Properties

Label 525.4.d.m
Level $525$
Weight $4$
Character orbit 525.d
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + 3 \beta_{3} q^{4} + ( -6 + 3 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} + ( 5 \beta_{1} + \beta_{2} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + 3 \beta_{3} q^{4} + ( -6 + 3 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} + ( 5 \beta_{1} + \beta_{2} ) q^{8} -9 q^{9} + ( -13 - 5 \beta_{3} ) q^{11} + ( -9 \beta_{1} - 9 \beta_{2} ) q^{12} + ( -17 \beta_{1} + 11 \beta_{2} ) q^{13} + ( 14 - 7 \beta_{3} ) q^{14} + ( -28 + 33 \beta_{3} ) q^{16} + ( 27 \beta_{1} + 53 \beta_{2} ) q^{17} + ( -9 \beta_{1} + 9 \beta_{2} ) q^{18} + 56 \beta_{3} q^{19} + 21 q^{21} + ( -8 \beta_{1} - 2 \beta_{2} ) q^{22} + ( -24 \beta_{1} + 115 \beta_{2} ) q^{23} + ( -12 + 15 \beta_{3} ) q^{24} + ( 124 - 45 \beta_{3} ) q^{26} + 27 \beta_{2} q^{27} + ( 21 \beta_{1} + 21 \beta_{2} ) q^{28} + ( -11 + 84 \beta_{3} ) q^{29} + ( -74 + 13 \beta_{3} ) q^{31} + ( -21 \beta_{1} + 135 \beta_{2} ) q^{32} + ( 15 \beta_{1} + 54 \beta_{2} ) q^{33} + ( -56 + \beta_{3} ) q^{34} -27 \beta_{3} q^{36} + ( -19 \beta_{1} - 66 \beta_{2} ) q^{37} + ( -56 \beta_{1} + 168 \beta_{2} ) q^{38} + ( 84 - 51 \beta_{3} ) q^{39} + ( 140 - 45 \beta_{3} ) q^{41} + ( 21 \beta_{1} - 21 \beta_{2} ) q^{42} + ( -58 \beta_{1} + 373 \beta_{2} ) q^{43} + ( -60 - 54 \beta_{3} ) q^{44} + ( 374 - 163 \beta_{3} ) q^{46} + ( -88 \beta_{1} - 120 \beta_{2} ) q^{47} + ( -99 \beta_{1} - 15 \beta_{2} ) q^{48} -49 q^{49} + ( 78 + 81 \beta_{3} ) q^{51} + ( 33 \beta_{1} - 171 \beta_{2} ) q^{52} + ( 89 \beta_{1} + 119 \beta_{2} ) q^{53} + ( 54 - 27 \beta_{3} ) q^{54} + ( 28 - 35 \beta_{3} ) q^{56} + ( -168 \beta_{1} - 168 \beta_{2} ) q^{57} + ( -95 \beta_{1} + 263 \beta_{2} ) q^{58} + ( 116 + 209 \beta_{3} ) q^{59} + ( -248 + 273 \beta_{3} ) q^{61} + ( -87 \beta_{1} + 113 \beta_{2} ) q^{62} -63 \beta_{2} q^{63} + ( 172 + 87 \beta_{3} ) q^{64} + ( 18 - 24 \beta_{3} ) q^{66} + ( -123 \beta_{1} - 640 \beta_{2} ) q^{67} + ( 159 \beta_{1} + 483 \beta_{2} ) q^{68} + ( 417 - 72 \beta_{3} ) q^{69} + ( 427 - 235 \beta_{3} ) q^{71} + ( -45 \beta_{1} - 9 \beta_{2} ) q^{72} + ( -88 \beta_{1} + 90 \beta_{2} ) q^{73} + ( -18 + 28 \beta_{3} ) q^{74} + ( 672 + 168 \beta_{3} ) q^{76} + ( -35 \beta_{1} - 126 \beta_{2} ) q^{77} + ( 135 \beta_{1} - 237 \beta_{2} ) q^{78} + ( 265 - 103 \beta_{3} ) q^{79} + 81 q^{81} + ( 185 \beta_{1} - 275 \beta_{2} ) q^{82} + ( 431 \beta_{1} + 821 \beta_{2} ) q^{83} + 63 \beta_{3} q^{84} + ( 1094 - 489 \beta_{3} ) q^{86} + ( -252 \beta_{1} - 219 \beta_{2} ) q^{87} + ( -70 \beta_{1} - 118 \beta_{2} ) q^{88} + ( 28 - 522 \beta_{3} ) q^{89} + ( -196 + 119 \beta_{3} ) q^{91} + ( 345 \beta_{1} + 57 \beta_{2} ) q^{92} + ( -39 \beta_{1} + 183 \beta_{2} ) q^{93} + ( 288 - 56 \beta_{3} ) q^{94} + ( 468 - 63 \beta_{3} ) q^{96} + ( 704 \beta_{1} + 266 \beta_{2} ) q^{97} + ( -49 \beta_{1} + 49 \beta_{2} ) q^{98} + ( 117 + 45 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} - 18q^{6} - 36q^{9} + O(q^{10}) \) \( 4q + 6q^{4} - 18q^{6} - 36q^{9} - 62q^{11} + 42q^{14} - 46q^{16} + 112q^{19} + 84q^{21} - 18q^{24} + 406q^{26} + 124q^{29} - 270q^{31} - 222q^{34} - 54q^{36} + 234q^{39} + 470q^{41} - 348q^{44} + 1170q^{46} - 196q^{49} + 474q^{51} + 162q^{54} + 42q^{56} + 882q^{59} - 446q^{61} + 862q^{64} + 24q^{66} + 1524q^{69} + 1238q^{71} - 16q^{74} + 3024q^{76} + 854q^{79} + 324q^{81} + 126q^{84} + 3398q^{86} - 932q^{89} - 546q^{91} + 1040q^{94} + 1746q^{96} + 558q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 5\)
\(\nu^{3}\)\(=\)\(4 \beta_{2} - 5 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
2.56155i
1.56155i
1.56155i
2.56155i
3.56155i 3.00000i −4.68466 0 −10.6847 7.00000i 11.8078i −9.00000 0
274.2 0.561553i 3.00000i 7.68466 0 1.68466 7.00000i 8.80776i −9.00000 0
274.3 0.561553i 3.00000i 7.68466 0 1.68466 7.00000i 8.80776i −9.00000 0
274.4 3.56155i 3.00000i −4.68466 0 −10.6847 7.00000i 11.8078i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.4.d.m 4
5.b even 2 1 inner 525.4.d.m 4
5.c odd 4 1 525.4.a.j 2
5.c odd 4 1 525.4.a.m yes 2
15.e even 4 1 1575.4.a.o 2
15.e even 4 1 1575.4.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.j 2 5.c odd 4 1
525.4.a.m yes 2 5.c odd 4 1
525.4.d.m 4 1.a even 1 1 trivial
525.4.d.m 4 5.b even 2 1 inner
1575.4.a.o 2 15.e even 4 1
1575.4.a.x 2 15.e even 4 1

Hecke kernels

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

\( T_{2}^{4} + 13 T_{2}^{2} + 4 \)
\( T_{11}^{2} + 31 T_{11} + 134 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 19 T^{2} + 180 T^{4} - 1216 T^{6} + 4096 T^{8} \)
$3$ \( ( 1 + 9 T^{2} )^{2} \)
$5$ 1
$7$ \( ( 1 + 49 T^{2} )^{2} \)
$11$ \( ( 1 + 31 T + 2796 T^{2} + 41261 T^{3} + 1771561 T^{4} )^{2} \)
$13$ \( 1 - 5571 T^{2} + 15544460 T^{4} - 26890152939 T^{6} + 23298085122481 T^{8} \)
$17$ \( 1 - 10335 T^{2} + 55642016 T^{4} - 249461775615 T^{6} + 582622237229761 T^{8} \)
$19$ \( ( 1 - 56 T + 1174 T^{2} - 384104 T^{3} + 47045881 T^{4} )^{2} \)
$23$ \( 1 - 11514 T^{2} + 171279659 T^{4} - 1704485225946 T^{6} + 21914624432020321 T^{8} \)
$29$ \( ( 1 - 62 T + 19751 T^{2} - 1512118 T^{3} + 594823321 T^{4} )^{2} \)
$31$ \( ( 1 + 135 T + 63420 T^{2} + 4021785 T^{3} + 887503681 T^{4} )^{2} \)
$37$ \( 1 - 193159 T^{2} + 14439461800 T^{4} - 495593147436031 T^{6} + 6582952005840035281 T^{8} \)
$41$ \( ( 1 - 235 T + 143042 T^{2} - 16196435 T^{3} + 4750104241 T^{4} )^{2} \)
$43$ \( 1 + 33774 T^{2} + 3686087315 T^{4} + 213497715616926 T^{6} + 39959630797262576401 T^{8} \)
$47$ \( 1 - 337916 T^{2} + 49344837574 T^{4} - 3642469327114364 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 - 517079 T^{2} + 110424015304 T^{4} - 11460725688222191 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \)
$59$ \( ( 1 - 441 T + 273734 T^{2} - 90572139 T^{3} + 42180533641 T^{4} )^{2} \)
$61$ \( ( 1 + 223 T + 149646 T^{2} + 50616763 T^{3} + 51520374361 T^{4} )^{2} \)
$67$ \( 1 - 405131 T^{2} + 135876758064 T^{4} - 36647494826509139 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \)
$71$ \( ( 1 - 619 T + 576906 T^{2} - 221546909 T^{3} + 128100283921 T^{4} )^{2} \)
$73$ \( 1 - 1454332 T^{2} + 829074972646 T^{4} - 220090207987333948 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \)
$79$ \( ( 1 - 427 T + 986572 T^{2} - 210527653 T^{3} + 243087455521 T^{4} )^{2} \)
$83$ \( 1 + 25081 T^{2} - 503757220916 T^{4} + 8199991504467889 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \)
$89$ \( ( 1 + 466 T + 306170 T^{2} + 328515554 T^{3} + 496981290961 T^{4} )^{2} \)
$97$ \( 1 + 576836 T^{2} + 1686814161670 T^{4} + 480488239435224644 T^{6} + \)\(69\!\cdots\!41\)\( T^{8} \)
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