Properties

Label 525.4.a.i
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: \(\Q(\sqrt{41}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{2} -3 q^{3} + ( 3 + 3 \beta ) q^{4} + ( 3 + 3 \beta ) q^{6} -7 q^{7} + ( -25 - \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( -1 - \beta ) q^{2} -3 q^{3} + ( 3 + 3 \beta ) q^{4} + ( 3 + 3 \beta ) q^{6} -7 q^{7} + ( -25 - \beta ) q^{8} + 9 q^{9} + ( 32 - 2 \beta ) q^{11} + ( -9 - 9 \beta ) q^{12} + ( -2 + 10 \beta ) q^{13} + ( 7 + 7 \beta ) q^{14} + ( 11 + 3 \beta ) q^{16} + ( -26 + 12 \beta ) q^{17} + ( -9 - 9 \beta ) q^{18} + ( -60 - 2 \beta ) q^{19} + 21 q^{21} + ( -12 - 28 \beta ) q^{22} + ( -32 + 48 \beta ) q^{23} + ( 75 + 3 \beta ) q^{24} + ( -98 - 18 \beta ) q^{26} -27 q^{27} + ( -21 - 21 \beta ) q^{28} + ( 170 + 12 \beta ) q^{29} + ( 52 - 38 \beta ) q^{31} + ( 159 - 9 \beta ) q^{32} + ( -96 + 6 \beta ) q^{33} + ( -94 + 2 \beta ) q^{34} + ( 27 + 27 \beta ) q^{36} + ( 134 - 80 \beta ) q^{37} + ( 80 + 64 \beta ) q^{38} + ( 6 - 30 \beta ) q^{39} + ( 62 - 108 \beta ) q^{41} + ( -21 - 21 \beta ) q^{42} + ( 248 - 100 \beta ) q^{43} + ( 36 + 84 \beta ) q^{44} + ( -448 - 64 \beta ) q^{46} + ( 164 - 140 \beta ) q^{47} + ( -33 - 9 \beta ) q^{48} + 49 q^{49} + ( 78 - 36 \beta ) q^{51} + ( 294 + 54 \beta ) q^{52} + ( -462 - 58 \beta ) q^{53} + ( 27 + 27 \beta ) q^{54} + ( 175 + 7 \beta ) q^{56} + ( 180 + 6 \beta ) q^{57} + ( -290 - 194 \beta ) q^{58} + ( 220 + 76 \beta ) q^{59} + ( -398 - 84 \beta ) q^{61} + ( 328 + 24 \beta ) q^{62} -63 q^{63} + ( -157 - 165 \beta ) q^{64} + ( 36 + 84 \beta ) q^{66} + ( 64 + 228 \beta ) q^{67} + ( 282 - 6 \beta ) q^{68} + ( 96 - 144 \beta ) q^{69} + ( 112 + 86 \beta ) q^{71} + ( -225 - 9 \beta ) q^{72} + ( -182 + 38 \beta ) q^{73} + ( 666 + 26 \beta ) q^{74} + ( -240 - 192 \beta ) q^{76} + ( -224 + 14 \beta ) q^{77} + ( 294 + 54 \beta ) q^{78} + ( 920 - 8 \beta ) q^{79} + 81 q^{81} + ( 1018 + 154 \beta ) q^{82} + ( 228 + 224 \beta ) q^{83} + ( 63 + 63 \beta ) q^{84} + ( 752 - 48 \beta ) q^{86} + ( -510 - 36 \beta ) q^{87} + ( -780 + 20 \beta ) q^{88} + ( 230 + 336 \beta ) q^{89} + ( 14 - 70 \beta ) q^{91} + ( 1344 + 192 \beta ) q^{92} + ( -156 + 114 \beta ) q^{93} + ( 1236 + 116 \beta ) q^{94} + ( -477 + 27 \beta ) q^{96} + ( 474 - 278 \beta ) q^{97} + ( -49 - 49 \beta ) q^{98} + ( 288 - 18 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} - 6q^{3} + 9q^{4} + 9q^{6} - 14q^{7} - 51q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 3q^{2} - 6q^{3} + 9q^{4} + 9q^{6} - 14q^{7} - 51q^{8} + 18q^{9} + 62q^{11} - 27q^{12} + 6q^{13} + 21q^{14} + 25q^{16} - 40q^{17} - 27q^{18} - 122q^{19} + 42q^{21} - 52q^{22} - 16q^{23} + 153q^{24} - 214q^{26} - 54q^{27} - 63q^{28} + 352q^{29} + 66q^{31} + 309q^{32} - 186q^{33} - 186q^{34} + 81q^{36} + 188q^{37} + 224q^{38} - 18q^{39} + 16q^{41} - 63q^{42} + 396q^{43} + 156q^{44} - 960q^{46} + 188q^{47} - 75q^{48} + 98q^{49} + 120q^{51} + 642q^{52} - 982q^{53} + 81q^{54} + 357q^{56} + 366q^{57} - 774q^{58} + 516q^{59} - 880q^{61} + 680q^{62} - 126q^{63} - 479q^{64} + 156q^{66} + 356q^{67} + 558q^{68} + 48q^{69} + 310q^{71} - 459q^{72} - 326q^{73} + 1358q^{74} - 672q^{76} - 434q^{77} + 642q^{78} + 1832q^{79} + 162q^{81} + 2190q^{82} + 680q^{83} + 189q^{84} + 1456q^{86} - 1056q^{87} - 1540q^{88} + 796q^{89} - 42q^{91} + 2880q^{92} - 198q^{93} + 2588q^{94} - 927q^{96} + 670q^{97} - 147q^{98} + 558q^{99} + O(q^{100}) \)

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
3.70156
−2.70156
−4.70156 −3.00000 14.1047 0 14.1047 −7.00000 −28.7016 9.00000 0
1.2 1.70156 −3.00000 −5.10469 0 −5.10469 −7.00000 −22.2984 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.i 2
3.b odd 2 1 1575.4.a.y 2
5.b even 2 1 105.4.a.g 2
5.c odd 4 2 525.4.d.j 4
15.d odd 2 1 315.4.a.g 2
20.d odd 2 1 1680.4.a.y 2
35.c odd 2 1 735.4.a.q 2
105.g even 2 1 2205.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 5.b even 2 1
315.4.a.g 2 15.d odd 2 1
525.4.a.i 2 1.a even 1 1 trivial
525.4.d.j 4 5.c odd 4 2
735.4.a.q 2 35.c odd 2 1
1575.4.a.y 2 3.b odd 2 1
1680.4.a.y 2 20.d odd 2 1
2205.4.a.v 2 105.g even 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(1\)

Hecke kernels

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

\( T_{2}^{2} + 3 T_{2} - 8 \)
\( T_{11}^{2} - 62 T_{11} + 920 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 8 T^{2} + 24 T^{3} + 64 T^{4} \)
$3$ \( ( 1 + 3 T )^{2} \)
$5$ 1
$7$ \( ( 1 + 7 T )^{2} \)
$11$ \( 1 - 62 T + 3582 T^{2} - 82522 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 6 T + 3378 T^{2} - 13182 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 40 T + 8750 T^{2} + 196520 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 122 T + 17398 T^{2} + 836798 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 16 T + 782 T^{2} + 194672 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 352 T + 78278 T^{2} - 8584928 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 66 T + 45870 T^{2} - 1966206 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 188 T + 44542 T^{2} - 9522764 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 - 16 T + 18350 T^{2} - 1102736 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 - 396 T + 95718 T^{2} - 31484772 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 188 T + 15582 T^{2} - 19518724 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 982 T + 504354 T^{2} + 146197214 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 516 T + 418118 T^{2} - 105975564 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 880 T + 575238 T^{2} + 199743280 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 356 T + 100374 T^{2} - 107071628 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 310 T + 664038 T^{2} - 110952410 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 + 326 T + 789802 T^{2} + 126819542 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 1832 T + 1824478 T^{2} - 903247448 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 680 T + 744870 T^{2} - 388815160 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 - 796 T + 411158 T^{2} - 561155324 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 - 670 T + 1145410 T^{2} - 611490910 T^{3} + 832972004929 T^{4} \)
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