Properties

Label 525.4.a.p
Level $525$
Weight $4$
Character orbit 525.a
Self dual yes
Analytic conductor $30.976$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - \beta ) q^{2} + 3 q^{3} + ( 12 - 7 \beta ) q^{4} + ( 12 - 3 \beta ) q^{6} + 7 q^{7} + ( 44 - 25 \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( 4 - \beta ) q^{2} + 3 q^{3} + ( 12 - 7 \beta ) q^{4} + ( 12 - 3 \beta ) q^{6} + 7 q^{7} + ( 44 - 25 \beta ) q^{8} + 9 q^{9} + ( -18 + 10 \beta ) q^{11} + ( 36 - 21 \beta ) q^{12} + ( 4 - 22 \beta ) q^{13} + ( 28 - 7 \beta ) q^{14} + ( 180 - 63 \beta ) q^{16} + ( -22 + 28 \beta ) q^{17} + ( 36 - 9 \beta ) q^{18} + ( 74 + 26 \beta ) q^{19} + 21 q^{21} + ( -112 + 48 \beta ) q^{22} + ( -120 + 56 \beta ) q^{23} + ( 132 - 75 \beta ) q^{24} + ( 104 - 70 \beta ) q^{26} + 27 q^{27} + ( 84 - 49 \beta ) q^{28} + ( -58 + 84 \beta ) q^{29} + ( 174 - 18 \beta ) q^{31} + ( 620 - 169 \beta ) q^{32} + ( -54 + 30 \beta ) q^{33} + ( -200 + 106 \beta ) q^{34} + ( 108 - 63 \beta ) q^{36} + ( 54 + 24 \beta ) q^{37} + ( 192 + 4 \beta ) q^{38} + ( 12 - 66 \beta ) q^{39} + ( 170 - 140 \beta ) q^{41} + ( 84 - 21 \beta ) q^{42} + ( -148 - 68 \beta ) q^{43} + ( -496 + 176 \beta ) q^{44} + ( -704 + 288 \beta ) q^{46} + ( -200 + 108 \beta ) q^{47} + ( 540 - 189 \beta ) q^{48} + 49 q^{49} + ( -66 + 84 \beta ) q^{51} + ( 664 - 138 \beta ) q^{52} + ( -124 + 214 \beta ) q^{53} + ( 108 - 27 \beta ) q^{54} + ( 308 - 175 \beta ) q^{56} + ( 222 + 78 \beta ) q^{57} + ( -568 + 310 \beta ) q^{58} + ( 200 - 36 \beta ) q^{59} + ( 270 + 252 \beta ) q^{61} + ( 768 - 228 \beta ) q^{62} + 63 q^{63} + ( 1716 - 623 \beta ) q^{64} + ( -336 + 144 \beta ) q^{66} + ( 380 + 28 \beta ) q^{67} + ( -1048 + 294 \beta ) q^{68} + ( -360 + 168 \beta ) q^{69} + ( 62 + 330 \beta ) q^{71} + ( 396 - 225 \beta ) q^{72} + ( -300 - 178 \beta ) q^{73} + ( 120 + 18 \beta ) q^{74} + ( 160 - 388 \beta ) q^{76} + ( -126 + 70 \beta ) q^{77} + ( 312 - 210 \beta ) q^{78} + ( 248 - 88 \beta ) q^{79} + 81 q^{81} + ( 1240 - 590 \beta ) q^{82} + ( -436 - 264 \beta ) q^{83} + ( 252 - 147 \beta ) q^{84} + ( -320 - 56 \beta ) q^{86} + ( -174 + 252 \beta ) q^{87} + ( -1792 + 640 \beta ) q^{88} + ( -346 + 728 \beta ) q^{89} + ( 28 - 154 \beta ) q^{91} + ( -3008 + 1120 \beta ) q^{92} + ( 522 - 54 \beta ) q^{93} + ( -1232 + 524 \beta ) q^{94} + ( 1860 - 507 \beta ) q^{96} + ( 176 + 146 \beta ) q^{97} + ( 196 - 49 \beta ) q^{98} + ( -162 + 90 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 7q^{2} + 6q^{3} + 17q^{4} + 21q^{6} + 14q^{7} + 63q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 7q^{2} + 6q^{3} + 17q^{4} + 21q^{6} + 14q^{7} + 63q^{8} + 18q^{9} - 26q^{11} + 51q^{12} - 14q^{13} + 49q^{14} + 297q^{16} - 16q^{17} + 63q^{18} + 174q^{19} + 42q^{21} - 176q^{22} - 184q^{23} + 189q^{24} + 138q^{26} + 54q^{27} + 119q^{28} - 32q^{29} + 330q^{31} + 1071q^{32} - 78q^{33} - 294q^{34} + 153q^{36} + 132q^{37} + 388q^{38} - 42q^{39} + 200q^{41} + 147q^{42} - 364q^{43} - 816q^{44} - 1120q^{46} - 292q^{47} + 891q^{48} + 98q^{49} - 48q^{51} + 1190q^{52} - 34q^{53} + 189q^{54} + 441q^{56} + 522q^{57} - 826q^{58} + 364q^{59} + 792q^{61} + 1308q^{62} + 126q^{63} + 2809q^{64} - 528q^{66} + 788q^{67} - 1802q^{68} - 552q^{69} + 454q^{71} + 567q^{72} - 778q^{73} + 258q^{74} - 68q^{76} - 182q^{77} + 414q^{78} + 408q^{79} + 162q^{81} + 1890q^{82} - 1136q^{83} + 357q^{84} - 696q^{86} - 96q^{87} - 2944q^{88} + 36q^{89} - 98q^{91} - 4896q^{92} + 990q^{93} - 1940q^{94} + 3213q^{96} + 498q^{97} + 343q^{98} - 234q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
1.43845 3.00000 −5.93087 0 4.31534 7.00000 −20.0388 9.00000 0
1.2 5.56155 3.00000 22.9309 0 16.6847 7.00000 83.0388 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.p 2
3.b odd 2 1 1575.4.a.m 2
5.b even 2 1 105.4.a.c 2
5.c odd 4 2 525.4.d.i 4
15.d odd 2 1 315.4.a.m 2
20.d odd 2 1 1680.4.a.bk 2
35.c odd 2 1 735.4.a.k 2
105.g even 2 1 2205.4.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.c 2 5.b even 2 1
315.4.a.m 2 15.d odd 2 1
525.4.a.p 2 1.a even 1 1 trivial
525.4.d.i 4 5.c odd 4 2
735.4.a.k 2 35.c odd 2 1
1575.4.a.m 2 3.b odd 2 1
1680.4.a.bk 2 20.d odd 2 1
2205.4.a.bh 2 105.g even 2 1

Hecke kernels

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

\( T_{2}^{2} - 7 T_{2} + 8 \)
\( T_{11}^{2} + 26 T_{11} - 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 7 T + 24 T^{2} - 56 T^{3} + 64 T^{4} \)
$3$ \( ( 1 - 3 T )^{2} \)
$5$ 1
$7$ \( ( 1 - 7 T )^{2} \)
$11$ \( 1 + 26 T + 2406 T^{2} + 34606 T^{3} + 1771561 T^{4} \)
$13$ \( 1 + 14 T + 2386 T^{2} + 30758 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 16 T + 6558 T^{2} + 78608 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 174 T + 18414 T^{2} - 1193466 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 184 T + 19470 T^{2} + 2238728 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 32 T + 19046 T^{2} + 780448 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 330 T + 85430 T^{2} - 9831030 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 132 T + 103214 T^{2} - 6686196 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 - 200 T + 64542 T^{2} - 13784200 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 364 T + 172486 T^{2} + 28940548 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 + 292 T + 179390 T^{2} + 30316316 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 34 T + 103410 T^{2} + 5061818 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 364 T + 438374 T^{2} - 74757956 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 792 T + 340886 T^{2} - 179768952 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 788 T + 753430 T^{2} - 237001244 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 454 T + 304526 T^{2} - 162491594 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 + 778 T + 794698 T^{2} + 302655226 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 408 T + 994782 T^{2} - 201159912 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 + 1136 T + 1169990 T^{2} + 649550032 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 - 36 T - 842170 T^{2} - 25378884 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 - 498 T + 1796754 T^{2} - 454511154 T^{3} + 832972004929 T^{4} \)
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