# Properties

 Label 75.10.b.f Level 75 Weight 10 Character orbit 75.b Analytic conductor 38.628 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$38.6276877123$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{241})$$ Defining polynomial: $$x^{4} + 121 x^{2} + 3600$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q + ( -16 \beta_{1} - \beta_{2} ) q^{2} -81 \beta_{1} q^{3} + ( -255 - 31 \beta_{3} ) q^{4} + ( -1215 - 81 \beta_{3} ) q^{6} + ( -7168 \beta_{1} - 224 \beta_{2} ) q^{7} + ( 13186 \beta_{1} + 239 \beta_{2} ) q^{8} -6561 q^{9} +O(q^{10})$$ $$q + ( -16 \beta_{1} - \beta_{2} ) q^{2} -81 \beta_{1} q^{3} + ( -255 - 31 \beta_{3} ) q^{4} + ( -1215 - 81 \beta_{3} ) q^{6} + ( -7168 \beta_{1} - 224 \beta_{2} ) q^{7} + ( 13186 \beta_{1} + 239 \beta_{2} ) q^{8} -6561 q^{9} + ( -9572 - 2368 \beta_{3} ) q^{11} + ( 23166 \beta_{1} + 2511 \beta_{2} ) q^{12} + ( 9470 \beta_{1} - 5344 \beta_{2} ) q^{13} + ( -225568 - 10528 \beta_{3} ) q^{14} + ( 193183 + 899 \beta_{3} ) q^{16} + ( 74718 \beta_{1} - 7520 \beta_{2} ) q^{17} + ( 104976 \beta_{1} + 6561 \beta_{2} ) q^{18} + ( 45084 + 5728 \beta_{3} ) q^{19} + ( -562464 - 18144 \beta_{3} ) q^{21} + ( 1474496 \beta_{1} + 47460 \beta_{2} ) q^{22} + ( -354496 \beta_{1} + 26272 \beta_{2} ) q^{23} + ( 1048707 + 19359 \beta_{3} ) q^{24} + ( -2674238 - 70690 \beta_{3} ) q^{26} + 531441 \beta_{1} q^{27} + ( 5813696 \beta_{1} + 279328 \beta_{2} ) q^{28} + ( 1254818 + 168576 \beta_{3} ) q^{29} + ( 5467584 - 152736 \beta_{3} ) q^{31} + ( 3158662 \beta_{1} - 85199 \beta_{2} ) q^{32} + ( 967140 \beta_{1} + 191808 \beta_{2} ) q^{33} + ( -2842270 - 38082 \beta_{3} ) q^{34} + ( 1673055 + 203391 \beta_{3} ) q^{36} + ( -10884918 \beta_{1} + 198496 \beta_{2} ) q^{37} + ( -3917568 \beta_{1} - 136732 \beta_{2} ) q^{38} + ( 1199934 - 432864 \beta_{3} ) q^{39} + ( 13276234 - 492096 \beta_{3} ) q^{41} + ( 19123776 \beta_{1} + 852768 \beta_{2} ) q^{42} + ( -3946748 \beta_{1} - 702336 \beta_{2} ) q^{43} + ( 42227996 + 973980 \beta_{3} ) q^{44} + ( 8527904 + 39584 \beta_{3} ) q^{46} + ( 14804856 \beta_{1} - 1970528 \beta_{2} ) q^{47} + ( -15720642 \beta_{1} - 72819 \beta_{2} ) q^{48} + ( -35060921 - 3161088 \beta_{3} ) q^{49} + ( 6661278 - 609120 \beta_{3} ) q^{51} + ( 87081468 \beta_{1} + 1069150 \beta_{2} ) q^{52} + ( 1476694 \beta_{1} - 177728 \beta_{2} ) q^{53} + ( 7971615 + 531441 \beta_{3} ) q^{54} + ( 118920480 + 4613280 \beta_{3} ) q^{56} + ( -4115772 \beta_{1} - 463968 \beta_{2} ) q^{57} + ( -114142496 \beta_{1} - 3952034 \beta_{2} ) q^{58} + ( 15573156 + 4348352 \beta_{3} ) q^{59} + ( 170273566 + 950208 \beta_{3} ) q^{61} + ( -2254656 \beta_{1} - 3023808 \beta_{2} ) q^{62} + ( 47029248 \beta_{1} + 1469664 \beta_{2} ) q^{63} + ( 101389753 + 2340965 \beta_{3} ) q^{64} + ( 115589916 + 3844260 \beta_{3} ) q^{66} + ( 143584628 \beta_{1} - 1026560 \beta_{2} ) q^{67} + ( 104981692 \beta_{1} - 398658 \beta_{2} ) q^{68} + ( -30842208 + 2128032 \beta_{3} ) q^{69} + ( 107415672 - 4545280 \beta_{3} ) q^{71} + ( -86513346 \beta_{1} - 1568079 \beta_{2} ) q^{72} + ( -115747446 \beta_{1} + 1168192 \beta_{2} ) q^{73} + ( -58666378 - 7907478 \beta_{3} ) q^{74} + ( -107738276 - 3035812 \beta_{3} ) q^{76} + ( 373080064 \beta_{1} + 19117952 \beta_{2} ) q^{77} + ( 222339168 \beta_{1} + 5725890 \beta_{2} ) q^{78} + ( 21902080 - 19049120 \beta_{3} ) q^{79} + 43046721 q^{81} + ( 62169824 \beta_{1} - 5402698 \beta_{2} ) q^{82} + ( -182410932 \beta_{1} + 7260288 \beta_{2} ) q^{83} + ( 448283808 + 22625568 \beta_{3} ) q^{84} + ( -429332292 - 14481788 \beta_{3} ) q^{86} + ( -115294914 \beta_{1} - 13654656 \beta_{2} ) q^{87} + ( -464186824 \beta_{1} - 33512156 \beta_{2} ) q^{88} + ( 218344134 - 9049152 \beta_{3} ) q^{89} + ( -545935936 - 34987456 \beta_{3} ) q^{91} + ( -340036288 \beta_{1} + 4290016 \beta_{2} ) q^{92} + ( -430502688 \beta_{1} + 12371616 \beta_{2} ) q^{93} + ( -816395416 - 14753064 \beta_{3} ) q^{94} + ( 262752741 - 6901119 \beta_{3} ) q^{96} + ( -879104002 \beta_{1} + 13450880 \beta_{2} ) q^{97} + ( 2324861840 \beta_{1} + 85638329 \beta_{2} ) q^{98} + ( 62801892 + 15536448 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 1082q^{4} - 5022q^{6} - 26244q^{9} + O(q^{10})$$ $$4q - 1082q^{4} - 5022q^{6} - 26244q^{9} - 43024q^{11} - 923328q^{14} + 774530q^{16} + 191792q^{19} - 2286144q^{21} + 4233546q^{24} - 10838332q^{26} + 5356424q^{29} + 21564864q^{31} - 11445244q^{34} + 7099002q^{36} + 3934008q^{39} + 52120744q^{41} + 170859944q^{44} + 34190784q^{46} - 146565860q^{49} + 25426872q^{51} + 32949342q^{54} + 484908480q^{56} + 70989328q^{59} + 682994680q^{61} + 410240942q^{64} + 470048184q^{66} - 119112768q^{69} + 420572128q^{71} - 250480468q^{74} - 437024728q^{76} + 49510080q^{79} + 172186884q^{81} + 1838386368q^{84} - 1746292744q^{86} + 855278232q^{89} - 2253718656q^{91} - 3295087792q^{94} + 1037208726q^{96} + 282280464q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 121 x^{2} + 3600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 61 \nu$$$$)/60$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 241 \nu$$$$)/60$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{2} + 182$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 182$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-61 \beta_{2} + 241 \beta_{1}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 7.26209i 8.26209i − 8.26209i − 7.26209i
38.7863i 81.0000i −992.374 0 −3141.69 12272.1i 18631.9i −6561.00 0
49.2 7.78626i 81.0000i 451.374 0 630.687 1839.88i 7501.08i −6561.00 0
49.3 7.78626i 81.0000i 451.374 0 630.687 1839.88i 7501.08i −6561.00 0
49.4 38.7863i 81.0000i −992.374 0 −3141.69 12272.1i 18631.9i −6561.00 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 75.10.b.f 4
3.b odd 2 1 225.10.b.i 4
5.b even 2 1 inner 75.10.b.f 4
5.c odd 4 1 15.10.a.d 2
5.c odd 4 1 75.10.a.f 2
15.d odd 2 1 225.10.b.i 4
15.e even 4 1 45.10.a.d 2
15.e even 4 1 225.10.a.k 2
20.e even 4 1 240.10.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.d 2 5.c odd 4 1
45.10.a.d 2 15.e even 4 1
75.10.a.f 2 5.c odd 4 1
75.10.b.f 4 1.a even 1 1 trivial
75.10.b.f 4 5.b even 2 1 inner
225.10.a.k 2 15.e even 4 1
225.10.b.i 4 3.b odd 2 1
225.10.b.i 4 15.d odd 2 1
240.10.a.r 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 1565 T_{2}^{2} + 91204$$ acting on $$S_{10}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 483 T^{2} + 61508 T^{4} - 126615552 T^{6} + 68719476736 T^{8}$$
$3$ $$( 1 + 6561 T^{2} )^{2}$$
$5$ 1
$7$ $$1 - 7424284 T^{2} - 2147813645596122 T^{4} -$$$$12\!\cdots\!16$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 + 21512 T + 1790961254 T^{2} + 50724170728792 T^{3} + 5559917313492231481 T^{4} )^{2}$$
$13$ $$1 - 11151625772 T^{2} +$$$$24\!\cdots\!78$$$$T^{4} -$$$$12\!\cdots\!88$$$$T^{6} +$$$$12\!\cdots\!41$$$$T^{8}$$
$17$ $$1 - 400705004220 T^{2} +$$$$67\!\cdots\!18$$$$T^{4} -$$$$56\!\cdots\!80$$$$T^{6} +$$$$19\!\cdots\!81$$$$T^{8}$$
$19$ $$( 1 - 95896 T + 629883192438 T^{2} - 30944459466214984 T^{3} +$$$$10\!\cdots\!41$$$$T^{4} )^{2}$$
$23$ $$1 - 6185762667356 T^{2} +$$$$15\!\cdots\!18$$$$T^{4} -$$$$20\!\cdots\!64$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 - 2678212 T + 15397908029438 T^{2} - 38853212438324066228 T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$31$ $$( 1 - 10782432 T + 69294691361342 T^{2} -$$$$28\!\cdots\!72$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{2}$$
$37$ $$1 - 235813135792844 T^{2} +$$$$37\!\cdots\!18$$$$T^{4} -$$$$39\!\cdots\!76$$$$T^{6} +$$$$28\!\cdots\!41$$$$T^{8}$$
$41$ $$( 1 - 26060372 T + 693239183881142 T^{2} -$$$$85\!\cdots\!92$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4} )^{2}$$
$43$ $$1 - 1449556389870860 T^{2} +$$$$10\!\cdots\!98$$$$T^{4} -$$$$36\!\cdots\!40$$$$T^{6} +$$$$63\!\cdots\!01$$$$T^{8}$$
$47$ $$1 + 233226346198980 T^{2} +$$$$41\!\cdots\!78$$$$T^{4} +$$$$29\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!21$$$$T^{8}$$
$53$ $$1 - 13159896059574156 T^{2} +$$$$65\!\cdots\!18$$$$T^{4} -$$$$14\!\cdots\!84$$$$T^{6} +$$$$11\!\cdots\!21$$$$T^{8}$$
$59$ $$( 1 - 35494664 T + 7388006896329158 T^{2} -$$$$30\!\cdots\!96$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4} )^{2}$$
$61$ $$( 1 - 341497340 T + 52053805546777278 T^{2} -$$$$39\!\cdots\!40$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4} )^{2}$$
$67$ $$1 - 66154849722487660 T^{2} +$$$$25\!\cdots\!18$$$$T^{4} -$$$$48\!\cdots\!40$$$$T^{6} +$$$$54\!\cdots\!81$$$$T^{8}$$
$71$ $$( 1 - 210286064 T + 91549406631588686 T^{2} -$$$$96\!\cdots\!84$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$73$ $$1 - 206940304125633116 T^{2} +$$$$17\!\cdots\!78$$$$T^{4} -$$$$71\!\cdots\!04$$$$T^{6} +$$$$12\!\cdots\!61$$$$T^{8}$$
$79$ $$( 1 - 24755040 T + 43090694479668638 T^{2} -$$$$29\!\cdots\!60$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4} )^{2}$$
$83$ $$1 - 621372529743013292 T^{2} +$$$$16\!\cdots\!98$$$$T^{4} -$$$$21\!\cdots\!28$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 - 427639116 T + 702028302670151638 T^{2} -$$$$14\!\cdots\!44$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4} )^{2}$$
$97$ $$1 - 1275322199616361340 T^{2} +$$$$12\!\cdots\!78$$$$T^{4} -$$$$73\!\cdots\!60$$$$T^{6} +$$$$33\!\cdots\!21$$$$T^{8}$$