Properties

 Label 400.6.c.f Level $400$ Weight $6$ Character orbit 400.c Analytic conductor $64.154$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$64.1535279252$$ 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.

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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 400.6.c.f 2
4.b odd 2 1 100.6.c.b 2
5.b even 2 1 inner 400.6.c.f 2
5.c odd 4 1 16.6.a.b 1
5.c odd 4 1 400.6.a.d 1
12.b even 2 1 900.6.d.a 2
15.e even 4 1 144.6.a.c 1
20.d odd 2 1 100.6.c.b 2
20.e even 4 1 4.6.a.a 1
20.e even 4 1 100.6.a.b 1
35.f even 4 1 784.6.a.d 1
40.i odd 4 1 64.6.a.b 1
40.k even 4 1 64.6.a.f 1
60.h even 2 1 900.6.d.a 2
60.l odd 4 1 36.6.a.a 1
60.l odd 4 1 900.6.a.h 1
80.i odd 4 1 256.6.b.c 2
80.j even 4 1 256.6.b.g 2
80.s even 4 1 256.6.b.g 2
80.t odd 4 1 256.6.b.c 2
120.q odd 4 1 576.6.a.bc 1
120.w even 4 1 576.6.a.bd 1
140.j odd 4 1 196.6.a.e 1
140.w even 12 2 196.6.e.g 2
140.x odd 12 2 196.6.e.d 2
180.v odd 12 2 324.6.e.d 2
180.x even 12 2 324.6.e.a 2
220.i odd 4 1 484.6.a.a 1
260.l odd 4 1 676.6.d.a 2
260.p even 4 1 676.6.a.a 1
260.s odd 4 1 676.6.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 20.e even 4 1
16.6.a.b 1 5.c odd 4 1
36.6.a.a 1 60.l odd 4 1
64.6.a.b 1 40.i odd 4 1
64.6.a.f 1 40.k even 4 1
100.6.a.b 1 20.e even 4 1
100.6.c.b 2 4.b odd 2 1
100.6.c.b 2 20.d odd 2 1
144.6.a.c 1 15.e even 4 1
196.6.a.e 1 140.j odd 4 1
196.6.e.d 2 140.x odd 12 2
196.6.e.g 2 140.w even 12 2
256.6.b.c 2 80.i odd 4 1
256.6.b.c 2 80.t odd 4 1
256.6.b.g 2 80.j even 4 1
256.6.b.g 2 80.s even 4 1
324.6.e.a 2 180.x even 12 2
324.6.e.d 2 180.v odd 12 2
400.6.a.d 1 5.c odd 4 1
400.6.c.f 2 1.a even 1 1 trivial
400.6.c.f 2 5.b even 2 1 inner
484.6.a.a 1 220.i odd 4 1
576.6.a.bc 1 120.q odd 4 1
576.6.a.bd 1 120.w even 4 1
676.6.a.a 1 260.p even 4 1
676.6.d.a 2 260.l odd 4 1
676.6.d.a 2 260.s odd 4 1
784.6.a.d 1 35.f even 4 1
900.6.a.h 1 60.l odd 4 1
900.6.d.a 2 12.b even 2 1
900.6.d.a 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 144$$ acting on $$S_{6}^{\mathrm{new}}(400, [\chi])$$.

Hecke characteristic polynomials

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