# Properties

 Label 75.10.b.d 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + 81 i q^{3} + 496 q^{4} -324 q^{6} + 7680 i q^{7} + 4032 i q^{8} -6561 q^{9} +O(q^{10})$$ $$q + 4 i q^{2} + 81 i q^{3} + 496 q^{4} -324 q^{6} + 7680 i q^{7} + 4032 i q^{8} -6561 q^{9} -86404 q^{11} + 40176 i q^{12} -149978 i q^{13} -30720 q^{14} + 237824 q^{16} + 207622 i q^{17} -26244 i q^{18} -716284 q^{19} -622080 q^{21} -345616 i q^{22} + 1369920 i q^{23} -326592 q^{24} + 599912 q^{26} -531441 i q^{27} + 3809280 i q^{28} + 3194402 q^{29} -2349000 q^{31} + 3015680 i q^{32} -6998724 i q^{33} -830488 q^{34} -3254256 q^{36} -18735710 i q^{37} -2865136 i q^{38} + 12148218 q^{39} -29282630 q^{41} -2488320 i q^{42} -1516724 i q^{43} -42856384 q^{44} -5479680 q^{46} -615752 i q^{47} + 19263744 i q^{48} -18628793 q^{49} -16817382 q^{51} -74389088 i q^{52} + 4747430 i q^{53} + 2125764 q^{54} -30965760 q^{56} -58019004 i q^{57} + 12777608 i q^{58} -60616076 q^{59} -126745682 q^{61} -9396000 i q^{62} -50388480 i q^{63} + 109703168 q^{64} + 27994896 q^{66} + 111182652 i q^{67} + 102980512 i q^{68} -110963520 q^{69} -175551608 q^{71} -26453952 i q^{72} -61233350 i q^{73} + 74942840 q^{74} -355276864 q^{76} -663582720 i q^{77} + 48592872 i q^{78} -234431160 q^{79} + 43046721 q^{81} -117130520 i q^{82} + 118910388 i q^{83} -308551680 q^{84} + 6066896 q^{86} + 258746562 i q^{87} -348380928 i q^{88} + 316534326 q^{89} + 1151831040 q^{91} + 679480320 i q^{92} -190269000 i q^{93} + 2463008 q^{94} -244270080 q^{96} -242912258 i q^{97} -74515172 i q^{98} + 566896644 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 992q^{4} - 648q^{6} - 13122q^{9} + O(q^{10})$$ $$2q + 992q^{4} - 648q^{6} - 13122q^{9} - 172808q^{11} - 61440q^{14} + 475648q^{16} - 1432568q^{19} - 1244160q^{21} - 653184q^{24} + 1199824q^{26} + 6388804q^{29} - 4698000q^{31} - 1660976q^{34} - 6508512q^{36} + 24296436q^{39} - 58565260q^{41} - 85712768q^{44} - 10959360q^{46} - 37257586q^{49} - 33634764q^{51} + 4251528q^{54} - 61931520q^{56} - 121232152q^{59} - 253491364q^{61} + 219406336q^{64} + 55989792q^{66} - 221927040q^{69} - 351103216q^{71} + 149885680q^{74} - 710553728q^{76} - 468862320q^{79} + 86093442q^{81} - 617103360q^{84} + 12133792q^{86} + 633068652q^{89} + 2303662080q^{91} + 4926016q^{94} - 488540160q^{96} + 1133793288q^{99} + O(q^{100})$$

## 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
4.00000i 81.0000i 496.000 0 −324.000 7680.00i 4032.00i −6561.00 0
49.2 4.00000i 81.0000i 496.000 0 −324.000 7680.00i 4032.00i −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.d 2
3.b odd 2 1 225.10.b.e 2
5.b even 2 1 inner 75.10.b.d 2
5.c odd 4 1 15.10.a.a 1
5.c odd 4 1 75.10.a.c 1
15.d odd 2 1 225.10.b.e 2
15.e even 4 1 45.10.a.b 1
15.e even 4 1 225.10.a.c 1
20.e even 4 1 240.10.a.c 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 16$$ acting on $$S_{10}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 1008 T^{2} + 262144 T^{4}$$
$3$ $$1 + 6561 T^{2}$$
$5$ 1
$7$ $$1 - 21724814 T^{2} + 1628413597910449 T^{4}$$
$11$ $$( 1 + 86404 T + 2357947691 T^{2} )^{2}$$
$13$ $$1 + 1284401738 T^{2} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 - 194068858110 T^{2} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$( 1 + 716284 T + 322687697779 T^{2} )^{2}$$
$23$ $$1 - 1725624516526 T^{2} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$( 1 - 3194402 T + 14507145975869 T^{2} )^{2}$$
$31$ $$( 1 + 2349000 T + 26439622160671 T^{2} )^{2}$$
$37$ $$1 + 91103349613946 T^{2} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$( 1 + 29282630 T + 327381934393961 T^{2} )^{2}$$
$43$ $$1 - 1002884772181510 T^{2} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 2237881795680030 T^{2} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 - 6576989091999366 T^{2} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$( 1 + 60616076 T + 8662995818654939 T^{2} )^{2}$$
$61$ $$( 1 + 126745682 T + 11694146092834141 T^{2} )^{2}$$
$67$ $$1 - 42051486686836790 T^{2} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$( 1 + 175551608 T + 45848500718449031 T^{2} )^{2}$$
$73$ $$1 - 113993650264313326 T^{2} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$( 1 + 234431160 T + 119851595982618319 T^{2} )^{2}$$
$83$ $$1 - 359740830160770262 T^{2} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$( 1 - 316534326 T + 350356403707485209 T^{2} )^{2}$$
$97$ $$1 - 1461455752222471870 T^{2} +$$$$57\!\cdots\!89$$$$T^{4}$$