# Properties

 Label 100.4.c.b 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} - 26 i q^{7} + 26 q^{9} +O(q^{10})$$ q + i * q^3 - 26*i * q^7 + 26 * q^9 $$q + i q^{3} - 26 i q^{7} + 26 q^{9} + 45 q^{11} + 44 i q^{13} - 117 i q^{17} + 91 q^{19} + 26 q^{21} - 18 i q^{23} + 53 i q^{27} - 144 q^{29} + 26 q^{31} + 45 i q^{33} + 214 i q^{37} - 44 q^{39} - 459 q^{41} - 460 i q^{43} + 468 i q^{47} - 333 q^{49} + 117 q^{51} + 558 i q^{53} + 91 i q^{57} + 72 q^{59} - 118 q^{61} - 676 i q^{63} - 251 i q^{67} + 18 q^{69} + 108 q^{71} + 299 i q^{73} - 1170 i q^{77} + 898 q^{79} + 649 q^{81} + 927 i q^{83} - 144 i q^{87} - 351 q^{89} + 1144 q^{91} + 26 i q^{93} - 386 i q^{97} + 1170 q^{99} +O(q^{100})$$ q + i * q^3 - 26*i * q^7 + 26 * q^9 + 45 * q^11 + 44*i * q^13 - 117*i * q^17 + 91 * q^19 + 26 * q^21 - 18*i * q^23 + 53*i * q^27 - 144 * q^29 + 26 * q^31 + 45*i * q^33 + 214*i * q^37 - 44 * q^39 - 459 * q^41 - 460*i * q^43 + 468*i * q^47 - 333 * q^49 + 117 * q^51 + 558*i * q^53 + 91*i * q^57 + 72 * q^59 - 118 * q^61 - 676*i * q^63 - 251*i * q^67 + 18 * q^69 + 108 * q^71 + 299*i * q^73 - 1170*i * q^77 + 898 * q^79 + 649 * q^81 + 927*i * q^83 - 144*i * q^87 - 351 * q^89 + 1144 * q^91 + 26*i * q^93 - 386*i * q^97 + 1170 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 52 q^{9}+O(q^{10})$$ 2 * q + 52 * q^9 $$2 q + 52 q^{9} + 90 q^{11} + 182 q^{19} + 52 q^{21} - 288 q^{29} + 52 q^{31} - 88 q^{39} - 918 q^{41} - 666 q^{49} + 234 q^{51} + 144 q^{59} - 236 q^{61} + 36 q^{69} + 216 q^{71} + 1796 q^{79} + 1298 q^{81} - 702 q^{89} + 2288 q^{91} + 2340 q^{99}+O(q^{100})$$ 2 * q + 52 * q^9 + 90 * q^11 + 182 * q^19 + 52 * q^21 - 288 * q^29 + 52 * q^31 - 88 * q^39 - 918 * q^41 - 666 * q^49 + 234 * q^51 + 144 * q^59 - 236 * q^61 + 36 * q^69 + 216 * q^71 + 1796 * q^79 + 1298 * q^81 - 702 * q^89 + 2288 * q^91 + 2340 * 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 1.00000i 0 0 0 26.0000i 0 26.0000 0
49.2 0 1.00000i 0 0 0 26.0000i 0 26.0000 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.b 2
3.b odd 2 1 900.4.d.a 2
4.b odd 2 1 400.4.c.l 2
5.b even 2 1 inner 100.4.c.b 2
5.c odd 4 1 100.4.a.b 1
5.c odd 4 1 100.4.a.c yes 1
15.d odd 2 1 900.4.d.a 2
15.e even 4 1 900.4.a.c 1
15.e even 4 1 900.4.a.p 1
20.d odd 2 1 400.4.c.l 2
20.e even 4 1 400.4.a.i 1
20.e even 4 1 400.4.a.l 1
40.i odd 4 1 1600.4.a.x 1
40.i odd 4 1 1600.4.a.bc 1
40.k even 4 1 1600.4.a.y 1
40.k even 4 1 1600.4.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.a.b 1 5.c odd 4 1
100.4.a.c yes 1 5.c odd 4 1
100.4.c.b 2 1.a even 1 1 trivial
100.4.c.b 2 5.b even 2 1 inner
400.4.a.i 1 20.e even 4 1
400.4.a.l 1 20.e even 4 1
400.4.c.l 2 4.b odd 2 1
400.4.c.l 2 20.d odd 2 1
900.4.a.c 1 15.e even 4 1
900.4.a.p 1 15.e even 4 1
900.4.d.a 2 3.b odd 2 1
900.4.d.a 2 15.d odd 2 1
1600.4.a.x 1 40.i odd 4 1
1600.4.a.y 1 40.k even 4 1
1600.4.a.bc 1 40.i odd 4 1
1600.4.a.bd 1 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 676$$
$11$ $$(T - 45)^{2}$$
$13$ $$T^{2} + 1936$$
$17$ $$T^{2} + 13689$$
$19$ $$(T - 91)^{2}$$
$23$ $$T^{2} + 324$$
$29$ $$(T + 144)^{2}$$
$31$ $$(T - 26)^{2}$$
$37$ $$T^{2} + 45796$$
$41$ $$(T + 459)^{2}$$
$43$ $$T^{2} + 211600$$
$47$ $$T^{2} + 219024$$
$53$ $$T^{2} + 311364$$
$59$ $$(T - 72)^{2}$$
$61$ $$(T + 118)^{2}$$
$67$ $$T^{2} + 63001$$
$71$ $$(T - 108)^{2}$$
$73$ $$T^{2} + 89401$$
$79$ $$(T - 898)^{2}$$
$83$ $$T^{2} + 859329$$
$89$ $$(T + 351)^{2}$$
$97$ $$T^{2} + 148996$$