Properties

Label 525.2.r.f
Level 525
Weight 2
Character orbit 525.r
Analytic conductor 4.192
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: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} -\zeta_{12} q^{3} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{4} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} + ( -2 - \zeta_{12}^{2} ) q^{7} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + ( 2 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} -\zeta_{12} q^{3} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{4} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} + ( -2 - \zeta_{12}^{2} ) q^{7} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{8} + \zeta_{12}^{2} q^{9} + ( 1 + \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{12} + ( 1 - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{13} + ( -5 + 3 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{14} + ( 4 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{16} + ( -1 - 5 \zeta_{12} - \zeta_{12}^{2} ) q^{17} + ( 1 - \zeta_{12} + \zeta_{12}^{2} ) q^{18} + ( -2 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{19} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{21} -2 \zeta_{12}^{3} q^{22} + ( -2 + 3 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{23} + ( 6 - 2 \zeta_{12} - 6 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{24} + ( -3 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{26} -\zeta_{12}^{3} q^{27} + ( -6 + 8 \zeta_{12} + 4 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{28} + ( -1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{29} + ( -3 - 2 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{31} + ( -8 + 8 \zeta_{12} - 8 \zeta_{12}^{2} ) q^{32} + ( -2 - \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{33} + ( 2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{34} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{36} + ( 6 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{37} + ( 3 - 7 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{38} + ( -4 - \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{39} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{41} + ( -2 + 5 \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{42} + ( -3 + 6 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{43} + 4 \zeta_{12}^{2} q^{44} + ( -6 + 4 \zeta_{12} + 6 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{46} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{47} + ( 4 - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{48} + ( 3 + 5 \zeta_{12}^{2} ) q^{49} + ( \zeta_{12} + 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{51} + ( 6 - 2 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{52} + ( -6 + 2 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{53} + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{54} + ( -8 + 6 \zeta_{12} + 10 \zeta_{12}^{2} - 18 \zeta_{12}^{3} ) q^{56} + ( -2 + 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{57} + ( 4 - 8 \zeta_{12} - 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{58} + ( 5 + 3 \zeta_{12} - 5 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{59} -4 \zeta_{12}^{2} q^{61} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{62} + ( 1 - 3 \zeta_{12}^{2} ) q^{63} + ( -16 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{64} + ( -2 + 2 \zeta_{12}^{2} ) q^{66} + ( -5 + 6 \zeta_{12} - 5 \zeta_{12}^{2} ) q^{67} + ( 16 - 4 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{68} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{69} + ( 1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{71} + ( 4 - 6 \zeta_{12} - 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{72} + ( -5 + 4 \zeta_{12} - 5 \zeta_{12}^{2} ) q^{73} + ( 7 - \zeta_{12} - 7 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{74} + ( 14 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{76} + ( -3 - 4 \zeta_{12} + 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{77} + ( -3 + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{78} + ( 6 \zeta_{12} + 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{82} + ( -7 + 14 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{83} + ( -10 + 6 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{84} + ( -\zeta_{12} + 7 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{86} + ( 3 + \zeta_{12} + 3 \zeta_{12}^{2} ) q^{87} + ( 4 + 4 \zeta_{12}^{2} ) q^{88} + ( -7 \zeta_{12} + 3 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{89} + ( -4 - 4 \zeta_{12} + 5 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{91} + ( -8 + 16 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{92} + ( 4 + 3 \zeta_{12} - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{93} + ( -2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{94} + ( 8 \zeta_{12} - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{96} + ( 4 - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{97} + ( 11 - 8 \zeta_{12} + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{98} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{2} + 4q^{4} + 4q^{6} - 10q^{7} + 2q^{9} + O(q^{10}) \) \( 4q + 6q^{2} + 4q^{4} + 4q^{6} - 10q^{7} + 2q^{9} + 2q^{11} + 12q^{12} - 18q^{14} - 16q^{16} - 6q^{17} + 6q^{18} + 2q^{19} - 6q^{23} + 12q^{24} + 2q^{26} - 16q^{28} - 4q^{29} - 6q^{31} - 48q^{32} - 6q^{33} + 8q^{34} + 8q^{36} + 18q^{37} + 18q^{38} - 8q^{39} + 4q^{41} - 10q^{42} + 8q^{44} - 12q^{46} + 22q^{49} + 10q^{51} + 36q^{52} - 36q^{53} + 2q^{54} - 12q^{56} + 12q^{58} + 10q^{59} - 8q^{61} - 2q^{63} - 64q^{64} - 4q^{66} - 30q^{67} + 48q^{68} - 12q^{69} + 4q^{71} + 12q^{72} - 30q^{73} + 14q^{74} + 56q^{76} - 8q^{77} + 6q^{79} - 2q^{81} - 36q^{84} + 14q^{86} + 18q^{87} + 24q^{88} + 6q^{89} - 6q^{91} + 12q^{93} - 4q^{94} - 16q^{96} + 48q^{98} + 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.633975 0.366025i −0.866025 0.500000i −0.732051 + 1.26795i 0 −0.732051 −2.50000 0.866025i 2.53590i 0.500000 + 0.866025i 0
424.2 2.36603 1.36603i 0.866025 + 0.500000i 2.73205 4.73205i 0 2.73205 −2.50000 0.866025i 9.46410i 0.500000 + 0.866025i 0
499.1 0.633975 + 0.366025i −0.866025 + 0.500000i −0.732051 1.26795i 0 −0.732051 −2.50000 + 0.866025i 2.53590i 0.500000 0.866025i 0
499.2 2.36603 + 1.36603i 0.866025 0.500000i 2.73205 + 4.73205i 0 2.73205 −2.50000 + 0.866025i 9.46410i 0.500000 0.866025i 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
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.r.f 4
5.b even 2 1 525.2.r.a 4
5.c odd 4 1 105.2.i.d 4
5.c odd 4 1 525.2.i.f 4
7.c even 3 1 525.2.r.a 4
15.e even 4 1 315.2.j.c 4
20.e even 4 1 1680.2.bg.o 4
35.f even 4 1 735.2.i.l 4
35.j even 6 1 inner 525.2.r.f 4
35.k even 12 1 735.2.a.h 2
35.k even 12 1 735.2.i.l 4
35.k even 12 1 3675.2.a.be 2
35.l odd 12 1 105.2.i.d 4
35.l odd 12 1 525.2.i.f 4
35.l odd 12 1 735.2.a.g 2
35.l odd 12 1 3675.2.a.bg 2
105.w odd 12 1 2205.2.a.ba 2
105.x even 12 1 315.2.j.c 4
105.x even 12 1 2205.2.a.z 2
140.w even 12 1 1680.2.bg.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 5.c odd 4 1
105.2.i.d 4 35.l odd 12 1
315.2.j.c 4 15.e even 4 1
315.2.j.c 4 105.x even 12 1
525.2.i.f 4 5.c odd 4 1
525.2.i.f 4 35.l odd 12 1
525.2.r.a 4 5.b even 2 1
525.2.r.a 4 7.c even 3 1
525.2.r.f 4 1.a even 1 1 trivial
525.2.r.f 4 35.j even 6 1 inner
735.2.a.g 2 35.l odd 12 1
735.2.a.h 2 35.k even 12 1
735.2.i.l 4 35.f even 4 1
735.2.i.l 4 35.k even 12 1
1680.2.bg.o 4 20.e even 4 1
1680.2.bg.o 4 140.w even 12 1
2205.2.a.z 2 105.x even 12 1
2205.2.a.ba 2 105.w odd 12 1
3675.2.a.be 2 35.k even 12 1
3675.2.a.bg 2 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} - 6 T_{2}^{3} + 14 T_{2}^{2} - 12 T_{2} + 4 \)
\( T_{11}^{4} - 2 T_{11}^{3} + 6 T_{11}^{2} + 4 T_{11} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 2 T^{2} )^{2}( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} ) \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( \)
$7$ \( ( 1 + 5 T + 7 T^{2} )^{2} \)
$11$ \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 14 T^{2} + 195 T^{4} - 2366 T^{6} + 28561 T^{8} \)
$17$ \( 1 + 6 T + 24 T^{2} + 72 T^{3} + 59 T^{4} + 1224 T^{5} + 6936 T^{6} + 29478 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 2 T - 23 T^{2} + 22 T^{3} + 292 T^{4} + 418 T^{5} - 8303 T^{6} - 13718 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 6 T + 52 T^{2} + 240 T^{3} + 1347 T^{4} + 5520 T^{5} + 27508 T^{6} + 73002 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 + 2 T + 32 T^{2} + 58 T^{3} + 841 T^{4} )^{2} \)
$31$ \( 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 558 T^{5} - 22103 T^{6} + 178746 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 18 T + 205 T^{2} - 1746 T^{3} + 12036 T^{4} - 64602 T^{5} + 280645 T^{6} - 911754 T^{7} + 1874161 T^{8} \)
$41$ \( ( 1 - 2 T + 80 T^{2} - 82 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 - 110 T^{2} + 6291 T^{4} - 203390 T^{6} + 3418801 T^{8} \)
$47$ \( 1 + 90 T^{2} + 5891 T^{4} + 198810 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 36 T + 642 T^{2} + 7560 T^{3} + 64187 T^{4} + 400680 T^{5} + 1803378 T^{6} + 5359572 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 10 T - 16 T^{2} + 20 T^{3} + 4075 T^{4} + 1180 T^{5} - 55696 T^{6} - 2053790 T^{7} + 12117361 T^{8} \)
$61$ \( ( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 30 T + 473 T^{2} + 5190 T^{3} + 45540 T^{4} + 347730 T^{5} + 2123297 T^{6} + 9022890 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 - 2 T + 116 T^{2} - 142 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 + 30 T + 505 T^{2} + 6150 T^{3} + 58596 T^{4} + 448950 T^{5} + 2691145 T^{6} + 11670510 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 6 T - 23 T^{2} + 594 T^{3} - 5604 T^{4} + 46926 T^{5} - 143543 T^{6} - 2958234 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 20 T^{2} + 8586 T^{4} - 137780 T^{6} + 47458321 T^{8} \)
$89$ \( 1 - 6 T - 4 T^{2} + 828 T^{3} - 9525 T^{4} + 73692 T^{5} - 31684 T^{6} - 4229814 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 164 T^{2} + 13254 T^{4} - 1543076 T^{6} + 88529281 T^{8} \)
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