# Properties

 Label 100.6.c.b 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4) 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.

 $$f(q)$$ $$=$$ $$q + 12 i q^{3} -88 i q^{7} + 99 q^{9} +O(q^{10})$$ $$q + 12 i q^{3} -88 i q^{7} + 99 q^{9} + 540 q^{11} + 418 i q^{13} + 594 i q^{17} -836 q^{19} + 1056 q^{21} + 4104 i q^{23} + 4104 i q^{27} + 594 q^{29} + 4256 q^{31} + 6480 i q^{33} -298 i q^{37} -5016 q^{39} + 17226 q^{41} + 12100 i q^{43} -1296 i q^{47} + 9063 q^{49} -7128 q^{51} -19494 i q^{53} -10032 i q^{57} + 7668 q^{59} -34738 q^{61} -8712 i q^{63} + 21812 i q^{67} -49248 q^{69} -46872 q^{71} -67562 i q^{73} -47520 i q^{77} + 76912 q^{79} -25191 q^{81} -67716 i q^{83} + 7128 i q^{87} -29754 q^{89} + 36784 q^{91} + 51072 i q^{93} -122398 i q^{97} + 53460 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 198 q^{9} + O(q^{10})$$ $$2 q + 198 q^{9} + 1080 q^{11} - 1672 q^{19} + 2112 q^{21} + 1188 q^{29} + 8512 q^{31} - 10032 q^{39} + 34452 q^{41} + 18126 q^{49} - 14256 q^{51} + 15336 q^{59} - 69476 q^{61} - 98496 q^{69} - 93744 q^{71} + 153824 q^{79} - 50382 q^{81} - 59508 q^{89} + 73568 q^{91} + 106920 q^{99} + O(q^{100})$$

## 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 12.0000i 0 0 0 88.0000i 0 99.0000 0
49.2 0 12.0000i 0 0 0 88.0000i 0 99.0000 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 100.6.c.b 2
3.b odd 2 1 900.6.d.a 2
4.b odd 2 1 400.6.c.f 2
5.b even 2 1 inner 100.6.c.b 2
5.c odd 4 1 4.6.a.a 1
5.c odd 4 1 100.6.a.b 1
15.d odd 2 1 900.6.d.a 2
15.e even 4 1 36.6.a.a 1
15.e even 4 1 900.6.a.h 1
20.d odd 2 1 400.6.c.f 2
20.e even 4 1 16.6.a.b 1
20.e even 4 1 400.6.a.d 1
35.f even 4 1 196.6.a.e 1
35.k even 12 2 196.6.e.d 2
35.l odd 12 2 196.6.e.g 2
40.i odd 4 1 64.6.a.f 1
40.k even 4 1 64.6.a.b 1
45.k odd 12 2 324.6.e.a 2
45.l even 12 2 324.6.e.d 2
55.e even 4 1 484.6.a.a 1
60.l odd 4 1 144.6.a.c 1
65.f even 4 1 676.6.d.a 2
65.h odd 4 1 676.6.a.a 1
65.k even 4 1 676.6.d.a 2
80.i odd 4 1 256.6.b.g 2
80.j even 4 1 256.6.b.c 2
80.s even 4 1 256.6.b.c 2
80.t odd 4 1 256.6.b.g 2
120.q odd 4 1 576.6.a.bd 1
120.w even 4 1 576.6.a.bc 1
140.j odd 4 1 784.6.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 5.c odd 4 1
16.6.a.b 1 20.e even 4 1
36.6.a.a 1 15.e even 4 1
64.6.a.b 1 40.k even 4 1
64.6.a.f 1 40.i odd 4 1
100.6.a.b 1 5.c odd 4 1
100.6.c.b 2 1.a even 1 1 trivial
100.6.c.b 2 5.b even 2 1 inner
144.6.a.c 1 60.l odd 4 1
196.6.a.e 1 35.f even 4 1
196.6.e.d 2 35.k even 12 2
196.6.e.g 2 35.l odd 12 2
256.6.b.c 2 80.j even 4 1
256.6.b.c 2 80.s even 4 1
256.6.b.g 2 80.i odd 4 1
256.6.b.g 2 80.t odd 4 1
324.6.e.a 2 45.k odd 12 2
324.6.e.d 2 45.l even 12 2
400.6.a.d 1 20.e even 4 1
400.6.c.f 2 4.b odd 2 1
400.6.c.f 2 20.d odd 2 1
484.6.a.a 1 55.e even 4 1
576.6.a.bc 1 120.w even 4 1
576.6.a.bd 1 120.q odd 4 1
676.6.a.a 1 65.h odd 4 1
676.6.d.a 2 65.f even 4 1
676.6.d.a 2 65.k even 4 1
784.6.a.d 1 140.j odd 4 1
900.6.a.h 1 15.e even 4 1
900.6.d.a 2 3.b odd 2 1
900.6.d.a 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 144$$ acting on $$S_{6}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$144 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7744 + T^{2}$$
$11$ $$( -540 + T )^{2}$$
$13$ $$174724 + T^{2}$$
$17$ $$352836 + T^{2}$$
$19$ $$( 836 + T )^{2}$$
$23$ $$16842816 + T^{2}$$
$29$ $$( -594 + T )^{2}$$
$31$ $$( -4256 + T )^{2}$$
$37$ $$88804 + T^{2}$$
$41$ $$( -17226 + T )^{2}$$
$43$ $$146410000 + T^{2}$$
$47$ $$1679616 + T^{2}$$
$53$ $$380016036 + T^{2}$$
$59$ $$( -7668 + T )^{2}$$
$61$ $$( 34738 + T )^{2}$$
$67$ $$475763344 + T^{2}$$
$71$ $$( 46872 + T )^{2}$$
$73$ $$4564623844 + T^{2}$$
$79$ $$( -76912 + T )^{2}$$
$83$ $$4585456656 + T^{2}$$
$89$ $$( 29754 + T )^{2}$$
$97$ $$14981270404 + T^{2}$$