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 $1$

Related objects

Downloads

Learn more

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: \( x^{2} - x - 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + 3 q^{3} + (3 \beta - 3) q^{4} + ( - 3 \beta - 3) q^{6} + 7 q^{7} + (5 \beta - 1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + 3 q^{3} + (3 \beta - 3) q^{4} + ( - 3 \beta - 3) q^{6} + 7 q^{7} + (5 \beta - 1) q^{8} + 9 q^{9} + (5 \beta - 18) q^{11} + (9 \beta - 9) q^{12} + ( - 17 \beta - 11) q^{13} + ( - 7 \beta - 7) q^{14} + ( - 33 \beta + 5) q^{16} + ( - 27 \beta + 53) q^{17} + ( - 9 \beta - 9) q^{18} + (56 \beta - 56) q^{19} + 21 q^{21} + (8 \beta - 2) q^{22} + ( - 24 \beta - 115) q^{23} + (15 \beta - 3) q^{24} + (45 \beta + 79) q^{26} + 27 q^{27} + (21 \beta - 21) q^{28} + (84 \beta - 73) q^{29} + ( - 13 \beta - 61) q^{31} + (21 \beta + 135) q^{32} + (15 \beta - 54) q^{33} + (\beta + 55) q^{34} + (27 \beta - 27) q^{36} + (19 \beta - 66) q^{37} + ( - 56 \beta - 168) q^{38} + ( - 51 \beta - 33) q^{39} + (45 \beta + 95) q^{41} + ( - 21 \beta - 21) q^{42} + ( - 58 \beta - 373) q^{43} + ( - 54 \beta + 114) q^{44} + (163 \beta + 211) q^{46} + (88 \beta - 120) q^{47} + ( - 99 \beta + 15) q^{48} + 49 q^{49} + ( - 81 \beta + 159) q^{51} + ( - 33 \beta - 171) q^{52} + (89 \beta - 119) q^{53} + ( - 27 \beta - 27) q^{54} + (35 \beta - 7) q^{56} + (168 \beta - 168) q^{57} + ( - 95 \beta - 263) q^{58} + (209 \beta - 325) q^{59} + ( - 273 \beta + 25) q^{61} + (87 \beta + 113) q^{62} + 63 q^{63} + (87 \beta - 259) q^{64} + (24 \beta - 6) q^{66} + (123 \beta - 640) q^{67} + (159 \beta - 483) q^{68} + ( - 72 \beta - 345) q^{69} + (235 \beta + 192) q^{71} + (45 \beta - 9) q^{72} + ( - 88 \beta - 90) q^{73} + (28 \beta - 10) q^{74} + ( - 168 \beta + 840) q^{76} + (35 \beta - 126) q^{77} + (135 \beta + 237) q^{78} + ( - 103 \beta - 162) q^{79} + 81 q^{81} + ( - 185 \beta - 275) q^{82} + (431 \beta - 821) q^{83} + (63 \beta - 63) q^{84} + (489 \beta + 605) q^{86} + (252 \beta - 219) q^{87} + ( - 70 \beta + 118) q^{88} + ( - 522 \beta + 494) q^{89} + ( - 119 \beta - 77) q^{91} + ( - 345 \beta + 57) q^{92} + ( - 39 \beta - 183) q^{93} + ( - 56 \beta - 232) q^{94} + (63 \beta + 405) q^{96} + ( - 704 \beta + 266) q^{97} + ( - 49 \beta - 49) q^{98} + (45 \beta - 162) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9} - 31 q^{11} - 9 q^{12} - 39 q^{13} - 21 q^{14} - 23 q^{16} + 79 q^{17} - 27 q^{18} - 56 q^{19} + 42 q^{21} + 4 q^{22} - 254 q^{23} + 9 q^{24} + 203 q^{26} + 54 q^{27} - 21 q^{28} - 62 q^{29} - 135 q^{31} + 291 q^{32} - 93 q^{33} + 111 q^{34} - 27 q^{36} - 113 q^{37} - 392 q^{38} - 117 q^{39} + 235 q^{41} - 63 q^{42} - 804 q^{43} + 174 q^{44} + 585 q^{46} - 152 q^{47} - 69 q^{48} + 98 q^{49} + 237 q^{51} - 375 q^{52} - 149 q^{53} - 81 q^{54} + 21 q^{56} - 168 q^{57} - 621 q^{58} - 441 q^{59} - 223 q^{61} + 313 q^{62} + 126 q^{63} - 431 q^{64} + 12 q^{66} - 1157 q^{67} - 807 q^{68} - 762 q^{69} + 619 q^{71} + 27 q^{72} - 268 q^{73} + 8 q^{74} + 1512 q^{76} - 217 q^{77} + 609 q^{78} - 427 q^{79} + 162 q^{81} - 735 q^{82} - 1211 q^{83} - 63 q^{84} + 1699 q^{86} - 186 q^{87} + 166 q^{88} + 466 q^{89} - 273 q^{91} - 231 q^{92} - 405 q^{93} - 520 q^{94} + 873 q^{96} - 172 q^{97} - 147 q^{98} - 279 q^{99}+O(q^{100}) \) 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
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2}^{2} + 3T_{2} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 31T_{11} + 134 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 2 \) 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
$11$ \( T^{2} + 31T + 134 \) Copy content Toggle raw display
$13$ \( T^{2} + 39T - 848 \) Copy content Toggle raw display
$17$ \( T^{2} - 79T - 1538 \) Copy content Toggle raw display
$19$ \( T^{2} + 56T - 12544 \) Copy content Toggle raw display
$23$ \( T^{2} + 254T + 13681 \) Copy content Toggle raw display
$29$ \( T^{2} + 62T - 29027 \) Copy content Toggle raw display
$31$ \( T^{2} + 135T + 3838 \) Copy content Toggle raw display
$37$ \( T^{2} + 113T + 1658 \) Copy content Toggle raw display
$41$ \( T^{2} - 235T + 5200 \) Copy content Toggle raw display
$43$ \( T^{2} + 804T + 147307 \) Copy content Toggle raw display
$47$ \( T^{2} + 152T - 27136 \) Copy content Toggle raw display
$53$ \( T^{2} + 149T - 28114 \) Copy content Toggle raw display
$59$ \( T^{2} + 441T - 137024 \) Copy content Toggle raw display
$61$ \( T^{2} + 223T - 304316 \) Copy content Toggle raw display
$67$ \( T^{2} + 1157 T + 270364 \) Copy content Toggle raw display
$71$ \( T^{2} - 619T - 138916 \) Copy content Toggle raw display
$73$ \( T^{2} + 268T - 14956 \) Copy content Toggle raw display
$79$ \( T^{2} + 427T + 494 \) Copy content Toggle raw display
$83$ \( T^{2} + 1211 T - 422854 \) Copy content Toggle raw display
$89$ \( T^{2} - 466 T - 1103768 \) Copy content Toggle raw display
$97$ \( T^{2} + 172 T - 2098972 \) Copy content Toggle raw display
show more
show less