Properties

Label 525.4.a.l
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + 3 q^{3} + (2 \beta + 1) q^{4} + (3 \beta + 3) q^{6} + 7 q^{7} + ( - 5 \beta + 9) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + 3 q^{3} + (2 \beta + 1) q^{4} + (3 \beta + 3) q^{6} + 7 q^{7} + ( - 5 \beta + 9) q^{8} + 9 q^{9} + (20 \beta - 8) q^{11} + (6 \beta + 3) q^{12} + (2 \beta + 38) q^{13} + (7 \beta + 7) q^{14} + ( - 12 \beta - 39) q^{16} + (2 \beta + 62) q^{17} + (9 \beta + 9) q^{18} + ( - 16 \beta - 48) q^{19} + 21 q^{21} + (12 \beta + 152) q^{22} + (34 \beta + 8) q^{23} + ( - 15 \beta + 27) q^{24} + (40 \beta + 54) q^{26} + 27 q^{27} + (14 \beta + 7) q^{28} + ( - 54 \beta + 94) q^{29} + (18 \beta - 60) q^{31} + ( - 11 \beta - 207) q^{32} + (60 \beta - 24) q^{33} + (64 \beta + 78) q^{34} + (18 \beta + 9) q^{36} + (66 \beta + 66) q^{37} + ( - 64 \beta - 176) q^{38} + (6 \beta + 114) q^{39} + (80 \beta + 50) q^{41} + (21 \beta + 21) q^{42} + ( - 62 \beta + 268) q^{43} + (4 \beta + 312) q^{44} + (42 \beta + 280) q^{46} + (42 \beta + 464) q^{47} + ( - 36 \beta - 117) q^{48} + 49 q^{49} + (6 \beta + 186) q^{51} + (78 \beta + 70) q^{52} + ( - 64 \beta - 442) q^{53} + (27 \beta + 27) q^{54} + ( - 35 \beta + 63) q^{56} + ( - 48 \beta - 144) q^{57} + (40 \beta - 338) q^{58} + ( - 204 \beta + 52) q^{59} + ( - 42 \beta - 234) q^{61} + ( - 42 \beta + 84) q^{62} + 63 q^{63} + ( - 122 \beta + 17) q^{64} + (36 \beta + 456) q^{66} + ( - 38 \beta + 844) q^{67} + (126 \beta + 94) q^{68} + (102 \beta + 24) q^{69} + ( - 150 \beta - 68) q^{71} + ( - 45 \beta + 81) q^{72} + ( - 322 \beta - 254) q^{73} + (132 \beta + 594) q^{74} + ( - 112 \beta - 304) q^{76} + (140 \beta - 56) q^{77} + (120 \beta + 162) q^{78} + ( - 232 \beta - 216) q^{79} + 81 q^{81} + (130 \beta + 690) q^{82} + (84 \beta + 292) q^{83} + (42 \beta + 21) q^{84} + (206 \beta - 228) q^{86} + ( - 162 \beta + 282) q^{87} + (220 \beta - 872) q^{88} + (112 \beta - 702) q^{89} + (14 \beta + 266) q^{91} + (50 \beta + 552) q^{92} + (54 \beta - 180) q^{93} + (506 \beta + 800) q^{94} + ( - 33 \beta - 621) q^{96} + ( - 46 \beta + 594) q^{97} + (49 \beta + 49) q^{98} + (180 \beta - 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 6 q^{3} + 2 q^{4} + 6 q^{6} + 14 q^{7} + 18 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 6 q^{3} + 2 q^{4} + 6 q^{6} + 14 q^{7} + 18 q^{8} + 18 q^{9} - 16 q^{11} + 6 q^{12} + 76 q^{13} + 14 q^{14} - 78 q^{16} + 124 q^{17} + 18 q^{18} - 96 q^{19} + 42 q^{21} + 304 q^{22} + 16 q^{23} + 54 q^{24} + 108 q^{26} + 54 q^{27} + 14 q^{28} + 188 q^{29} - 120 q^{31} - 414 q^{32} - 48 q^{33} + 156 q^{34} + 18 q^{36} + 132 q^{37} - 352 q^{38} + 228 q^{39} + 100 q^{41} + 42 q^{42} + 536 q^{43} + 624 q^{44} + 560 q^{46} + 928 q^{47} - 234 q^{48} + 98 q^{49} + 372 q^{51} + 140 q^{52} - 884 q^{53} + 54 q^{54} + 126 q^{56} - 288 q^{57} - 676 q^{58} + 104 q^{59} - 468 q^{61} + 168 q^{62} + 126 q^{63} + 34 q^{64} + 912 q^{66} + 1688 q^{67} + 188 q^{68} + 48 q^{69} - 136 q^{71} + 162 q^{72} - 508 q^{73} + 1188 q^{74} - 608 q^{76} - 112 q^{77} + 324 q^{78} - 432 q^{79} + 162 q^{81} + 1380 q^{82} + 584 q^{83} + 42 q^{84} - 456 q^{86} + 564 q^{87} - 1744 q^{88} - 1404 q^{89} + 532 q^{91} + 1104 q^{92} - 360 q^{93} + 1600 q^{94} - 1242 q^{96} + 1188 q^{97} + 98 q^{98} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−1.41421
1.41421
−1.82843 3.00000 −4.65685 0 −5.48528 7.00000 23.1421 9.00000 0
1.2 3.82843 3.00000 6.65685 0 11.4853 7.00000 −5.14214 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.l 2
3.b odd 2 1 1575.4.a.q 2
5.b even 2 1 105.4.a.e 2
5.c odd 4 2 525.4.d.l 4
15.d odd 2 1 315.4.a.k 2
20.d odd 2 1 1680.4.a.bo 2
35.c odd 2 1 735.4.a.o 2
105.g even 2 1 2205.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 5.b even 2 1
315.4.a.k 2 15.d odd 2 1
525.4.a.l 2 1.a even 1 1 trivial
525.4.d.l 4 5.c odd 4 2
735.4.a.o 2 35.c odd 2 1
1575.4.a.q 2 3.b odd 2 1
1680.4.a.bo 2 20.d odd 2 1
2205.4.a.bb 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2}^{2} - 2T_{2} - 7 \) Copy content Toggle raw display
\( T_{11}^{2} + 16T_{11} - 3136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16T - 3136 \) Copy content Toggle raw display
$13$ \( T^{2} - 76T + 1412 \) Copy content Toggle raw display
$17$ \( T^{2} - 124T + 3812 \) Copy content Toggle raw display
$19$ \( T^{2} + 96T + 256 \) Copy content Toggle raw display
$23$ \( T^{2} - 16T - 9184 \) Copy content Toggle raw display
$29$ \( T^{2} - 188T - 14492 \) Copy content Toggle raw display
$31$ \( T^{2} + 120T + 1008 \) Copy content Toggle raw display
$37$ \( T^{2} - 132T - 30492 \) Copy content Toggle raw display
$41$ \( T^{2} - 100T - 48700 \) Copy content Toggle raw display
$43$ \( T^{2} - 536T + 41072 \) Copy content Toggle raw display
$47$ \( T^{2} - 928T + 201184 \) Copy content Toggle raw display
$53$ \( T^{2} + 884T + 162596 \) Copy content Toggle raw display
$59$ \( T^{2} - 104T - 330224 \) Copy content Toggle raw display
$61$ \( T^{2} + 468T + 40644 \) Copy content Toggle raw display
$67$ \( T^{2} - 1688 T + 700784 \) Copy content Toggle raw display
$71$ \( T^{2} + 136T - 175376 \) Copy content Toggle raw display
$73$ \( T^{2} + 508T - 764956 \) Copy content Toggle raw display
$79$ \( T^{2} + 432T - 383936 \) Copy content Toggle raw display
$83$ \( T^{2} - 584T + 28816 \) Copy content Toggle raw display
$89$ \( T^{2} + 1404 T + 392452 \) Copy content Toggle raw display
$97$ \( T^{2} - 1188 T + 335908 \) Copy content Toggle raw display
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