# Properties

 Label 75.10.b.b Level $75$ Weight $10$ Character orbit 75.b Analytic conductor $38.628$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 22 i q^{2} + 81 i q^{3} + 28 q^{4} -1782 q^{6} -5988 i q^{7} + 11880 i q^{8} -6561 q^{9} +O(q^{10})$$ $$q + 22 i q^{2} + 81 i q^{3} + 28 q^{4} -1782 q^{6} -5988 i q^{7} + 11880 i q^{8} -6561 q^{9} -14648 q^{11} + 2268 i q^{12} -37906 i q^{13} + 131736 q^{14} -247024 q^{16} -441098 i q^{17} -144342 i q^{18} -441820 q^{19} + 485028 q^{21} -322256 i q^{22} -2264136 i q^{23} -962280 q^{24} + 833932 q^{26} -531441 i q^{27} -167664 i q^{28} + 1049350 q^{29} -7910568 q^{31} + 648032 i q^{32} -1186488 i q^{33} + 9704156 q^{34} -183708 q^{36} -20992558 i q^{37} -9720040 i q^{38} + 3070386 q^{39} + 13285562 q^{41} + 10670616 i q^{42} + 23130764 i q^{43} -410144 q^{44} + 49810992 q^{46} -13873688 i q^{47} -20008944 i q^{48} + 4497463 q^{49} + 35728938 q^{51} -1061368 i q^{52} + 57635174 i q^{53} + 11691702 q^{54} + 71137440 q^{56} -35787420 i q^{57} + 23085700 i q^{58} + 32042120 q^{59} + 110664022 q^{61} -174032496 i q^{62} + 39287268 i q^{63} -140732992 q^{64} + 26102736 q^{66} -118568268 i q^{67} -12350744 i q^{68} + 183395016 q^{69} + 276679712 q^{71} -77944680 i q^{72} + 264023294 i q^{73} + 461836276 q^{74} -12370960 q^{76} + 87712224 i q^{77} + 67548492 i q^{78} -448202760 q^{79} + 43046721 q^{81} + 292282364 i q^{82} -851015796 i q^{83} + 13580784 q^{84} -508876808 q^{86} + 84997350 i q^{87} -174018240 i q^{88} -189894930 q^{89} -226981128 q^{91} -63395808 i q^{92} -640756008 i q^{93} + 305221136 q^{94} -52490592 q^{96} -1014149278 i q^{97} + 98944186 i q^{98} + 96105528 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 56q^{4} - 3564q^{6} - 13122q^{9} + O(q^{10})$$ $$2q + 56q^{4} - 3564q^{6} - 13122q^{9} - 29296q^{11} + 263472q^{14} - 494048q^{16} - 883640q^{19} + 970056q^{21} - 1924560q^{24} + 1667864q^{26} + 2098700q^{29} - 15821136q^{31} + 19408312q^{34} - 367416q^{36} + 6140772q^{39} + 26571124q^{41} - 820288q^{44} + 99621984q^{46} + 8994926q^{49} + 71457876q^{51} + 23383404q^{54} + 142274880q^{56} + 64084240q^{59} + 221328044q^{61} - 281465984q^{64} + 52205472q^{66} + 366790032q^{69} + 553359424q^{71} + 923672552q^{74} - 24741920q^{76} - 896405520q^{79} + 86093442q^{81} + 27161568q^{84} - 1017753616q^{86} - 379789860q^{89} - 453962256q^{91} + 610442272q^{94} - 104981184q^{96} + 192211056q^{99} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
22.0000i 81.0000i 28.0000 0 −1782.00 5988.00i 11880.0i −6561.00 0
49.2 22.0000i 81.0000i 28.0000 0 −1782.00 5988.00i 11880.0i −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.b 2
3.b odd 2 1 225.10.b.b 2
5.b even 2 1 inner 75.10.b.b 2
5.c odd 4 1 15.10.a.b 1
5.c odd 4 1 75.10.a.a 1
15.d odd 2 1 225.10.b.b 2
15.e even 4 1 45.10.a.a 1
15.e even 4 1 225.10.a.f 1
20.e even 4 1 240.10.a.g 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 484$$ acting on $$S_{10}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 540 T^{2} + 262144 T^{4}$$
$3$ $$1 + 6561 T^{2}$$
$5$ 1
$7$ $$1 - 44851070 T^{2} + 1628413597910449 T^{4}$$
$11$ $$( 1 + 14648 T + 2357947691 T^{2} )^{2}$$
$13$ $$1 - 19772133910 T^{2} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 - 42608307390 T^{2} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$( 1 + 441820 T + 322687697779 T^{2} )^{2}$$
$23$ $$1 + 1524006503570 T^{2} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$( 1 - 1049350 T + 14507145975869 T^{2} )^{2}$$
$31$ $$( 1 + 7910568 T + 26439622160671 T^{2} )^{2}$$
$37$ $$1 + 180764011793210 T^{2} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$( 1 - 13285562 T + 327381934393961 T^{2} )^{2}$$
$43$ $$1 - 470152980649990 T^{2} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 2045781727484190 T^{2} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 - 3277713901593990 T^{2} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$( 1 - 32042120 T + 8662995818654939 T^{2} )^{2}$$
$61$ $$( 1 - 110664022 T + 11694146092834141 T^{2} )^{2}$$
$67$ $$1 - 40354634616070070 T^{2} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$( 1 - 276679712 T + 45848500718449031 T^{2} )^{2}$$
$73$ $$1 - 48034873641925390 T^{2} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$( 1 + 448202760 T + 119851595982618319 T^{2} )^{2}$$
$83$ $$1 + 350347374506432810 T^{2} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$( 1 + 189894930 T + 350356403707485209 T^{2} )^{2}$$
$97$ $$1 - 491963359241209150 T^{2} +$$$$57\!\cdots\!89$$$$T^{4}$$