# Properties

 Label 3525.1.bg.a Level $3525$ Weight $1$ Character orbit 3525.bg Analytic conductor $1.759$ Analytic rank $0$ Dimension $88$ Projective image $D_{46}$ CM discriminant -15 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3525.bg (of order $$92$$, degree $$44$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.75920416953$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$2$$ over $$\Q(\zeta_{92})$$ Coefficient field: $$\Q(\zeta_{184})$$ Defining polynomial: $$x^{88} - x^{84} + x^{80} - x^{76} + x^{72} - x^{68} + x^{64} - x^{60} + x^{56} - x^{52} + x^{48} - x^{44} + x^{40} - x^{36} + x^{32} - x^{28} + x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{46}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{46} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{184}^{33} + \zeta_{184}^{61} ) q^{2} + \zeta_{184}^{19} q^{3} + ( \zeta_{184}^{2} - \zeta_{184}^{30} + \zeta_{184}^{66} ) q^{4} + ( -\zeta_{184}^{52} + \zeta_{184}^{80} ) q^{6} + ( \zeta_{184}^{7} - \zeta_{184}^{35} + \zeta_{184}^{63} - \zeta_{184}^{91} ) q^{8} + \zeta_{184}^{38} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{184}^{33} + \zeta_{184}^{61} ) q^{2} + \zeta_{184}^{19} q^{3} + ( \zeta_{184}^{2} - \zeta_{184}^{30} + \zeta_{184}^{66} ) q^{4} + ( -\zeta_{184}^{52} + \zeta_{184}^{80} ) q^{6} + ( \zeta_{184}^{7} - \zeta_{184}^{35} + \zeta_{184}^{63} - \zeta_{184}^{91} ) q^{8} + \zeta_{184}^{38} q^{9} + ( \zeta_{184}^{21} - \zeta_{184}^{49} + \zeta_{184}^{85} ) q^{12} + ( \zeta_{184}^{4} - \zeta_{184}^{32} - \zeta_{184}^{40} + \zeta_{184}^{60} + \zeta_{184}^{68} ) q^{16} + ( -\zeta_{184}^{17} - \zeta_{184}^{41} ) q^{17} + ( -\zeta_{184}^{7} - \zeta_{184}^{71} ) q^{18} + ( -\zeta_{184}^{58} - \zeta_{184}^{62} ) q^{19} + ( -\zeta_{184}^{23} - \zeta_{184}^{67} ) q^{23} + ( \zeta_{184}^{18} + \zeta_{184}^{26} - \zeta_{184}^{54} + \zeta_{184}^{82} ) q^{24} + \zeta_{184}^{57} q^{27} + ( -\zeta_{184}^{44} - \zeta_{184}^{56} ) q^{31} + ( \zeta_{184} + \zeta_{184}^{9} - \zeta_{184}^{29} - \zeta_{184}^{37} + \zeta_{184}^{65} + \zeta_{184}^{73} ) q^{32} + ( \zeta_{184}^{10} + \zeta_{184}^{50} + \zeta_{184}^{74} - \zeta_{184}^{78} ) q^{34} + ( -\zeta_{184}^{12} + \zeta_{184}^{40} - \zeta_{184}^{68} ) q^{36} + ( -\zeta_{184}^{3} + \zeta_{184}^{27} + \zeta_{184}^{31} + \zeta_{184}^{91} ) q^{38} + ( -\zeta_{184}^{8} + \zeta_{184}^{36} + \zeta_{184}^{56} - \zeta_{184}^{84} ) q^{46} -\zeta_{184}^{13} q^{47} + ( \zeta_{184}^{23} - \zeta_{184}^{51} - \zeta_{184}^{59} + \zeta_{184}^{79} + \zeta_{184}^{87} ) q^{48} -\zeta_{184}^{70} q^{49} + ( -\zeta_{184}^{36} - \zeta_{184}^{60} ) q^{51} + ( \zeta_{184}^{35} + \zeta_{184}^{51} ) q^{53} + ( -\zeta_{184}^{26} - \zeta_{184}^{90} ) q^{54} + ( -\zeta_{184}^{77} - \zeta_{184}^{81} ) q^{57} + ( \zeta_{184}^{20} + \zeta_{184}^{52} ) q^{61} + ( \zeta_{184}^{13} + \zeta_{184}^{25} + \zeta_{184}^{77} + \zeta_{184}^{89} ) q^{62} + ( \zeta_{184}^{6} + \zeta_{184}^{14} - \zeta_{184}^{34} - \zeta_{184}^{42} + \zeta_{184}^{62} + \zeta_{184}^{70} - \zeta_{184}^{90} ) q^{64} + ( \zeta_{184}^{15} - \zeta_{184}^{19} - \zeta_{184}^{43} + \zeta_{184}^{47} + \zeta_{184}^{71} - \zeta_{184}^{83} ) q^{68} + ( -\zeta_{184}^{42} - \zeta_{184}^{86} ) q^{69} + ( -\zeta_{184}^{9} + \zeta_{184}^{37} + \zeta_{184}^{45} - \zeta_{184}^{73} ) q^{72} + ( -1 + \zeta_{184}^{32} + \zeta_{184}^{36} - \zeta_{184}^{60} - \zeta_{184}^{64} + \zeta_{184}^{88} ) q^{76} + ( \zeta_{184}^{22} - \zeta_{184}^{34} ) q^{79} + \zeta_{184}^{76} q^{81} + ( \zeta_{184}^{43} + \zeta_{184}^{83} ) q^{83} + ( -\zeta_{184}^{5} - \zeta_{184}^{25} + \zeta_{184}^{41} + \zeta_{184}^{53} - \zeta_{184}^{69} - \zeta_{184}^{89} ) q^{92} + ( -\zeta_{184}^{63} - \zeta_{184}^{75} ) q^{93} + ( \zeta_{184}^{46} - \zeta_{184}^{74} ) q^{94} + ( -1 + \zeta_{184}^{20} + \zeta_{184}^{28} - \zeta_{184}^{48} - \zeta_{184}^{56} + \zeta_{184}^{84} ) q^{96} + ( -\zeta_{184}^{11} + \zeta_{184}^{39} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$88q - 8q^{6} + O(q^{10})$$ $$88q - 8q^{6} + 20q^{16} - 12q^{36} - 8q^{51} + 8q^{61} - 92q^{76} + 4q^{81} - 68q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times$$.

 $$n$$ $$1552$$ $$2026$$ $$2351$$ $$\chi(n)$$ $$\zeta_{184}^{46}$$ $$-\zeta_{184}^{64}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 0.985460 + 0.169910i −0.985460 − 0.169910i 0.366854 − 0.930278i −0.366854 + 0.930278i 0.0341411 − 0.999417i −0.0341411 + 0.999417i −0.102264 + 0.994757i 0.102264 − 0.994757i 0.999417 − 0.0341411i −0.999417 + 0.0341411i 0.366854 + 0.930278i −0.366854 − 0.930278i 0.971567 − 0.236764i −0.971567 + 0.236764i −0.657204 + 0.753713i 0.657204 − 0.753713i −0.657204 − 0.753713i 0.657204 + 0.753713i 0.985460 − 0.169910i −0.985460 + 0.169910i
−1.34526 0.231946i −0.994757 0.102264i 0.813657 + 0.289174i 0 1.31448 + 0.368301i 0 0.162405 + 0.0913155i 0.979084 + 0.203456i 0
107.2 1.34526 + 0.231946i 0.994757 + 0.102264i 0.813657 + 0.289174i 0 1.31448 + 0.368301i 0 −0.162405 0.0913155i 0.979084 + 0.203456i 0
182.1 −0.626895 + 1.58970i −0.753713 + 0.657204i −1.40330 1.31059i 0 −0.572255 1.61017i 0 1.41995 0.675300i 0.136167 0.990686i 0
182.2 0.626895 1.58970i 0.753713 0.657204i −1.40330 1.31059i 0 −0.572255 1.61017i 0 −1.41995 + 0.675300i 0.136167 0.990686i 0
218.1 −0.0314143 + 0.919594i −0.604236 + 0.796805i 0.153003 + 0.0104657i 0 −0.713755 0.580683i 0 −0.108527 + 1.05568i −0.269797 0.962917i 0
218.2 0.0314143 0.919594i 0.604236 0.796805i 0.153003 + 0.0104657i 0 −0.713755 0.580683i 0 0.108527 1.05568i −0.269797 0.962917i 0
257.1 −0.202623 + 1.97098i 0.930278 + 0.366854i −2.86464 0.595279i 0 −0.911560 + 1.75923i 0 1.15433 3.63700i 0.730836 + 0.682553i 0
257.2 0.202623 1.97098i −0.930278 0.366854i −2.86464 0.595279i 0 −0.911560 + 1.75923i 0 −1.15433 + 3.63700i 0.730836 + 0.682553i 0
293.1 −0.919594 + 0.0314143i 0.796805 0.604236i −0.153003 + 0.0104657i 0 −0.713755 + 0.580683i 0 1.05568 0.108527i 0.269797 0.962917i 0
293.2 0.919594 0.0314143i −0.796805 + 0.604236i −0.153003 + 0.0104657i 0 −0.713755 + 0.580683i 0 −1.05568 + 0.108527i 0.269797 0.962917i 0
368.1 −0.626895 1.58970i −0.753713 0.657204i −1.40330 + 1.31059i 0 −0.572255 + 1.61017i 0 1.41995 + 0.675300i 0.136167 + 0.990686i 0
368.2 0.626895 + 1.58970i 0.753713 + 0.657204i −1.40330 + 1.31059i 0 −0.572255 + 1.61017i 0 −1.41995 0.675300i 0.136167 + 0.990686i 0
407.1 −0.395342 + 0.0963423i −0.169910 + 0.985460i −0.740871 + 0.383889i 0 −0.0277687 0.405963i 0 0.562608 0.490569i −0.942261 0.334880i 0
407.2 0.395342 0.0963423i 0.169910 0.985460i −0.740871 + 0.383889i 0 −0.0277687 0.405963i 0 −0.562608 + 0.490569i −0.942261 0.334880i 0
443.1 −0.757993 + 0.869303i 0.871660 0.490110i −0.0449678 0.327165i 0 −0.234658 + 1.12924i 0 −0.645929 0.423665i 0.519584 0.854419i 0
443.2 0.757993 0.869303i −0.871660 + 0.490110i −0.0449678 0.327165i 0 −0.234658 + 1.12924i 0 0.645929 + 0.423665i 0.519584 0.854419i 0
557.1 −0.757993 0.869303i 0.871660 + 0.490110i −0.0449678 + 0.327165i 0 −0.234658 1.12924i 0 −0.645929 + 0.423665i 0.519584 + 0.854419i 0
557.2 0.757993 + 0.869303i −0.871660 0.490110i −0.0449678 + 0.327165i 0 −0.234658 1.12924i 0 0.645929 0.423665i 0.519584 + 0.854419i 0
593.1 −1.34526 + 0.231946i −0.994757 + 0.102264i 0.813657 0.289174i 0 1.31448 0.368301i 0 0.162405 0.0913155i 0.979084 0.203456i 0
593.2 1.34526 0.231946i 0.994757 0.102264i 0.813657 0.289174i 0 1.31448 0.368301i 0 −0.162405 + 0.0913155i 0.979084 0.203456i 0
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3518.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.e even 4 2 inner
47.d odd 46 1 inner
141.g even 46 1 inner
235.j odd 46 1 inner
235.l even 92 2 inner
705.o even 46 1 inner
705.u odd 92 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3525.1.bg.a 88
3.b odd 2 1 inner 3525.1.bg.a 88
5.b even 2 1 inner 3525.1.bg.a 88
5.c odd 4 2 inner 3525.1.bg.a 88
15.d odd 2 1 CM 3525.1.bg.a 88
15.e even 4 2 inner 3525.1.bg.a 88
47.d odd 46 1 inner 3525.1.bg.a 88
141.g even 46 1 inner 3525.1.bg.a 88
235.j odd 46 1 inner 3525.1.bg.a 88
235.l even 92 2 inner 3525.1.bg.a 88
705.o even 46 1 inner 3525.1.bg.a 88
705.u odd 92 2 inner 3525.1.bg.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3525.1.bg.a 88 1.a even 1 1 trivial
3525.1.bg.a 88 3.b odd 2 1 inner
3525.1.bg.a 88 5.b even 2 1 inner
3525.1.bg.a 88 5.c odd 4 2 inner
3525.1.bg.a 88 15.d odd 2 1 CM
3525.1.bg.a 88 15.e even 4 2 inner
3525.1.bg.a 88 47.d odd 46 1 inner
3525.1.bg.a 88 141.g even 46 1 inner
3525.1.bg.a 88 235.j odd 46 1 inner
3525.1.bg.a 88 235.l even 92 2 inner
3525.1.bg.a 88 705.o even 46 1 inner
3525.1.bg.a 88 705.u odd 92 2 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3525, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 1137 T^{4} + 8264000 T^{8} - 294683395 T^{12} + 8870688801 T^{16} + 68169538352 T^{20} + 96930025810 T^{24} - 551442427592 T^{28} + 113665829572 T^{32} + 377888692286 T^{36} + 18356065784 T^{40} - 30478741181 T^{44} + 10622984092 T^{48} - 470719342 T^{52} + 528730429 T^{56} - 33024752 T^{60} + 8727860 T^{64} - 550120 T^{68} + 45526 T^{72} - 2624 T^{76} + 164 T^{80} - 16 T^{84} + T^{88}$$
$3$ $$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} - T^{68} + T^{72} - T^{76} + T^{80} - T^{84} + T^{88}$$
$5$ $$T^{88}$$
$7$ $$T^{88}$$
$11$ $$T^{88}$$
$13$ $$T^{88}$$
$17$ $$1 - 5786 T^{4} + 8741227 T^{8} - 643329102 T^{12} + 16557089437 T^{16} - 123508804027 T^{20} + 452419620952 T^{24} - 615482026148 T^{28} + 601520127784 T^{32} - 102247162748 T^{36} + 129244723278 T^{40} + 46445370872 T^{44} + 24486698278 T^{48} + 8439258455 T^{52} + 2300383100 T^{56} + 481285700 T^{60} + 77622060 T^{64} + 9543108 T^{68} + 878977 T^{72} + 58740 T^{76} + 2694 T^{80} + 76 T^{84} + T^{88}$$
$19$ $$( 529 + 6348 T^{2} + 111090 T^{4} + 283544 T^{6} + 68241 T^{8} - 1451576 T^{10} + 2401660 T^{12} - 43907 T^{14} + 450708 T^{16} + 1964177 T^{18} + 26450 T^{20} - 3220 T^{22} + 235014 T^{24} - 6670 T^{26} + 3289 T^{30} + 805 T^{32} - 46 T^{38} + T^{44} )^{2}$$
$23$ $$279841 + 89269279 T^{4} + 10007394001 T^{8} + 87511597679 T^{12} + 1097738167361 T^{16} - 2352578792281 T^{20} + 10455672600769 T^{24} - 18664460469072 T^{28} + 20013710005800 T^{32} - 12441921687228 T^{36} + 4425247256996 T^{40} - 868349980034 T^{44} + 88093456946 T^{48} - 4131342938 T^{52} + 73726730 T^{56} - 75647 T^{60} + 101039 T^{64} + 33649 T^{68} + 8855 T^{72} + 1771 T^{76} + 253 T^{80} + 23 T^{84} + T^{88}$$
$29$ $$T^{88}$$
$31$ $$( 23 + 184 T + 598 T^{2} + 690 T^{3} + 161 T^{4} - 23 T^{7} + 736 T^{8} - 1449 T^{9} + 138 T^{10} + 69 T^{15} + 23 T^{16} + T^{22} )^{4}$$
$37$ $$T^{88}$$
$41$ $$T^{88}$$
$43$ $$T^{88}$$
$47$ $$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} - T^{68} + T^{72} - T^{76} + T^{80} - T^{84} + T^{88}$$
$53$ $$1 - 4429 T^{4} + 8119123 T^{8} + 340515977 T^{12} + 13440263044 T^{16} + 95910349481 T^{20} + 164928251587 T^{24} - 95488416633 T^{28} + 437630135119 T^{32} - 521671377684 T^{36} + 163471192098 T^{40} - 14590332625 T^{44} + 1799819427 T^{48} + 1601290720 T^{52} + 363520118 T^{56} - 14599130 T^{60} + 913196 T^{64} - 67258 T^{68} + 40420 T^{72} - 4096 T^{76} + 256 T^{80} - 16 T^{84} + T^{88}$$
$59$ $$T^{88}$$
$61$ $$( 1 - 12 T + 52 T^{2} - 26 T^{3} + 174 T^{4} + 396 T^{5} - 198 T^{6} - 39 T^{7} + 1158 T^{8} - 579 T^{9} + 301 T^{10} + 896 T^{11} - 448 T^{12} + 224 T^{13} + 348 T^{14} - 174 T^{15} + 87 T^{16} + 60 T^{17} - 30 T^{18} + 15 T^{19} + 4 T^{20} - 2 T^{21} + T^{22} )^{4}$$
$67$ $$T^{88}$$
$71$ $$T^{88}$$
$73$ $$T^{88}$$
$79$ $$( 1 - 98 T^{2} + 2865 T^{4} - 1458 T^{6} + 73976 T^{8} + 347068 T^{10} + 1033655 T^{12} + 1576555 T^{14} + 1011811 T^{16} + 144194 T^{18} - 9863 T^{20} + 2598 T^{22} - 638 T^{24} - 496 T^{26} + 5598 T^{28} - 9622 T^{30} + 4050 T^{32} - 1024 T^{34} + 256 T^{36} - 64 T^{38} + 16 T^{40} - 4 T^{42} + T^{44} )^{2}$$
$83$ $$1 + 4909 T^{4} + 8229730 T^{8} - 121875701 T^{12} + 11734273234 T^{16} - 118127155444 T^{20} + 427063160391 T^{24} - 467948735011 T^{28} + 184567084138 T^{32} + 84218725373 T^{36} + 139331375624 T^{40} + 77719412538 T^{44} + 11893551380 T^{48} - 50642420 T^{52} + 6008210 T^{56} - 980140 T^{60} + 1479640 T^{64} - 690512 T^{68} + 65444 T^{72} - 4096 T^{76} + 256 T^{80} - 16 T^{84} + T^{88}$$
$89$ $$T^{88}$$
$97$ $$T^{88}$$