# Properties

 Label 100.4.c.a Level $100$ Weight $4$ Character orbit 100.c 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.90019100057$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 20) 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{3} + 8 \beta q^{7} + 11 q^{9}+O(q^{10})$$ q + 2*b * q^3 + 8*b * q^7 + 11 * q^9 $$q + 2 \beta q^{3} + 8 \beta q^{7} + 11 q^{9} - 60 q^{11} + 43 \beta q^{13} - 9 \beta q^{17} - 44 q^{19} - 64 q^{21} + 24 \beta q^{23} + 76 \beta q^{27} + 186 q^{29} + 176 q^{31} - 120 \beta q^{33} - 127 \beta q^{37} - 344 q^{39} + 186 q^{41} - 50 \beta q^{43} - 84 \beta q^{47} + 87 q^{49} + 72 q^{51} - 249 \beta q^{53} - 88 \beta q^{57} + 252 q^{59} - 58 q^{61} + 88 \beta q^{63} + 518 \beta q^{67} - 192 q^{69} + 168 q^{71} + 253 \beta q^{73} - 480 \beta q^{77} - 272 q^{79} - 311 q^{81} + 474 \beta q^{83} + 372 \beta q^{87} + 1014 q^{89} - 1376 q^{91} + 352 \beta q^{93} + 383 \beta q^{97} - 660 q^{99} +O(q^{100})$$ q + 2*b * q^3 + 8*b * q^7 + 11 * q^9 - 60 * q^11 + 43*b * q^13 - 9*b * q^17 - 44 * q^19 - 64 * q^21 + 24*b * q^23 + 76*b * q^27 + 186 * q^29 + 176 * q^31 - 120*b * q^33 - 127*b * q^37 - 344 * q^39 + 186 * q^41 - 50*b * q^43 - 84*b * q^47 + 87 * q^49 + 72 * q^51 - 249*b * q^53 - 88*b * q^57 + 252 * q^59 - 58 * q^61 + 88*b * q^63 + 518*b * q^67 - 192 * q^69 + 168 * q^71 + 253*b * q^73 - 480*b * q^77 - 272 * q^79 - 311 * q^81 + 474*b * q^83 + 372*b * q^87 + 1014 * q^89 - 1376 * q^91 + 352*b * q^93 + 383*b * q^97 - 660 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 22 q^{9}+O(q^{10})$$ 2 * q + 22 * q^9 $$2 q + 22 q^{9} - 120 q^{11} - 88 q^{19} - 128 q^{21} + 372 q^{29} + 352 q^{31} - 688 q^{39} + 372 q^{41} + 174 q^{49} + 144 q^{51} + 504 q^{59} - 116 q^{61} - 384 q^{69} + 336 q^{71} - 544 q^{79} - 622 q^{81} + 2028 q^{89} - 2752 q^{91} - 1320 q^{99}+O(q^{100})$$ 2 * q + 22 * q^9 - 120 * q^11 - 88 * q^19 - 128 * q^21 + 372 * q^29 + 352 * q^31 - 688 * q^39 + 372 * q^41 + 174 * q^49 + 144 * q^51 + 504 * q^59 - 116 * q^61 - 384 * q^69 + 336 * q^71 - 544 * q^79 - 622 * q^81 + 2028 * q^89 - 2752 * q^91 - 1320 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$77$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
0 4.00000i 0 0 0 16.0000i 0 11.0000 0
49.2 0 4.00000i 0 0 0 16.0000i 0 11.0000 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 100.4.c.a 2
3.b odd 2 1 900.4.d.k 2
4.b odd 2 1 400.4.c.j 2
5.b even 2 1 inner 100.4.c.a 2
5.c odd 4 1 20.4.a.a 1
5.c odd 4 1 100.4.a.a 1
15.d odd 2 1 900.4.d.k 2
15.e even 4 1 180.4.a.a 1
15.e even 4 1 900.4.a.m 1
20.d odd 2 1 400.4.c.j 2
20.e even 4 1 80.4.a.c 1
20.e even 4 1 400.4.a.o 1
35.f even 4 1 980.4.a.c 1
35.k even 12 2 980.4.i.n 2
35.l odd 12 2 980.4.i.e 2
40.i odd 4 1 320.4.a.d 1
40.i odd 4 1 1600.4.a.bl 1
40.k even 4 1 320.4.a.k 1
40.k even 4 1 1600.4.a.p 1
45.k odd 12 2 1620.4.i.d 2
45.l even 12 2 1620.4.i.j 2
55.e even 4 1 2420.4.a.d 1
60.l odd 4 1 720.4.a.k 1
80.i odd 4 1 1280.4.d.n 2
80.j even 4 1 1280.4.d.c 2
80.s even 4 1 1280.4.d.c 2
80.t odd 4 1 1280.4.d.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 5.c odd 4 1
80.4.a.c 1 20.e even 4 1
100.4.a.a 1 5.c odd 4 1
100.4.c.a 2 1.a even 1 1 trivial
100.4.c.a 2 5.b even 2 1 inner
180.4.a.a 1 15.e even 4 1
320.4.a.d 1 40.i odd 4 1
320.4.a.k 1 40.k even 4 1
400.4.a.o 1 20.e even 4 1
400.4.c.j 2 4.b odd 2 1
400.4.c.j 2 20.d odd 2 1
720.4.a.k 1 60.l odd 4 1
900.4.a.m 1 15.e even 4 1
900.4.d.k 2 3.b odd 2 1
900.4.d.k 2 15.d odd 2 1
980.4.a.c 1 35.f even 4 1
980.4.i.e 2 35.l odd 12 2
980.4.i.n 2 35.k even 12 2
1280.4.d.c 2 80.j even 4 1
1280.4.d.c 2 80.s even 4 1
1280.4.d.n 2 80.i odd 4 1
1280.4.d.n 2 80.t odd 4 1
1600.4.a.p 1 40.k even 4 1
1600.4.a.bl 1 40.i odd 4 1
1620.4.i.d 2 45.k odd 12 2
1620.4.i.j 2 45.l even 12 2
2420.4.a.d 1 55.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 16$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 256$$
$11$ $$(T + 60)^{2}$$
$13$ $$T^{2} + 7396$$
$17$ $$T^{2} + 324$$
$19$ $$(T + 44)^{2}$$
$23$ $$T^{2} + 2304$$
$29$ $$(T - 186)^{2}$$
$31$ $$(T - 176)^{2}$$
$37$ $$T^{2} + 64516$$
$41$ $$(T - 186)^{2}$$
$43$ $$T^{2} + 10000$$
$47$ $$T^{2} + 28224$$
$53$ $$T^{2} + 248004$$
$59$ $$(T - 252)^{2}$$
$61$ $$(T + 58)^{2}$$
$67$ $$T^{2} + 1073296$$
$71$ $$(T - 168)^{2}$$
$73$ $$T^{2} + 256036$$
$79$ $$(T + 272)^{2}$$
$83$ $$T^{2} + 898704$$
$89$ $$(T - 1014)^{2}$$
$97$ $$T^{2} + 586756$$