# Properties

 Label 75.4.a.a Level $75$ Weight $4$ Character orbit 75.a Self dual yes Analytic conductor $4.425$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.42514325043$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + 3 q^{3} + q^{4} - 9 q^{6} - 20 q^{7} + 21 q^{8} + 9 q^{9}+O(q^{10})$$ q - 3 * q^2 + 3 * q^3 + q^4 - 9 * q^6 - 20 * q^7 + 21 * q^8 + 9 * q^9 $$q - 3 q^{2} + 3 q^{3} + q^{4} - 9 q^{6} - 20 q^{7} + 21 q^{8} + 9 q^{9} - 24 q^{11} + 3 q^{12} - 74 q^{13} + 60 q^{14} - 71 q^{16} - 54 q^{17} - 27 q^{18} - 124 q^{19} - 60 q^{21} + 72 q^{22} + 120 q^{23} + 63 q^{24} + 222 q^{26} + 27 q^{27} - 20 q^{28} - 78 q^{29} + 200 q^{31} + 45 q^{32} - 72 q^{33} + 162 q^{34} + 9 q^{36} + 70 q^{37} + 372 q^{38} - 222 q^{39} + 330 q^{41} + 180 q^{42} - 92 q^{43} - 24 q^{44} - 360 q^{46} + 24 q^{47} - 213 q^{48} + 57 q^{49} - 162 q^{51} - 74 q^{52} - 450 q^{53} - 81 q^{54} - 420 q^{56} - 372 q^{57} + 234 q^{58} + 24 q^{59} - 322 q^{61} - 600 q^{62} - 180 q^{63} + 433 q^{64} + 216 q^{66} + 196 q^{67} - 54 q^{68} + 360 q^{69} - 288 q^{71} + 189 q^{72} + 430 q^{73} - 210 q^{74} - 124 q^{76} + 480 q^{77} + 666 q^{78} - 520 q^{79} + 81 q^{81} - 990 q^{82} - 156 q^{83} - 60 q^{84} + 276 q^{86} - 234 q^{87} - 504 q^{88} + 1026 q^{89} + 1480 q^{91} + 120 q^{92} + 600 q^{93} - 72 q^{94} + 135 q^{96} + 286 q^{97} - 171 q^{98} - 216 q^{99}+O(q^{100})$$ q - 3 * q^2 + 3 * q^3 + q^4 - 9 * q^6 - 20 * q^7 + 21 * q^8 + 9 * q^9 - 24 * q^11 + 3 * q^12 - 74 * q^13 + 60 * q^14 - 71 * q^16 - 54 * q^17 - 27 * q^18 - 124 * q^19 - 60 * q^21 + 72 * q^22 + 120 * q^23 + 63 * q^24 + 222 * q^26 + 27 * q^27 - 20 * q^28 - 78 * q^29 + 200 * q^31 + 45 * q^32 - 72 * q^33 + 162 * q^34 + 9 * q^36 + 70 * q^37 + 372 * q^38 - 222 * q^39 + 330 * q^41 + 180 * q^42 - 92 * q^43 - 24 * q^44 - 360 * q^46 + 24 * q^47 - 213 * q^48 + 57 * q^49 - 162 * q^51 - 74 * q^52 - 450 * q^53 - 81 * q^54 - 420 * q^56 - 372 * q^57 + 234 * q^58 + 24 * q^59 - 322 * q^61 - 600 * q^62 - 180 * q^63 + 433 * q^64 + 216 * q^66 + 196 * q^67 - 54 * q^68 + 360 * q^69 - 288 * q^71 + 189 * q^72 + 430 * q^73 - 210 * q^74 - 124 * q^76 + 480 * q^77 + 666 * q^78 - 520 * q^79 + 81 * q^81 - 990 * q^82 - 156 * q^83 - 60 * q^84 + 276 * q^86 - 234 * q^87 - 504 * q^88 + 1026 * q^89 + 1480 * q^91 + 120 * q^92 + 600 * q^93 - 72 * q^94 + 135 * q^96 + 286 * q^97 - 171 * q^98 - 216 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−3.00000 3.00000 1.00000 0 −9.00000 −20.0000 21.0000 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.4.a.a 1
3.b odd 2 1 225.4.a.g 1
4.b odd 2 1 1200.4.a.o 1
5.b even 2 1 15.4.a.b 1
5.c odd 4 2 75.4.b.a 2
15.d odd 2 1 45.4.a.b 1
15.e even 4 2 225.4.b.d 2
20.d odd 2 1 240.4.a.f 1
20.e even 4 2 1200.4.f.m 2
35.c odd 2 1 735.4.a.i 1
40.e odd 2 1 960.4.a.l 1
40.f even 2 1 960.4.a.bi 1
45.h odd 6 2 405.4.e.k 2
45.j even 6 2 405.4.e.d 2
55.d odd 2 1 1815.4.a.a 1
60.h even 2 1 720.4.a.r 1
105.g even 2 1 2205.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 5.b even 2 1
45.4.a.b 1 15.d odd 2 1
75.4.a.a 1 1.a even 1 1 trivial
75.4.b.a 2 5.c odd 4 2
225.4.a.g 1 3.b odd 2 1
225.4.b.d 2 15.e even 4 2
240.4.a.f 1 20.d odd 2 1
405.4.e.d 2 45.j even 6 2
405.4.e.k 2 45.h odd 6 2
720.4.a.r 1 60.h even 2 1
735.4.a.i 1 35.c odd 2 1
960.4.a.l 1 40.e odd 2 1
960.4.a.bi 1 40.f even 2 1
1200.4.a.o 1 4.b odd 2 1
1200.4.f.m 2 20.e even 4 2
1815.4.a.a 1 55.d odd 2 1
2205.4.a.c 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(75))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T + 20$$
$11$ $$T + 24$$
$13$ $$T + 74$$
$17$ $$T + 54$$
$19$ $$T + 124$$
$23$ $$T - 120$$
$29$ $$T + 78$$
$31$ $$T - 200$$
$37$ $$T - 70$$
$41$ $$T - 330$$
$43$ $$T + 92$$
$47$ $$T - 24$$
$53$ $$T + 450$$
$59$ $$T - 24$$
$61$ $$T + 322$$
$67$ $$T - 196$$
$71$ $$T + 288$$
$73$ $$T - 430$$
$79$ $$T + 520$$
$83$ $$T + 156$$
$89$ $$T - 1026$$
$97$ $$T - 286$$