# Properties

 Label 100.6.c.a Level $100$ Weight $6$ Character orbit 100.c Analytic conductor $16.038$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 100.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.0383819813$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 22 i q^{3} -218 i q^{7} -241 q^{9} +O(q^{10})$$ $$q + 22 i q^{3} -218 i q^{7} -241 q^{9} -480 q^{11} -622 i q^{13} -186 i q^{17} + 1204 q^{19} + 4796 q^{21} -3186 i q^{23} + 44 i q^{27} -5526 q^{29} + 9356 q^{31} -10560 i q^{33} -5618 i q^{37} + 13684 q^{39} -14394 q^{41} -370 i q^{43} -16146 i q^{47} -30717 q^{49} + 4092 q^{51} -4374 i q^{53} + 26488 i q^{57} + 11748 q^{59} + 13202 q^{61} + 52538 i q^{63} + 11542 i q^{67} + 70092 q^{69} -29532 q^{71} + 33698 i q^{73} + 104640 i q^{77} -31208 q^{79} -59531 q^{81} -38466 i q^{83} -121572 i q^{87} -119514 q^{89} -135596 q^{91} + 205832 i q^{93} -94658 i q^{97} + 115680 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 482q^{9} + O(q^{10})$$ $$2q - 482q^{9} - 960q^{11} + 2408q^{19} + 9592q^{21} - 11052q^{29} + 18712q^{31} + 27368q^{39} - 28788q^{41} - 61434q^{49} + 8184q^{51} + 23496q^{59} + 26404q^{61} + 140184q^{69} - 59064q^{71} - 62416q^{79} - 119062q^{81} - 239028q^{89} - 271192q^{91} + 231360q^{99} + O(q^{100})$$

## 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 22.0000i 0 0 0 218.000i 0 −241.000 0
49.2 0 22.0000i 0 0 0 218.000i 0 −241.000 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 100.6.c.a 2
3.b odd 2 1 900.6.d.h 2
4.b odd 2 1 400.6.c.c 2
5.b even 2 1 inner 100.6.c.a 2
5.c odd 4 1 20.6.a.a 1
5.c odd 4 1 100.6.a.a 1
15.d odd 2 1 900.6.d.h 2
15.e even 4 1 180.6.a.e 1
15.e even 4 1 900.6.a.b 1
20.d odd 2 1 400.6.c.c 2
20.e even 4 1 80.6.a.b 1
20.e even 4 1 400.6.a.m 1
35.f even 4 1 980.6.a.b 1
40.i odd 4 1 320.6.a.c 1
40.k even 4 1 320.6.a.n 1
60.l odd 4 1 720.6.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.a.a 1 5.c odd 4 1
80.6.a.b 1 20.e even 4 1
100.6.a.a 1 5.c odd 4 1
100.6.c.a 2 1.a even 1 1 trivial
100.6.c.a 2 5.b even 2 1 inner
180.6.a.e 1 15.e even 4 1
320.6.a.c 1 40.i odd 4 1
320.6.a.n 1 40.k even 4 1
400.6.a.m 1 20.e even 4 1
400.6.c.c 2 4.b odd 2 1
400.6.c.c 2 20.d odd 2 1
720.6.a.l 1 60.l odd 4 1
900.6.a.b 1 15.e even 4 1
900.6.d.h 2 3.b odd 2 1
900.6.d.h 2 15.d odd 2 1
980.6.a.b 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 484$$ acting on $$S_{6}^{\mathrm{new}}(100, [\chi])$$.