# Properties

 Label 75.4.a.e Level $75$ Weight $4$ Character orbit 75.a Self dual yes Analytic conductor $4.425$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{19})$$ Defining polynomial: $$x^{2} - 19$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} -3 q^{3} + ( 12 + 2 \beta ) q^{4} + ( -3 - 3 \beta ) q^{6} + ( -13 + 4 \beta ) q^{7} + ( 42 + 6 \beta ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} -3 q^{3} + ( 12 + 2 \beta ) q^{4} + ( -3 - 3 \beta ) q^{6} + ( -13 + 4 \beta ) q^{7} + ( 42 + 6 \beta ) q^{8} + 9 q^{9} + ( 14 - 4 \beta ) q^{11} + ( -36 - 6 \beta ) q^{12} + ( -9 - 16 \beta ) q^{13} + ( 63 - 9 \beta ) q^{14} + ( 60 + 32 \beta ) q^{16} + ( 34 - 20 \beta ) q^{17} + ( 9 + 9 \beta ) q^{18} + ( 3 + 4 \beta ) q^{19} + ( 39 - 12 \beta ) q^{21} + ( -62 + 10 \beta ) q^{22} + ( -66 - 12 \beta ) q^{23} + ( -126 - 18 \beta ) q^{24} + ( -313 - 25 \beta ) q^{26} -27 q^{27} + ( -4 + 22 \beta ) q^{28} + ( 46 + 28 \beta ) q^{29} + ( 61 - 28 \beta ) q^{31} + ( 332 + 44 \beta ) q^{32} + ( -42 + 12 \beta ) q^{33} + ( -346 + 14 \beta ) q^{34} + ( 108 + 18 \beta ) q^{36} + ( 142 + 24 \beta ) q^{37} + ( 79 + 7 \beta ) q^{38} + ( 27 + 48 \beta ) q^{39} + ( 196 + 52 \beta ) q^{41} + ( -189 + 27 \beta ) q^{42} + ( -345 - 4 \beta ) q^{43} + ( 16 - 20 \beta ) q^{44} + ( -294 - 78 \beta ) q^{46} + ( 310 - 32 \beta ) q^{47} + ( -180 - 96 \beta ) q^{48} + ( 130 - 104 \beta ) q^{49} + ( -102 + 60 \beta ) q^{51} + ( -716 - 210 \beta ) q^{52} + ( 424 + 28 \beta ) q^{53} + ( -27 - 27 \beta ) q^{54} + ( -90 + 90 \beta ) q^{56} + ( -9 - 12 \beta ) q^{57} + ( 578 + 74 \beta ) q^{58} + ( 62 - 64 \beta ) q^{59} + ( 375 - 56 \beta ) q^{61} + ( -471 + 33 \beta ) q^{62} + ( -117 + 36 \beta ) q^{63} + ( 688 + 120 \beta ) q^{64} + ( 186 - 30 \beta ) q^{66} + ( 179 + 100 \beta ) q^{67} + ( -352 - 172 \beta ) q^{68} + ( 198 + 36 \beta ) q^{69} + ( 412 - 20 \beta ) q^{71} + ( 378 + 54 \beta ) q^{72} + ( 54 + 8 \beta ) q^{73} + ( 598 + 166 \beta ) q^{74} + ( 188 + 54 \beta ) q^{76} + ( -486 + 108 \beta ) q^{77} + ( 939 + 75 \beta ) q^{78} + ( -440 - 160 \beta ) q^{79} + 81 q^{81} + ( 1184 + 248 \beta ) q^{82} + ( -78 + 192 \beta ) q^{83} + ( 12 - 66 \beta ) q^{84} + ( -421 - 349 \beta ) q^{86} + ( -138 - 84 \beta ) q^{87} + ( 132 - 84 \beta ) q^{88} + ( -432 + 144 \beta ) q^{89} + ( -1099 + 172 \beta ) q^{91} + ( -1248 - 276 \beta ) q^{92} + ( -183 + 84 \beta ) q^{93} + ( -298 + 278 \beta ) q^{94} + ( -996 - 132 \beta ) q^{96} -521 q^{97} + ( -1846 + 26 \beta ) q^{98} + ( 126 - 36 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 6 q^{3} + 24 q^{4} - 6 q^{6} - 26 q^{7} + 84 q^{8} + 18 q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} - 6 q^{3} + 24 q^{4} - 6 q^{6} - 26 q^{7} + 84 q^{8} + 18 q^{9} + 28 q^{11} - 72 q^{12} - 18 q^{13} + 126 q^{14} + 120 q^{16} + 68 q^{17} + 18 q^{18} + 6 q^{19} + 78 q^{21} - 124 q^{22} - 132 q^{23} - 252 q^{24} - 626 q^{26} - 54 q^{27} - 8 q^{28} + 92 q^{29} + 122 q^{31} + 664 q^{32} - 84 q^{33} - 692 q^{34} + 216 q^{36} + 284 q^{37} + 158 q^{38} + 54 q^{39} + 392 q^{41} - 378 q^{42} - 690 q^{43} + 32 q^{44} - 588 q^{46} + 620 q^{47} - 360 q^{48} + 260 q^{49} - 204 q^{51} - 1432 q^{52} + 848 q^{53} - 54 q^{54} - 180 q^{56} - 18 q^{57} + 1156 q^{58} + 124 q^{59} + 750 q^{61} - 942 q^{62} - 234 q^{63} + 1376 q^{64} + 372 q^{66} + 358 q^{67} - 704 q^{68} + 396 q^{69} + 824 q^{71} + 756 q^{72} + 108 q^{73} + 1196 q^{74} + 376 q^{76} - 972 q^{77} + 1878 q^{78} - 880 q^{79} + 162 q^{81} + 2368 q^{82} - 156 q^{83} + 24 q^{84} - 842 q^{86} - 276 q^{87} + 264 q^{88} - 864 q^{89} - 2198 q^{91} - 2496 q^{92} - 366 q^{93} - 596 q^{94} - 1992 q^{96} - 1042 q^{97} - 3692 q^{98} + 252 q^{99} + O(q^{100})$$

## 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
 −4.35890 4.35890
−3.35890 −3.00000 3.28220 0 10.0767 −30.4356 15.8466 9.00000 0
1.2 5.35890 −3.00000 20.7178 0 −16.0767 4.43560 68.1534 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.e yes 2
3.b odd 2 1 225.4.a.j 2
4.b odd 2 1 1200.4.a.bu 2
5.b even 2 1 75.4.a.d 2
5.c odd 4 2 75.4.b.c 4
15.d odd 2 1 225.4.a.n 2
15.e even 4 2 225.4.b.h 4
20.d odd 2 1 1200.4.a.bl 2
20.e even 4 2 1200.4.f.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.a.d 2 5.b even 2 1
75.4.a.e yes 2 1.a even 1 1 trivial
75.4.b.c 4 5.c odd 4 2
225.4.a.j 2 3.b odd 2 1
225.4.a.n 2 15.d odd 2 1
225.4.b.h 4 15.e even 4 2
1200.4.a.bl 2 20.d odd 2 1
1200.4.a.bu 2 4.b odd 2 1
1200.4.f.v 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2 T_{2} - 18$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(75))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-18 - 2 T + T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-135 + 26 T + T^{2}$$
$11$ $$-108 - 28 T + T^{2}$$
$13$ $$-4783 + 18 T + T^{2}$$
$17$ $$-6444 - 68 T + T^{2}$$
$19$ $$-295 - 6 T + T^{2}$$
$23$ $$1620 + 132 T + T^{2}$$
$29$ $$-12780 - 92 T + T^{2}$$
$31$ $$-11175 - 122 T + T^{2}$$
$37$ $$9220 - 284 T + T^{2}$$
$41$ $$-12960 - 392 T + T^{2}$$
$43$ $$118721 + 690 T + T^{2}$$
$47$ $$76644 - 620 T + T^{2}$$
$53$ $$164880 - 848 T + T^{2}$$
$59$ $$-73980 - 124 T + T^{2}$$
$61$ $$81041 - 750 T + T^{2}$$
$67$ $$-157959 - 358 T + T^{2}$$
$71$ $$162144 - 824 T + T^{2}$$
$73$ $$1700 - 108 T + T^{2}$$
$79$ $$-292800 + 880 T + T^{2}$$
$83$ $$-694332 + 156 T + T^{2}$$
$89$ $$-207360 + 864 T + T^{2}$$
$97$ $$( 521 + T )^{2}$$