Properties

 Label 100.4.a.d Level $100$ Weight $4$ Character orbit 100.a Self dual yes Analytic conductor $5.900$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 100.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$5.90019100057$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{19})$$ Defining polynomial: $$x^{2} - 19$$ x^2 - 19 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 20) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + \beta q^{7} + 49 q^{9}+O(q^{10})$$ q + b * q^3 + b * q^7 + 49 * q^9 $$q + \beta q^{3} + \beta q^{7} + 49 q^{9} + 20 q^{11} - 6 \beta q^{13} - 8 \beta q^{17} + 84 q^{19} + 76 q^{21} + 7 \beta q^{23} + 22 \beta q^{27} - 6 q^{29} - 224 q^{31} + 20 \beta q^{33} - 14 \beta q^{37} - 456 q^{39} + 266 q^{41} - 35 \beta q^{43} - 43 \beta q^{47} - 267 q^{49} - 608 q^{51} - 42 \beta q^{53} + 84 \beta q^{57} + 28 q^{59} + 182 q^{61} + 49 \beta q^{63} - 49 \beta q^{67} + 532 q^{69} + 408 q^{71} + 124 \beta q^{73} + 20 \beta q^{77} - 48 q^{79} + 349 q^{81} - 23 \beta q^{83} - 6 \beta q^{87} + 1526 q^{89} - 456 q^{91} - 224 \beta q^{93} - 64 \beta q^{97} + 980 q^{99} +O(q^{100})$$ q + b * q^3 + b * q^7 + 49 * q^9 + 20 * q^11 - 6*b * q^13 - 8*b * q^17 + 84 * q^19 + 76 * q^21 + 7*b * q^23 + 22*b * q^27 - 6 * q^29 - 224 * q^31 + 20*b * q^33 - 14*b * q^37 - 456 * q^39 + 266 * q^41 - 35*b * q^43 - 43*b * q^47 - 267 * q^49 - 608 * q^51 - 42*b * q^53 + 84*b * q^57 + 28 * q^59 + 182 * q^61 + 49*b * q^63 - 49*b * q^67 + 532 * q^69 + 408 * q^71 + 124*b * q^73 + 20*b * q^77 - 48 * q^79 + 349 * q^81 - 23*b * q^83 - 6*b * q^87 + 1526 * q^89 - 456 * q^91 - 224*b * q^93 - 64*b * q^97 + 980 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 98 q^{9}+O(q^{10})$$ 2 * q + 98 * q^9 $$2 q + 98 q^{9} + 40 q^{11} + 168 q^{19} + 152 q^{21} - 12 q^{29} - 448 q^{31} - 912 q^{39} + 532 q^{41} - 534 q^{49} - 1216 q^{51} + 56 q^{59} + 364 q^{61} + 1064 q^{69} + 816 q^{71} - 96 q^{79} + 698 q^{81} + 3052 q^{89} - 912 q^{91} + 1960 q^{99}+O(q^{100})$$ 2 * q + 98 * q^9 + 40 * q^11 + 168 * q^19 + 152 * q^21 - 12 * q^29 - 448 * q^31 - 912 * q^39 + 532 * q^41 - 534 * q^49 - 1216 * q^51 + 56 * q^59 + 364 * q^61 + 1064 * q^69 + 816 * q^71 - 96 * q^79 + 698 * q^81 + 3052 * q^89 - 912 * q^91 + 1960 * 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
 −4.35890 4.35890
0 −8.71780 0 0 0 −8.71780 0 49.0000 0
1.2 0 8.71780 0 0 0 8.71780 0 49.0000 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
$$2$$ $$-1$$
$$5$$ $$-1$$

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 100.4.a.d 2
3.b odd 2 1 900.4.a.s 2
4.b odd 2 1 400.4.a.w 2
5.b even 2 1 inner 100.4.a.d 2
5.c odd 4 2 20.4.c.a 2
8.b even 2 1 1600.4.a.cj 2
8.d odd 2 1 1600.4.a.ck 2
15.d odd 2 1 900.4.a.s 2
15.e even 4 2 180.4.d.a 2
20.d odd 2 1 400.4.a.w 2
20.e even 4 2 80.4.c.b 2
35.f even 4 2 980.4.e.a 2
40.e odd 2 1 1600.4.a.ck 2
40.f even 2 1 1600.4.a.cj 2
40.i odd 4 2 320.4.c.a 2
40.k even 4 2 320.4.c.b 2
60.l odd 4 2 720.4.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.c.a 2 5.c odd 4 2
80.4.c.b 2 20.e even 4 2
100.4.a.d 2 1.a even 1 1 trivial
100.4.a.d 2 5.b even 2 1 inner
180.4.d.a 2 15.e even 4 2
320.4.c.a 2 40.i odd 4 2
320.4.c.b 2 40.k even 4 2
400.4.a.w 2 4.b odd 2 1
400.4.a.w 2 20.d odd 2 1
720.4.f.a 2 60.l odd 4 2
900.4.a.s 2 3.b odd 2 1
900.4.a.s 2 15.d odd 2 1
980.4.e.a 2 35.f even 4 2
1600.4.a.cj 2 8.b even 2 1
1600.4.a.cj 2 40.f even 2 1
1600.4.a.ck 2 8.d odd 2 1
1600.4.a.ck 2 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 76$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(100))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 76$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 76$$
$11$ $$(T - 20)^{2}$$
$13$ $$T^{2} - 2736$$
$17$ $$T^{2} - 4864$$
$19$ $$(T - 84)^{2}$$
$23$ $$T^{2} - 3724$$
$29$ $$(T + 6)^{2}$$
$31$ $$(T + 224)^{2}$$
$37$ $$T^{2} - 14896$$
$41$ $$(T - 266)^{2}$$
$43$ $$T^{2} - 93100$$
$47$ $$T^{2} - 140524$$
$53$ $$T^{2} - 134064$$
$59$ $$(T - 28)^{2}$$
$61$ $$(T - 182)^{2}$$
$67$ $$T^{2} - 182476$$
$71$ $$(T - 408)^{2}$$
$73$ $$T^{2} - 1168576$$
$79$ $$(T + 48)^{2}$$
$83$ $$T^{2} - 40204$$
$89$ $$(T - 1526)^{2}$$
$97$ $$T^{2} - 311296$$