Properties

Label 525.4.a.j
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{2} + 3 q^{3} + ( -3 + 3 \beta ) q^{4} + ( -3 - 3 \beta ) q^{6} + 7 q^{7} + ( -1 + 5 \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( -1 - \beta ) q^{2} + 3 q^{3} + ( -3 + 3 \beta ) q^{4} + ( -3 - 3 \beta ) q^{6} + 7 q^{7} + ( -1 + 5 \beta ) q^{8} + 9 q^{9} + ( -18 + 5 \beta ) q^{11} + ( -9 + 9 \beta ) q^{12} + ( -11 - 17 \beta ) q^{13} + ( -7 - 7 \beta ) q^{14} + ( 5 - 33 \beta ) q^{16} + ( 53 - 27 \beta ) q^{17} + ( -9 - 9 \beta ) q^{18} + ( -56 + 56 \beta ) q^{19} + 21 q^{21} + ( -2 + 8 \beta ) q^{22} + ( -115 - 24 \beta ) q^{23} + ( -3 + 15 \beta ) q^{24} + ( 79 + 45 \beta ) q^{26} + 27 q^{27} + ( -21 + 21 \beta ) q^{28} + ( -73 + 84 \beta ) q^{29} + ( -61 - 13 \beta ) q^{31} + ( 135 + 21 \beta ) q^{32} + ( -54 + 15 \beta ) q^{33} + ( 55 + \beta ) q^{34} + ( -27 + 27 \beta ) q^{36} + ( -66 + 19 \beta ) q^{37} + ( -168 - 56 \beta ) q^{38} + ( -33 - 51 \beta ) q^{39} + ( 95 + 45 \beta ) q^{41} + ( -21 - 21 \beta ) q^{42} + ( -373 - 58 \beta ) q^{43} + ( 114 - 54 \beta ) q^{44} + ( 211 + 163 \beta ) q^{46} + ( -120 + 88 \beta ) q^{47} + ( 15 - 99 \beta ) q^{48} + 49 q^{49} + ( 159 - 81 \beta ) q^{51} + ( -171 - 33 \beta ) q^{52} + ( -119 + 89 \beta ) q^{53} + ( -27 - 27 \beta ) q^{54} + ( -7 + 35 \beta ) q^{56} + ( -168 + 168 \beta ) q^{57} + ( -263 - 95 \beta ) q^{58} + ( -325 + 209 \beta ) q^{59} + ( 25 - 273 \beta ) q^{61} + ( 113 + 87 \beta ) q^{62} + 63 q^{63} + ( -259 + 87 \beta ) q^{64} + ( -6 + 24 \beta ) q^{66} + ( -640 + 123 \beta ) q^{67} + ( -483 + 159 \beta ) q^{68} + ( -345 - 72 \beta ) q^{69} + ( 192 + 235 \beta ) q^{71} + ( -9 + 45 \beta ) q^{72} + ( -90 - 88 \beta ) q^{73} + ( -10 + 28 \beta ) q^{74} + ( 840 - 168 \beta ) q^{76} + ( -126 + 35 \beta ) q^{77} + ( 237 + 135 \beta ) q^{78} + ( -162 - 103 \beta ) q^{79} + 81 q^{81} + ( -275 - 185 \beta ) q^{82} + ( -821 + 431 \beta ) q^{83} + ( -63 + 63 \beta ) q^{84} + ( 605 + 489 \beta ) q^{86} + ( -219 + 252 \beta ) q^{87} + ( 118 - 70 \beta ) q^{88} + ( 494 - 522 \beta ) q^{89} + ( -77 - 119 \beta ) q^{91} + ( 57 - 345 \beta ) q^{92} + ( -183 - 39 \beta ) q^{93} + ( -232 - 56 \beta ) q^{94} + ( 405 + 63 \beta ) q^{96} + ( 266 - 704 \beta ) q^{97} + ( -49 - 49 \beta ) q^{98} + ( -162 + 45 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + 6q^{3} - 3q^{4} - 9q^{6} + 14q^{7} + 3q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 3q^{2} + 6q^{3} - 3q^{4} - 9q^{6} + 14q^{7} + 3q^{8} + 18q^{9} - 31q^{11} - 9q^{12} - 39q^{13} - 21q^{14} - 23q^{16} + 79q^{17} - 27q^{18} - 56q^{19} + 42q^{21} + 4q^{22} - 254q^{23} + 9q^{24} + 203q^{26} + 54q^{27} - 21q^{28} - 62q^{29} - 135q^{31} + 291q^{32} - 93q^{33} + 111q^{34} - 27q^{36} - 113q^{37} - 392q^{38} - 117q^{39} + 235q^{41} - 63q^{42} - 804q^{43} + 174q^{44} + 585q^{46} - 152q^{47} - 69q^{48} + 98q^{49} + 237q^{51} - 375q^{52} - 149q^{53} - 81q^{54} + 21q^{56} - 168q^{57} - 621q^{58} - 441q^{59} - 223q^{61} + 313q^{62} + 126q^{63} - 431q^{64} + 12q^{66} - 1157q^{67} - 807q^{68} - 762q^{69} + 619q^{71} + 27q^{72} - 268q^{73} + 8q^{74} + 1512q^{76} - 217q^{77} + 609q^{78} - 427q^{79} + 162q^{81} - 735q^{82} - 1211q^{83} - 63q^{84} + 1699q^{86} - 186q^{87} + 166q^{88} + 466q^{89} - 273q^{91} - 231q^{92} - 405q^{93} - 520q^{94} + 873q^{96} - 172q^{97} - 147q^{98} - 279q^{99} + O(q^{100}) \)

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
−3.56155 3.00000 4.68466 0 −10.6847 7.00000 11.8078 9.00000 0
1.2 0.561553 3.00000 −7.68466 0 1.68466 7.00000 −8.80776 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.j 2
3.b odd 2 1 1575.4.a.x 2
5.b even 2 1 525.4.a.m yes 2
5.c odd 4 2 525.4.d.m 4
15.d odd 2 1 1575.4.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.j 2 1.a even 1 1 trivial
525.4.a.m yes 2 5.b even 2 1
525.4.d.m 4 5.c odd 4 2
1575.4.a.o 2 15.d odd 2 1
1575.4.a.x 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2}^{2} + 3 T_{2} - 2 \)
\( T_{11}^{2} + 31 T_{11} + 134 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 14 T^{2} + 24 T^{3} + 64 T^{4} \)
$3$ \( ( 1 - 3 T )^{2} \)
$5$ 1
$7$ \( ( 1 - 7 T )^{2} \)
$11$ \( 1 + 31 T + 2796 T^{2} + 41261 T^{3} + 1771561 T^{4} \)
$13$ \( 1 + 39 T + 3546 T^{2} + 85683 T^{3} + 4826809 T^{4} \)
$17$ \( 1 - 79 T + 8288 T^{2} - 388127 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 56 T + 1174 T^{2} + 384104 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 254 T + 38015 T^{2} + 3090418 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 62 T + 19751 T^{2} + 1512118 T^{3} + 594823321 T^{4} \)
$31$ \( 1 + 135 T + 63420 T^{2} + 4021785 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 113 T + 102964 T^{2} + 5723789 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 - 235 T + 143042 T^{2} - 16196435 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 804 T + 306321 T^{2} + 63923628 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 + 152 T + 180510 T^{2} + 15781096 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 149 T + 269640 T^{2} + 22182673 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 441 T + 273734 T^{2} + 90572139 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 223 T + 149646 T^{2} + 50616763 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 1157 T + 871890 T^{2} + 347982791 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 619 T + 576906 T^{2} - 221546909 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 + 268 T + 763078 T^{2} + 104256556 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 427 T + 986572 T^{2} + 210527653 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 + 1211 T + 720720 T^{2} + 692434057 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 - 466 T + 306170 T^{2} - 328515554 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 + 172 T - 273626 T^{2} + 156979756 T^{3} + 832972004929 T^{4} \)
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