Properties

Label 525.4.a.k
Level $525$
Weight $4$
Character orbit 525.a
Self dual yes
Analytic conductor $30.976$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
Defining polynomial: \(x^{2} - x - 16\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{65})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} -3 q^{3} + ( 8 + \beta ) q^{4} + 3 \beta q^{6} + 7 q^{7} + ( -16 - \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q -\beta q^{2} -3 q^{3} + ( 8 + \beta ) q^{4} + 3 \beta q^{6} + 7 q^{7} + ( -16 - \beta ) q^{8} + 9 q^{9} + ( -10 - 2 \beta ) q^{11} + ( -24 - 3 \beta ) q^{12} + ( 12 - 2 \beta ) q^{13} -7 \beta q^{14} + ( -48 + 9 \beta ) q^{16} + ( -66 + 16 \beta ) q^{17} -9 \beta q^{18} + ( 58 - 14 \beta ) q^{19} -21 q^{21} + ( 32 + 12 \beta ) q^{22} + ( -140 + 20 \beta ) q^{23} + ( 48 + 3 \beta ) q^{24} + ( 32 - 10 \beta ) q^{26} -27 q^{27} + ( 56 + 7 \beta ) q^{28} + ( -74 - 48 \beta ) q^{29} + ( 54 + 42 \beta ) q^{31} + ( -16 + 47 \beta ) q^{32} + ( 30 + 6 \beta ) q^{33} + ( -256 + 50 \beta ) q^{34} + ( 72 + 9 \beta ) q^{36} + ( 30 + 36 \beta ) q^{37} + ( 224 - 44 \beta ) q^{38} + ( -36 + 6 \beta ) q^{39} + ( -138 + 100 \beta ) q^{41} + 21 \beta q^{42} + ( 156 + 32 \beta ) q^{43} + ( -112 - 28 \beta ) q^{44} + ( -320 + 120 \beta ) q^{46} + ( -304 + 48 \beta ) q^{47} + ( 144 - 27 \beta ) q^{48} + 49 q^{49} + ( 198 - 48 \beta ) q^{51} + ( 64 - 6 \beta ) q^{52} + ( -120 - 86 \beta ) q^{53} + 27 \beta q^{54} + ( -112 - 7 \beta ) q^{56} + ( -174 + 42 \beta ) q^{57} + ( 768 + 122 \beta ) q^{58} + ( -464 + 84 \beta ) q^{59} + ( -114 + 24 \beta ) q^{61} + ( -672 - 96 \beta ) q^{62} + 63 q^{63} + ( -368 - 103 \beta ) q^{64} + ( -96 - 36 \beta ) q^{66} + ( 84 - 64 \beta ) q^{67} + ( -272 + 78 \beta ) q^{68} + ( 420 - 60 \beta ) q^{69} + ( 814 + 42 \beta ) q^{71} + ( -144 - 9 \beta ) q^{72} + ( 92 + 202 \beta ) q^{73} + ( -576 - 66 \beta ) q^{74} + ( 240 - 68 \beta ) q^{76} + ( -70 - 14 \beta ) q^{77} + ( -96 + 30 \beta ) q^{78} + ( -392 - 104 \beta ) q^{79} + 81 q^{81} + ( -1600 + 38 \beta ) q^{82} + ( -356 - 216 \beta ) q^{83} + ( -168 - 21 \beta ) q^{84} + ( -512 - 188 \beta ) q^{86} + ( 222 + 144 \beta ) q^{87} + ( 192 + 44 \beta ) q^{88} + ( 290 + 8 \beta ) q^{89} + ( 84 - 14 \beta ) q^{91} + ( -800 + 40 \beta ) q^{92} + ( -162 - 126 \beta ) q^{93} + ( -768 + 256 \beta ) q^{94} + ( 48 - 141 \beta ) q^{96} + ( -104 - 314 \beta ) q^{97} -49 \beta q^{98} + ( -90 - 18 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 6q^{3} + 17q^{4} + 3q^{6} + 14q^{7} - 33q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - q^{2} - 6q^{3} + 17q^{4} + 3q^{6} + 14q^{7} - 33q^{8} + 18q^{9} - 22q^{11} - 51q^{12} + 22q^{13} - 7q^{14} - 87q^{16} - 116q^{17} - 9q^{18} + 102q^{19} - 42q^{21} + 76q^{22} - 260q^{23} + 99q^{24} + 54q^{26} - 54q^{27} + 119q^{28} - 196q^{29} + 150q^{31} + 15q^{32} + 66q^{33} - 462q^{34} + 153q^{36} + 96q^{37} + 404q^{38} - 66q^{39} - 176q^{41} + 21q^{42} + 344q^{43} - 252q^{44} - 520q^{46} - 560q^{47} + 261q^{48} + 98q^{49} + 348q^{51} + 122q^{52} - 326q^{53} + 27q^{54} - 231q^{56} - 306q^{57} + 1658q^{58} - 844q^{59} - 204q^{61} - 1440q^{62} + 126q^{63} - 839q^{64} - 228q^{66} + 104q^{67} - 466q^{68} + 780q^{69} + 1670q^{71} - 297q^{72} + 386q^{73} - 1218q^{74} + 412q^{76} - 154q^{77} - 162q^{78} - 888q^{79} + 162q^{81} - 3162q^{82} - 928q^{83} - 357q^{84} - 1212q^{86} + 588q^{87} + 428q^{88} + 588q^{89} + 154q^{91} - 1560q^{92} - 450q^{93} - 1280q^{94} - 45q^{96} - 522q^{97} - 49q^{98} - 198q^{99} + O(q^{100}) \)

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
4.53113
−3.53113
−4.53113 −3.00000 12.5311 0 13.5934 7.00000 −20.5311 9.00000 0
1.2 3.53113 −3.00000 4.46887 0 −10.5934 7.00000 −12.4689 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.k 2
3.b odd 2 1 1575.4.a.w 2
5.b even 2 1 105.4.a.f 2
5.c odd 4 2 525.4.d.h 4
15.d odd 2 1 315.4.a.i 2
20.d odd 2 1 1680.4.a.bg 2
35.c odd 2 1 735.4.a.p 2
105.g even 2 1 2205.4.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.f 2 5.b even 2 1
315.4.a.i 2 15.d odd 2 1
525.4.a.k 2 1.a even 1 1 trivial
525.4.d.h 4 5.c odd 4 2
735.4.a.p 2 35.c odd 2 1
1575.4.a.w 2 3.b odd 2 1
1680.4.a.bg 2 20.d odd 2 1
2205.4.a.z 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2}^{2} + T_{2} - 16 \)
\( T_{11}^{2} + 22 T_{11} + 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 8 T^{3} + 64 T^{4} \)
$3$ \( ( 1 + 3 T )^{2} \)
$5$ 1
$7$ \( ( 1 - 7 T )^{2} \)
$11$ \( 1 + 22 T + 2718 T^{2} + 29282 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 22 T + 4450 T^{2} - 48334 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 116 T + 9030 T^{2} + 569908 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 102 T + 13134 T^{2} - 699618 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 260 T + 34734 T^{2} + 3163420 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 196 T + 20942 T^{2} + 4780244 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 150 T + 36542 T^{2} - 4468650 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 96 T + 82550 T^{2} - 4862688 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 + 176 T - 16914 T^{2} + 12130096 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 - 344 T + 171958 T^{2} - 27350408 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 + 560 T + 248606 T^{2} + 58140880 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 326 T + 204138 T^{2} + 48533902 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 844 T + 474182 T^{2} + 173339876 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 204 T + 455006 T^{2} + 46304124 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 104 T + 537670 T^{2} - 31279352 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 1670 T + 1384382 T^{2} - 597711370 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 - 386 T + 152218 T^{2} - 150160562 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 888 T + 1007454 T^{2} + 437818632 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 + 928 T + 600710 T^{2} + 530618336 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 - 588 T + 1495334 T^{2} - 414521772 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 + 522 T + 291282 T^{2} + 476415306 T^{3} + 832972004929 T^{4} \)
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