# Properties

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

# Related objects

## 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: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{2} -3 \zeta_{8}^{2} q^{3} + ( -1 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{4} + ( 3 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{6} + 7 \zeta_{8}^{2} q^{7} + ( 10 \zeta_{8} - 9 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{2} -3 \zeta_{8}^{2} q^{3} + ( -1 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{4} + ( 3 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{6} + 7 \zeta_{8}^{2} q^{7} + ( 10 \zeta_{8} - 9 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{8} -9 q^{9} + ( -8 + 40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{11} + ( 12 \zeta_{8} + 3 \zeta_{8}^{2} + 12 \zeta_{8}^{3} ) q^{12} + ( -4 \zeta_{8} - 38 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{13} + ( -7 - 14 \zeta_{8} + 14 \zeta_{8}^{3} ) q^{14} + ( -39 - 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{16} + ( 4 \zeta_{8} + 62 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{17} + ( -18 \zeta_{8} - 9 \zeta_{8}^{2} - 18 \zeta_{8}^{3} ) q^{18} + ( 48 + 32 \zeta_{8} - 32 \zeta_{8}^{3} ) q^{19} + 21 q^{21} + ( 24 \zeta_{8} + 152 \zeta_{8}^{2} + 24 \zeta_{8}^{3} ) q^{22} + ( -68 \zeta_{8} - 8 \zeta_{8}^{2} - 68 \zeta_{8}^{3} ) q^{23} + ( -27 + 30 \zeta_{8} - 30 \zeta_{8}^{3} ) q^{24} + ( 54 + 80 \zeta_{8} - 80 \zeta_{8}^{3} ) q^{26} + 27 \zeta_{8}^{2} q^{27} + ( -28 \zeta_{8} - 7 \zeta_{8}^{2} - 28 \zeta_{8}^{3} ) q^{28} + ( -94 + 108 \zeta_{8} - 108 \zeta_{8}^{3} ) q^{29} + ( -60 + 36 \zeta_{8} - 36 \zeta_{8}^{3} ) q^{31} + ( -22 \zeta_{8} - 207 \zeta_{8}^{2} - 22 \zeta_{8}^{3} ) q^{32} + ( -120 \zeta_{8} + 24 \zeta_{8}^{2} - 120 \zeta_{8}^{3} ) q^{33} + ( -78 - 128 \zeta_{8} + 128 \zeta_{8}^{3} ) q^{34} + ( 9 + 36 \zeta_{8} - 36 \zeta_{8}^{3} ) q^{36} + ( 132 \zeta_{8} + 66 \zeta_{8}^{2} + 132 \zeta_{8}^{3} ) q^{37} + ( 128 \zeta_{8} + 176 \zeta_{8}^{2} + 128 \zeta_{8}^{3} ) q^{38} + ( -114 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{39} + ( 50 + 160 \zeta_{8} - 160 \zeta_{8}^{3} ) q^{41} + ( 42 \zeta_{8} + 21 \zeta_{8}^{2} + 42 \zeta_{8}^{3} ) q^{42} + ( 124 \zeta_{8} - 268 \zeta_{8}^{2} + 124 \zeta_{8}^{3} ) q^{43} + ( -312 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{44} + ( 280 + 84 \zeta_{8} - 84 \zeta_{8}^{3} ) q^{46} + ( 84 \zeta_{8} + 464 \zeta_{8}^{2} + 84 \zeta_{8}^{3} ) q^{47} + ( 72 \zeta_{8} + 117 \zeta_{8}^{2} + 72 \zeta_{8}^{3} ) q^{48} -49 q^{49} + ( 186 + 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{51} + ( 156 \zeta_{8} + 70 \zeta_{8}^{2} + 156 \zeta_{8}^{3} ) q^{52} + ( 128 \zeta_{8} + 442 \zeta_{8}^{2} + 128 \zeta_{8}^{3} ) q^{53} + ( -27 - 54 \zeta_{8} + 54 \zeta_{8}^{3} ) q^{54} + ( 63 - 70 \zeta_{8} + 70 \zeta_{8}^{3} ) q^{56} + ( -96 \zeta_{8} - 144 \zeta_{8}^{2} - 96 \zeta_{8}^{3} ) q^{57} + ( -80 \zeta_{8} + 338 \zeta_{8}^{2} - 80 \zeta_{8}^{3} ) q^{58} + ( -52 + 408 \zeta_{8} - 408 \zeta_{8}^{3} ) q^{59} + ( -234 - 84 \zeta_{8} + 84 \zeta_{8}^{3} ) q^{61} + ( -84 \zeta_{8} + 84 \zeta_{8}^{2} - 84 \zeta_{8}^{3} ) q^{62} -63 \zeta_{8}^{2} q^{63} + ( -17 + 244 \zeta_{8} - 244 \zeta_{8}^{3} ) q^{64} + ( 456 + 72 \zeta_{8} - 72 \zeta_{8}^{3} ) q^{66} + ( -76 \zeta_{8} + 844 \zeta_{8}^{2} - 76 \zeta_{8}^{3} ) q^{67} + ( -252 \zeta_{8} - 94 \zeta_{8}^{2} - 252 \zeta_{8}^{3} ) q^{68} + ( -24 - 204 \zeta_{8} + 204 \zeta_{8}^{3} ) q^{69} + ( -68 - 300 \zeta_{8} + 300 \zeta_{8}^{3} ) q^{71} + ( -90 \zeta_{8} + 81 \zeta_{8}^{2} - 90 \zeta_{8}^{3} ) q^{72} + ( 644 \zeta_{8} + 254 \zeta_{8}^{2} + 644 \zeta_{8}^{3} ) q^{73} + ( -594 - 264 \zeta_{8} + 264 \zeta_{8}^{3} ) q^{74} + ( -304 - 224 \zeta_{8} + 224 \zeta_{8}^{3} ) q^{76} + ( 280 \zeta_{8} - 56 \zeta_{8}^{2} + 280 \zeta_{8}^{3} ) q^{77} + ( -240 \zeta_{8} - 162 \zeta_{8}^{2} - 240 \zeta_{8}^{3} ) q^{78} + ( 216 + 464 \zeta_{8} - 464 \zeta_{8}^{3} ) q^{79} + 81 q^{81} + ( 260 \zeta_{8} + 690 \zeta_{8}^{2} + 260 \zeta_{8}^{3} ) q^{82} + ( -168 \zeta_{8} - 292 \zeta_{8}^{2} - 168 \zeta_{8}^{3} ) q^{83} + ( -21 - 84 \zeta_{8} + 84 \zeta_{8}^{3} ) q^{84} + ( -228 + 412 \zeta_{8} - 412 \zeta_{8}^{3} ) q^{86} + ( -324 \zeta_{8} + 282 \zeta_{8}^{2} - 324 \zeta_{8}^{3} ) q^{87} + ( -440 \zeta_{8} + 872 \zeta_{8}^{2} - 440 \zeta_{8}^{3} ) q^{88} + ( 702 - 224 \zeta_{8} + 224 \zeta_{8}^{3} ) q^{89} + ( 266 + 28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{91} + ( 100 \zeta_{8} + 552 \zeta_{8}^{2} + 100 \zeta_{8}^{3} ) q^{92} + ( -108 \zeta_{8} + 180 \zeta_{8}^{2} - 108 \zeta_{8}^{3} ) q^{93} + ( -800 - 1012 \zeta_{8} + 1012 \zeta_{8}^{3} ) q^{94} + ( -621 - 66 \zeta_{8} + 66 \zeta_{8}^{3} ) q^{96} + ( -92 \zeta_{8} + 594 \zeta_{8}^{2} - 92 \zeta_{8}^{3} ) q^{97} + ( -98 \zeta_{8} - 49 \zeta_{8}^{2} - 98 \zeta_{8}^{3} ) q^{98} + ( 72 - 360 \zeta_{8} + 360 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 12q^{6} - 36q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 12q^{6} - 36q^{9} - 32q^{11} - 28q^{14} - 156q^{16} + 192q^{19} + 84q^{21} - 108q^{24} + 216q^{26} - 376q^{29} - 240q^{31} - 312q^{34} + 36q^{36} - 456q^{39} + 200q^{41} - 1248q^{44} + 1120q^{46} - 196q^{49} + 744q^{51} - 108q^{54} + 252q^{56} - 208q^{59} - 936q^{61} - 68q^{64} + 1824q^{66} - 96q^{69} - 272q^{71} - 2376q^{74} - 1216q^{76} + 864q^{79} + 324q^{81} - 84q^{84} - 912q^{86} + 2808q^{89} + 1064q^{91} - 3200q^{94} - 2484q^{96} + 288q^{99} + O(q^{100})$$

## Character values

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

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

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
3.82843i 3.00000i −6.65685 0 11.4853 7.00000i 5.14214i −9.00000 0
274.2 1.82843i 3.00000i 4.65685 0 −5.48528 7.00000i 23.1421i −9.00000 0
274.3 1.82843i 3.00000i 4.65685 0 −5.48528 7.00000i 23.1421i −9.00000 0
274.4 3.82843i 3.00000i −6.65685 0 11.4853 7.00000i 5.14214i −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.l 4
5.b even 2 1 inner 525.4.d.l 4
5.c odd 4 1 105.4.a.e 2
5.c odd 4 1 525.4.a.l 2
15.e even 4 1 315.4.a.k 2
15.e even 4 1 1575.4.a.q 2
20.e even 4 1 1680.4.a.bo 2
35.f even 4 1 735.4.a.o 2
105.k odd 4 1 2205.4.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 5.c odd 4 1
315.4.a.k 2 15.e even 4 1
525.4.a.l 2 5.c odd 4 1
525.4.d.l 4 1.a even 1 1 trivial
525.4.d.l 4 5.b even 2 1 inner
735.4.a.o 2 35.f even 4 1
1575.4.a.q 2 15.e even 4 1
1680.4.a.bo 2 20.e even 4 1
2205.4.a.bb 2 105.k odd 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 18 T_{2}^{2} + 49$$ $$T_{11}^{2} + 16 T_{11} - 3136$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 14 T^{2} + 145 T^{4} - 896 T^{6} + 4096 T^{8}$$
$3$ $$( 1 + 9 T^{2} )^{2}$$
$5$ 1
$7$ $$( 1 + 49 T^{2} )^{2}$$
$11$ $$( 1 + 16 T - 474 T^{2} + 21296 T^{3} + 1771561 T^{4} )^{2}$$
$13$ $$1 - 5836 T^{2} + 17983510 T^{4} - 28169257324 T^{6} + 23298085122481 T^{8}$$
$17$ $$1 - 11900 T^{2} + 83185606 T^{4} - 287237071100 T^{6} + 582622237229761 T^{8}$$
$19$ $$( 1 - 96 T + 13974 T^{2} - 658464 T^{3} + 47045881 T^{4} )^{2}$$
$23$ $$1 - 30044 T^{2} + 519364774 T^{4} - 4447590249116 T^{6} + 21914624432020321 T^{8}$$
$29$ $$( 1 + 188 T + 34286 T^{2} + 4585132 T^{3} + 594823321 T^{4} )^{2}$$
$31$ $$( 1 + 120 T + 60590 T^{2} + 3574920 T^{3} + 887503681 T^{4} )^{2}$$
$37$ $$1 - 124204 T^{2} + 8380919670 T^{4} - 318673482903436 T^{6} + 6582952005840035281 T^{8}$$
$41$ $$( 1 - 100 T + 89142 T^{2} - 6892100 T^{3} + 4750104241 T^{4} )^{2}$$
$43$ $$1 - 112876 T^{2} + 6993047350 T^{4} - 713530175518924 T^{6} + 39959630797262576401 T^{8}$$
$47$ $$1 + 43524 T^{2} + 9878986694 T^{4} + 469154567979396 T^{6} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 - 139244 T^{2} + 23569194934 T^{4} - 3086254301046476 T^{6} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$( 1 + 104 T + 80534 T^{2} + 21359416 T^{3} + 42180533641 T^{4} )^{2}$$
$61$ $$( 1 + 468 T + 494606 T^{2} + 106227108 T^{3} + 51520374361 T^{4} )^{2}$$
$67$ $$1 + 244724 T^{2} + 162973601494 T^{4} + 22137337117926356 T^{6} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$( 1 + 136 T + 540446 T^{2} + 48675896 T^{3} + 128100283921 T^{4} )^{2}$$
$73$ $$1 + 231908 T^{2} + 102056920486 T^{4} + 35095617750229412 T^{6} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$( 1 - 432 T + 602142 T^{2} - 212992848 T^{3} + 243087455521 T^{4} )^{2}$$
$83$ $$1 - 2003724 T^{2} + 1638356284694 T^{4} - 655098272688426156 T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$89$ $$( 1 - 1404 T + 1802390 T^{2} - 989776476 T^{3} + 496981290961 T^{4} )^{2}$$
$97$ $$1 - 2911164 T^{2} + 3760771737350 T^{4} - 2424918113757127356 T^{6} +$$$$69\!\cdots\!41$$$$T^{8}$$