} + 1389341527610 T^{4} + 657402503393464 T^{5} + 684999557431306790 T^{6} + \)\(24\!\cdots\!40\)\( T^{7} + \)\(20\!\cdots\!05\)\( T^{8} + \)\(48\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!99\)\( T^{10} \)
$83$ \( 1 + 1520 T + 3526775 T^{2} + 3535501120 T^{3} + 4405187467130 T^{4} + 3049079895261856 T^{5} + 2518828926267861310 T^{6} + \)\(11\!\cdots\!80\)\( T^{7} + \)\(65\!\cdots\!25\)\( T^{8} + \)\(16\!\cdots\!20\)\( T^{9} + \)\(61\!\cdots\!07\)\( T^{10} \)
$89$ \( 1 - 286 T + 2177465 T^{2} - 600135040 T^{3} + 2162009088238 T^{4} - 560754412921604 T^{5} + 1524149384926054622 T^{6} - \)\(29\!\cdots\!40\)\( T^{7} + \)\(76\!\cdots\!85\)\( T^{8} - \)\(70\!\cdots\!06\)\( T^{9} + \)\(17\!\cdots\!49\)\( T^{10} \)
$97$ \( 1 + 278 T + 3429709 T^{2} + 1008794696 T^{3} + 5371131513042 T^{4} + 1398135665763908 T^{5} + 4902086711402581266 T^{6} + \)\(84\!\cdots\!84\)\( T^{7} + \)\(26\!\cdots\!53\)\( T^{8} + \)\(19\!\cdots\!98\)\( T^{9} + \)\(63\!\cdots\!93\)\( T^{10} \)
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